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Calculated Shard: 152 (from laksa000)

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curl -X POST \
  'http://laksa152.int.ahrefs:8124/' \
  -H 'Content-Type: text/plain' \
  -H 'X-ClickHouse-Database: crawler3' \
  -H 'Authorization: Basic YXBpOg==' \
  -d 'SELECT getAhrefsURLFromUnparsed(src_unparsed) AS found_url, ifNull(toUnixTimestamp(download_stamp), 0) AS crawl_time, ifNull(toUnixTimestamp(props_url_first_seen), 0) AS first_indexed_time, download_http_code AS http_code, src_unparsed AS src_unparsed, src_root_hash AS src_root_hash, history_drop_reason AS history_drop_reason, meta_title AS meta_title, meta_descriptions AS meta_descriptions, attrs_boilerpipe_text AS attrs_boilerpipe_text, attrs_markdown AS attrs_markdown, attrs_readable_markdown AS attrs_readable_markdown, meta_canonical AS meta_canonical, ml_categories_json AS ml_categories_json, ml_types_json AS ml_types_json, ml_intent_types_json AS ml_intent_types_json, meta_language AS meta_language, attrs_author AS attrs_author, ifNull(toUnixTimestamp(attrs_publish_time), 0) AS attrs_publish_time, ifNull(toUnixTimestamp(attrs_original_publish_time), 0) AS attrs_original_publish_time, ifNull(attrs_is_republished, 0) AS attrs_is_republished, ifNull(attrs_nr_words, 0) AS attrs_nr_words, ifNull(attrs_boilerpipe_nr_words, 0) AS attrs_boilerpipe_nr_words, ifNull(body_ext_links_number, 0) AS body_ext_links_number, ifNull(body_int_links_number, 0) AS body_int_links_number, ifNull(meta_nofollow, 0) AS meta_nofollow, ifNull(meta_noarchive, 0) AS meta_noarchive, ifNull(props_was_rendered, 0) AS props_was_rendered, ifNull(src_redirect, \'\') AS src_redirect, ifNull(download_time_msec, 0) AS download_time_msec, ifNull(download_ttfb_msec, 0) AS download_ttfb_msec, ifNull(download_size, 0) AS download_size FROM crawler3.page_info_local FINAL PREWHERE (src_root_hash, src_unparsed) IN ((getAhrefsRootHashFromUnparsed(getAhrefsUnparsedNoserviceFromURL(\'https://en.wikipedia.org/wiki/Quantum_mechanics\')), getAhrefsUnparsedNoserviceFromURL(\'https://en.wikipedia.org/wiki/Quantum_mechanics\'))) FORMAT JSONEachRow'

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{"found_url":"https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics","crawl_time":1776969125,"first_indexed_time":1375978975,"http_code":200,"src_unparsed":"org,wikipedia!en,\/wiki\/Quantum_mechanics s443","src_root_hash":"17790707453426894952","history_drop_reason":null,"meta_title":"Quantum mechanics - Wikipedia","meta_descriptions":[],"attrs_boilerpipe_text":"\"Quantum systems\" redirects here. For the company, see\nQuantum-Systems\n.\nWave functions\nof the\nelectron\nin a hydrogen atom at different energy levels. Quantum mechanics cannot predict the exact location of a particle in space, only the probability of finding it at different locations.\n[\n1\n]\nThe brighter areas represent a higher probability of finding the electron.\nQuantum mechanics\nis the fundamental physical\ntheory\nthat describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of\natoms\n.\n[\n2\n]\n: 1.1 \nIt is the foundation of all\nquantum physics\n, which includes\nquantum chemistry\n,\nquantum biology\n,\nquantum field theory\n,\nquantum technology\n, and\nquantum information science\n.\nQuantum mechanics can describe many systems that\nclassical physics\ncannot. Classical physics can describe many aspects of nature at an ordinary (\nmacroscopic\nand\n(optical) microscopic\n) scale, however is insufficient for describing them at very small\nsubmicroscopic\n(atomic and\nsubatomic\n) scales. Classical mechanics can be derived from quantum mechanics as an approximation that is valid at ordinary scales.\n[\n3\n]\nQuantum systems\nhave\nbound\nstates that are\nquantized\nto\ndiscrete values\nof\nenergy\n,\nmomentum\n,\nangular momentum\n, and other quantities, in contrast to classical systems where these quantities can be measured continuously. Measurements of quantum systems show characteristics of both\nparticles\nand\nwaves\n(\nwave–particle duality\n), and there are limits to how accurately the value of a physical quantity can be predicted prior to its measurement, given a complete set of initial conditions (the\nuncertainty principle\n).\nQuantum mechanics arose gradually from theories to explain observations that could not be reconciled with\nclassical physics\n, such as\nMax Planck\n's solution in 1900 to the\nblack-body radiation\nproblem, and the correspondence between energy and frequency in\nAlbert Einstein\n's\n1905 paper\n, which explained the\nphotoelectric effect\n. These early attempts to understand microscopic phenomena, now known as the \"\nold quantum theory\n\", led to the full development of quantum mechanics in the mid-1920s by\nNiels Bohr\n,\nErwin Schrödinger\n,\nWerner Heisenberg\n,\nMax Born\n,\nPaul Dirac\nand others. The modern theory is formulated in various\nspecially developed mathematical formalisms\n. In one of them, a mathematical entity called the\nwave function\nprovides information, in the form of\nprobability amplitudes\n, about what measurements of a particle's energy, momentum, and other physical properties may yield.\nOverview and fundamental concepts\nQuantum mechanics allows the calculation of properties and behaviour of\nphysical systems\n. It is typically applied to microscopic systems:\nmolecules\n,\natoms\nand\nsubatomic particles\n. It has been demonstrated to hold for complex molecules with thousands of atoms,\n[\n4\n]\nbut its application to human beings raises philosophical problems, such as\nWigner's friend\n, and its application to the universe as a whole remains speculative.\n[\n5\n]\nPredictions of quantum mechanics have been verified experimentally to an extremely high degree of\naccuracy\n. For example, the refinement of quantum mechanics for the interaction of light and matter, known as\nquantum electrodynamics\n(QED), has been\nshown to agree with experiment\nto within 1 part in 10\n12\nwhen predicting the magnetic properties of an electron.\n[\n6\n]\nA fundamental feature of the theory is that it usually cannot predict with certainty what will happen, but only gives probabilities. Mathematically, a probability is found by taking the square of the absolute value of a\ncomplex number\n, known as a probability amplitude. This is known as the\nBorn rule\n, named after physicist\nMax Born\n. For example, a quantum particle like an\nelectron\ncan be described by a wave function, which associates to each point in space a probability amplitude. Applying the Born rule to these amplitudes gives a\nprobability density function\nfor the position that the electron will be found to have when an experiment is performed to measure it. This is the best the theory can do; it cannot say for certain where the electron will be found. The\nSchrödinger equation\nrelates the collection of probability amplitudes that pertain to one moment of time to the collection of probability amplitudes that pertain to another.\n[\n7\n]\n: 67–87 \nOne consequence of the mathematical rules of quantum mechanics is a tradeoff in predictability between measurable quantities. The most famous form of this\nuncertainty principle\nsays that no matter how a quantum particle is prepared or how carefully experiments upon it are arranged, it is impossible to have a precise prediction for a measurement of its position and also at the same time for a measurement of its\nmomentum\n.\n[\n7\n]\n: 427–435 \nAn illustration of the\ndouble-slit experiment\nAnother consequence of the mathematical rules of quantum mechanics is the phenomenon of\nquantum interference\n, which is often illustrated with the\ndouble-slit experiment\n. In the basic version of this experiment, a\ncoherent light source\n, such as a\nlaser\nbeam, illuminates a plate pierced by two parallel slits, and the light passing through the slits is observed on a screen behind the plate.\n[\n8\n]\n: 102–111 \n[\n2\n]\n: 1.1–1.8 \nThe wave nature of light causes the light waves passing through the two slits to\ninterfere\n, producing bright and dark bands on the screen – a result that would not be expected if light consisted of classical particles.\n[\n8\n]\nHowever, the light is always found to be absorbed at the screen at discrete points, as individual particles rather than waves; the interference pattern appears via the varying density of these particle hits on the screen. Furthermore, versions of the experiment that include detectors at the slits find that each detected\nphoton\npasses through one slit (as would a classical particle), and not through both slits (as would a wave).\n[\n8\n]\n: 109 \n[\n9\n]\n[\n10\n]\nHowever,\nsuch experiments\ndemonstrate that particles do not form the interference pattern if one detects which slit they pass through.  This behavior is known as\nwave–particle duality\n. In addition to light,\nelectrons\n,\natoms\n, and\nmolecules\nare all found to exhibit the same dual behavior when fired towards a double slit.\n[\n2\n]\nA simplified diagram of\nquantum tunneling\n, a phenomenon by which a particle may move through a barrier which would be impossible under classical mechanics\nAnother non-classical phenomenon predicted by quantum mechanics is\nquantum tunnelling\n: a particle that goes up against a\npotential barrier\ncan cross it, even if its kinetic energy is smaller than the maximum of the potential.\n[\n11\n]\nIn classical mechanics this particle would be trapped. Quantum tunnelling has several important consequences, enabling\nradioactive decay\n,\nnuclear fusion\nin stars, and applications such as\nscanning tunnelling microscopy\n,\ntunnel diode\nand\ntunnel field-effect transistor\n.\n[\n12\n]\n[\n13\n]\nWhen quantum systems interact, the result can be the creation of\nquantum entanglement\n: their properties become so intertwined that a description of the whole solely in terms of the individual parts is no longer possible. Erwin Schrödinger called entanglement \"...\nthe\ncharacteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought\".\n[\n14\n]\nQuantum entanglement enables\nquantum computing\nand is part of quantum communication protocols, such as\nquantum key distribution\nand\nsuperdense coding\n.\n[\n15\n]\nContrary to popular misconception, entanglement does not allow sending signals\nfaster than light\n, as demonstrated by the\nno-communication theorem\n.\n[\n15\n]\nAnother possibility opened by entanglement is testing for \"\nhidden variables\n\", hypothetical properties more fundamental than the quantities addressed in quantum theory itself, knowledge of which would allow more exact predictions than quantum theory provides. A collection of results, most significantly\nBell's theorem\n, have demonstrated that broad classes of such hidden-variable theories are in fact incompatible with quantum physics. According to Bell's theorem, if nature actually operates in accord with any theory of\nlocal\nhidden variables, then the results of a\nBell test\nwill be constrained in a particular, quantifiable way. Many Bell tests have been performed and they have shown results incompatible with the constraints imposed by local hidden variables.\n[\n16\n]\n[\n17\n]\nIt is not possible to present these concepts in more than a superficial way without introducing the mathematics involved; understanding quantum mechanics requires not only manipulating complex numbers, but also\nlinear algebra\n,\ndifferential equations\n,\ngroup theory\n, and other more advanced subjects.\n[\n18\n]\n[\n19\n]\nAccordingly, this article will present a mathematical formulation of quantum mechanics and survey its application to some useful and oft-studied examples.\nMathematical formulation\nIn the mathematically rigorous formulation of quantum mechanics, the state of a quantum mechanical system is a vector\nbelonging to a (\nseparable\n) complex\nHilbert space\n. This vector is postulated to be normalized under the Hilbert space inner product, that is, it obeys\n, and it is well-defined up to a complex number of modulus 1 (the global phase), that is,\nand\nrepresent the same physical system. In other words, the possible states are points in the\nprojective space\nof a Hilbert space, usually called the\ncomplex projective space\n. The exact nature of this Hilbert space is dependent on the system – for example, for describing position and momentum the Hilbert space is the space of complex\nsquare-integrable\nfunctions\n,\n[\n20\n]\n: 13 \nwhile the Hilbert space for the\nspin\nof a single proton is simply the space of two-dimensional complex vectors\nwith the usual inner product.\n[\n20\n]\n: 20 \nPhysical quantities of interest – position, momentum, energy, spin – are represented by observables, which are\nHermitian\n(more precisely,\nself-adjoint\n) linear\noperators\nacting on the Hilbert space.\n[\n20\n]\n: 17 \nA quantum state can be an\neigenvector\nof an observable, in which case it is called an\neigenstate\n, and the associated\neigenvalue\ncorresponds to the value of the observable in that eigenstate. More generally, a quantum state will be a linear combination of the eigenstates, known as a\nquantum superposition\n. When an observable is measured, the result will be one of its eigenvalues with probability given by the\nBorn rule\n: in the simplest case the eigenvalue\nis non-degenerate and the probability is given by\n, where\nis its associated unit-length eigenvector. More generally, the eigenvalue is degenerate and the probability is given by\n, where\nis the projector onto its associated eigenspace.\n[\n21\n]\nIn the continuous case, these formulas give instead the\nprobability density\n.\nAfter the\nmeasurement\n, if result\nwas obtained, the quantum state is postulated to\ncollapse\nto\n, in the non-degenerate case, or to\n, in the general case. The\nprobabilistic\nnature of quantum mechanics thus stems from the act of measurement. This is one of the most debated aspects of quantum theory, with different\ninterpretations of quantum mechanics\ngiving radically different answers to questions regarding quantum-state collapse, as discussed\nbelow\n.\nTime evolution of a quantum state\nThe time evolution of a quantum state is described by the Schrödinger equation:\nHere\ndenotes the\nHamiltonian\n, the observable corresponding to the\ntotal energy\nof the system, and\nis the reduced\nPlanck constant\n. The constant\nis introduced so that the Hamiltonian is reduced to the\nclassical Hamiltonian\nin cases where the quantum system can be approximated by a classical system; the ability to make such an approximation in certain limits is called the\ncorrespondence principle\n.\nThe solution of this differential equation is given by\nThe operator\nis known as the time-evolution operator, and has the crucial property that it is\nunitary\n. This time evolution is\ndeterministic\nin the sense that – given an initial quantum state\n– it makes a definite prediction of what the quantum state\nwill be at any later time.\n[\n22\n]\nFig. 1:\nProbability densities\ncorresponding to the wave functions of an electron in a hydrogen atom possessing definite energy levels (increasing from the top of the image to the bottom:\nn\n= 1, 2, 3, ...) and angular momenta (increasing across from left to right:\ns\n,\np\n,\nd\n, ...). Denser areas correspond to higher probability density in a position measurement.\nSuch wave functions are directly comparable to\nChladni's figures\nof\nacoustic\nmodes of vibration in classical physics and are modes of oscillation as well, possessing a sharp energy and thus, a definite frequency. The\nangular momentum\nand energy are\nquantized\nand take\nonly\ndiscrete values like those shown – as is the case for\nresonant frequencies\nin acoustics.\nSome wave functions produce probability distributions that are independent of time, such as\neigenstates\nof the Hamiltonian.\n[\n7\n]\n: 133–137 \nMany systems that are treated dynamically in classical mechanics are described by such \"static\" wave functions. For example, a single electron in an unexcited atom is pictured classically as a particle moving in a circular trajectory around the\natomic nucleus\n, whereas in quantum mechanics, it is described by a static wave function surrounding the nucleus. For example, the electron wave function for an unexcited hydrogen atom is a spherically symmetric function known as an\ns\norbital\n(\nFig. 1\n).\nAnalytic solutions of the Schrödinger equation are known for\nvery few relatively simple model Hamiltonians\nincluding the\nquantum harmonic oscillator\n, the\nparticle in a box\n, the\ndihydrogen cation\n, and the\nhydrogen atom\n. Even the\nhelium\natom – which contains just two electrons – has defied all attempts at a fully analytic treatment, admitting no solution in\nclosed form\n.\n[\n23\n]\n[\n24\n]\n[\n25\n]\nHowever, there are techniques for finding approximate solutions. One method, called\nperturbation theory\n, uses the analytic result for a simple quantum mechanical model to create a result for a related but more complicated model by (for example) the addition of a weak\npotential energy\n.\n[\n7\n]\n: 793 \nAnother approximation method applies to systems for which quantum mechanics produces only small deviations from classical behavior. These deviations can then be computed based on the classical motion.\n[\n7\n]\n: 849 \nUncertainty principle\nOne consequence of the basic quantum formalism is the uncertainty principle. In its most familiar form, this states that no preparation of a quantum particle can imply simultaneously precise predictions both for a measurement of its position and for a measurement of its momentum.\n[\n26\n]\n[\n27\n]\nBoth position and momentum are observables, meaning that they are represented by\nHermitian operators\n. The position operator\nand momentum operator\ndo not commute, but rather satisfy the\ncanonical commutation relation\n:\nGiven a quantum state, the Born rule lets us compute expectation values for both\nand\n, and moreover for powers of them. Defining the uncertainty for an observable by a\nstandard deviation\n, we have\nand likewise for the momentum:\nThe uncertainty principle states that\nEither standard deviation can in principle be made arbitrarily small, but not both simultaneously.\n[\n28\n]\nThis inequality generalizes to arbitrary pairs of self-adjoint operators\nand\n. The\ncommutator\nof these two operators is\nand this provides the lower bound on the product of standard deviations:\nAnother consequence of the canonical commutation relation is that the position and momentum operators are\nFourier transforms\nof each other, so that a description of an object according to its momentum is the Fourier transform of its description according to its position. The fact that dependence in momentum is the Fourier transform of the dependence in position means that the momentum operator is equivalent (up to an\nfactor) to taking the derivative according to the position, since in Fourier analysis\ndifferentiation corresponds to multiplication in the dual space\n. This is why in quantum equations in position space, the momentum\nis replaced by\n, and in particular in the\nnon-relativistic Schrödinger equation in position space\nthe momentum-squared term is replaced with a Laplacian times\n.\n[\n26\n]\nComposite systems and entanglement\nWhen two different quantum systems are considered together, the Hilbert space of the combined system is the\ntensor product\nof the Hilbert spaces of the two components. For example, let\nA\nand\nB\nbe two quantum systems, with Hilbert spaces\nand\n, respectively. The Hilbert space of the composite system is then\nIf the state for the first system is the vector\nand the state for the second system is\n, then the state of the composite system is\nNot all states in the joint Hilbert space\ncan be written in this form, however, because the superposition principle implies that linear combinations of these \"separable\" or \"product states\" are also valid. For example, if\nand\nare both possible states for system\n, and likewise\nand\nare both possible states for system\n, then\nis a valid joint state that is not separable. States that are not separable are called\nentangled\n.\n[\n29\n]\n[\n30\n]\nIf the state for a composite system is entangled, it is impossible to describe either component system\nA\nor system\nB\nby a state vector. One can instead define\nreduced density matrices\nthat describe the statistics that can be obtained by making measurements on either component system alone. This necessarily causes a loss of information, though: knowing the reduced density matrices of the individual systems is not enough to reconstruct the state of the composite system.\n[\n29\n]\n[\n30\n]\nJust as density matrices specify the state of a subsystem of a larger system, analogously,\npositive operator-valued measures\n(POVMs) describe the effect on a subsystem of a measurement performed on a larger system. POVMs are extensively used in quantum information theory.\n[\n29\n]\n[\n31\n]\nAs described above, entanglement is a key feature of models of measurement processes in which an apparatus becomes entangled with the system being measured. Systems interacting with the environment in which they reside generally become entangled with that environment, a phenomenon known as\nquantum decoherence\n. This can explain why, in practice, quantum effects are difficult to observe in systems larger than microscopic.\n[\n32\n]\nEquivalence between formulations\nThere are many mathematically equivalent formulations of quantum mechanics. One of the oldest and most common is the \"\ntransformation theory\n\" proposed by\nPaul Dirac\n, which unifies and generalizes the two earliest formulations of quantum mechanics –\nmatrix mechanics\n(invented by\nWerner Heisenberg\n) and wave mechanics (invented by\nErwin Schrödinger\n).\n[\n33\n]\nAn alternative formulation of quantum mechanics is\nFeynman\n's\npath integral formulation\n, in which a quantum-mechanical amplitude is considered as a sum over all possible classical and non-classical paths between the initial and final states. This is the quantum-mechanical counterpart of the\naction principle\nin classical mechanics.\n[\n34\n]\nSymmetries and conservation laws\nThe Hamiltonian\nis known as the\ngenerator\nof time evolution, since it defines a unitary time-evolution operator\nfor each value of\n. From this relation between\nand\n, it follows that any observable\nthat commutes with\nwill be\nconserved\n: its expectation value will not change over time.\n[\n7\n]\n: 471 \nThis statement generalizes, as mathematically, any Hermitian operator\ncan generate a family of unitary operators parameterized by a variable\n. Under the evolution generated by\n, any observable\nthat commutes with\nwill be conserved. Moreover, if\nis conserved by evolution under\n, then\nis conserved under the evolution generated by\n. This implies a quantum version of the result proven by\nEmmy Noether\nin classical (\nLagrangian\n) mechanics: for every\ndifferentiable\nsymmetry\nof a Hamiltonian, there exists a corresponding\nconservation law\n.\nExamples\nFree particle\nPosition space probability density of a Gaussian\nwave packet\nmoving in one dimension in free space\nThe simplest example of a quantum system with a position degree of freedom is a free particle in a single spatial dimension. A free particle is one which is not subject to external influences, so that its Hamiltonian consists only of its kinetic energy:\nThe general solution of the Schrödinger equation is given by\nwhich is a superposition of all possible\nplane waves\n, which are eigenstates of the momentum operator with momentum\n. The coefficients of the superposition are\n, which is the Fourier transform of the initial quantum state\n.\nIt is not possible for the solution to be a single momentum eigenstate, or a single position eigenstate, as these are not normalizable quantum states.\n[\nnote 1\n]\nInstead, we can consider a Gaussian\nwave packet\n:\nwhich has Fourier transform, and therefore momentum distribution\nWe see that as we make\nsmaller the spread in position gets smaller, but the spread in momentum gets larger. Conversely, by making\nlarger we make the spread in momentum smaller, but the spread in position gets larger. This illustrates the uncertainty principle.\nAs we let the Gaussian wave packet evolve in time, we see that its center moves through space at a constant velocity (like a classical particle with no forces acting on it). However, the wave packet will also spread out as time progresses, which means that the position becomes more and more uncertain. The uncertainty in momentum, however, stays constant.\n[\n35\n]\nParticle in a box\n1-dimensional potential energy box (or infinite potential well)\nThe particle in a one-dimensional potential energy box is the most mathematically simple example where restraints lead to the quantization of energy levels. The box is defined as having zero potential energy everywhere\ninside\na certain region, and therefore infinite potential energy everywhere\noutside\nthat region.\n[\n26\n]\n: 77–78 \nFor the one-dimensional case in the\ndirection, the time-independent Schrödinger equation may be written\nWith the differential operator defined by\nthe previous equation is evocative of the\nclassic kinetic energy analogue\n,\nwith state\nin this case having energy\ncoincident with the kinetic energy of the particle.\nThe general solutions of the Schrödinger equation for the particle in a box are\nor, from\nEuler's formula\n,\nThe infinite potential walls of the box determine the values of\nand\nat\nand\nwhere\nmust be zero. Thus, at\n,\nand\n. At\n,\nin which\ncannot be zero as this would conflict with the postulate that\nhas norm 1. Therefore, since\n,\nmust be an integer multiple of\n,\nThis constraint on\nimplies a constraint on the energy levels, yielding\nA\nfinite potential well\nis the generalization of the infinite potential well problem to potential wells having finite depth. The finite potential well problem is mathematically more complicated than the infinite particle-in-a-box problem as the wave function is not pinned to zero at the walls of the well. Instead, the wave function must satisfy more complicated mathematical boundary conditions as it is nonzero in regions outside the well. Another related problem is that of the\nrectangular potential barrier\n, which furnishes a model for the\nquantum tunneling\neffect that plays an important role in the performance of modern technologies such as\nflash memory\nand\nscanning tunneling microscopy\n.\nHarmonic oscillator\nSome trajectories of a\nharmonic oscillator\n(i.e. a ball attached to a\nspring\n) in\nclassical mechanics\n(A-B) and quantum mechanics (C-H). In quantum mechanics, the position of the ball is represented by a\nwave\n(called the wave function), with the\nreal part\nshown in blue and the\nimaginary part\nshown in red. Some of the trajectories (such as C, D, E, and F) are\nstanding waves\n(or \"\nstationary states\n\"). Each standing-wave frequency is proportional to a possible\nenergy level\nof the oscillator. This \"energy quantization\" does not occur in classical physics, where the oscillator can have\nany\nenergy.\nAs in the classical case, the potential for the quantum harmonic oscillator is given by\n[\n7\n]\n: 234 \nThis problem can either be treated by directly solving the Schrödinger equation, which is not trivial, or by using the more elegant \"ladder method\" first proposed by Paul Dirac. The\neigenstates\nare given by\nwhere\nH\nn\nare the\nHermite polynomials\nand the corresponding energy levels are\nThis is another example illustrating the discretization of energy for\nbound states\n.\nMach–Zehnder interferometer\nSchematic of a Mach–Zehnder interferometer\nThe\nMach–Zehnder interferometer\n(MZI) illustrates the concepts of superposition and interference with linear algebra in dimension 2, rather than differential equations. It can be seen as a simplified version of the double-slit experiment, but it is of interest in its own right, for example in the\ndelayed choice quantum eraser\n, the\nElitzur–Vaidman bomb tester\n, and in studies of quantum entanglement.\n[\n36\n]\n[\n37\n]\nWe can model a photon going through the interferometer by considering that at each point it can be in a superposition of only two paths: the \"lower\" path which starts from the left, goes straight through both beam splitters, and ends at the top, and the \"upper\" path which starts from the bottom, goes straight through both beam splitters, and ends at the right. The quantum state of the photon is therefore a vector\nthat is a superposition of the \"lower\" path\nand the \"upper\" path\n, that is,\nfor complex\n. In order to respect the postulate that\nwe require that\n.\nBoth\nbeam splitters\nare modelled as the unitary matrix\n, which means that when a photon meets the beam splitter it will either stay on the same path with a probability amplitude of\n, or be reflected to the other path with a probability amplitude of\n. The phase shifter on the upper arm is modelled as the unitary matrix\n, which means that if the photon is on the \"upper\" path it will gain a relative phase of\n, and it will stay unchanged if it is in the lower path.\nA photon that enters the interferometer from the left will then be acted upon with a beam splitter\n, a phase shifter\n, and another beam splitter\n, and so end up in the state\nand the probabilities that it will be detected at the right or at the top are given respectively by\nOne can therefore use the Mach–Zehnder interferometer to estimate the\nphase shift\nby estimating these probabilities.\nIt is interesting to consider what would happen if the photon were definitely in either the \"lower\" or \"upper\" paths between the beam splitters. This can be accomplished by blocking one of the paths, or equivalently by removing the first beam splitter (and feeding the photon from the left or the bottom, as desired). In both cases, there will be no interference between the paths anymore, and the probabilities are given by\n, independently of the phase\n. From this we can conclude that the photon does not take one path or another after the first beam splitter, but rather that it is in a genuine quantum superposition of the two paths.\n[\n38\n]\nApplications\nQuantum mechanics has had enormous success in explaining many of the features of our universe, with regard to small-scale and discrete quantities and interactions which cannot be explained by\nclassical methods\n.\n[\nnote 2\n]\nQuantum mechanics is often the only theory that can reveal the individual behaviors of the subatomic particles that make up all forms of matter (electrons,\nprotons\n,\nneutrons\n,\nphotons\n, and others).\nSolid-state physics\nand\nmaterials science\nare dependent upon quantum mechanics.\n[\n39\n]\nIn many aspects, modern technology operates at a scale where quantum effects are significant. Important applications of quantum theory include\nquantum chemistry\n,\nquantum optics\n,\nquantum computing\n,\nsuperconducting magnets\n,\nlight-emitting diodes\n, the\noptical amplifier\nand the laser, the\ntransistor\nand\nsemiconductors\nsuch as the\nmicroprocessor\n,\nmedical and research imaging\nsuch as\nmagnetic resonance imaging\nand\nelectron microscopy\n.\n[\n40\n]\nExplanations for many biological and physical phenomena are rooted in the nature of the chemical bond, most notably the macro-molecule\nDNA\n.\nRelation to other scientific theories\nClassical mechanics\nThe rules of quantum mechanics assert that the state space of a system is a Hilbert space and that observables of the system are Hermitian operators acting on vectors in that space – although they do not tell us which Hilbert space or which operators. These can be chosen appropriately in order to obtain a quantitative description of a quantum system, a necessary step in making physical predictions. An important guide for making these choices is the\ncorrespondence principle\n, a heuristic which states that the predictions of quantum mechanics reduce to those of\nclassical mechanics\nin the regime of large\nquantum numbers\n.\n[\n41\n]\nOne can also start from an established classical model of a particular system, and then try to guess the underlying quantum model that would give rise to the classical model in the correspondence limit. This approach is known as\nquantization\n.\n[\n42\n]\n: 299 \n[\n43\n]\nWhen quantum mechanics was originally formulated, it was applied to models whose correspondence limit was\nnon-relativistic\nclassical mechanics. For instance, the well-known model of the\nquantum harmonic oscillator\nuses an explicitly non-relativistic expression for the\nkinetic energy\nof the oscillator, and is thus a quantum version of the\nclassical harmonic oscillator\n.\n[\n7\n]\n: 234 \nComplications arise with\nchaotic systems\n, which do not have good quantum numbers, and\nquantum chaos\nstudies the relationship between classical and quantum descriptions in these systems.\n[\n42\n]\n: 353 \nQuantum decoherence\nis a mechanism through which quantum systems lose\ncoherence\n, and thus become incapable of displaying many typically quantum effects:\nquantum superpositions\nbecome simply probabilistic mixtures, and quantum entanglement becomes simply classical correlations.\n[\n7\n]\n: 687–730 \nQuantum coherence is not typically evident at macroscopic scales, though at temperatures approaching\nabsolute zero\nquantum behavior may manifest macroscopically.\n[\nnote 3\n]\nMany macroscopic properties of a classical system are a direct consequence of the quantum behavior of its parts. For example, the stability of bulk matter (consisting of atoms and\nmolecules\nwhich would quickly collapse under electric forces alone), the rigidity of solids, and the mechanical, thermal, chemical, optical and magnetic properties of matter are all results of the interaction of\nelectric charges\nunder the rules of quantum mechanics.\n[\n44\n]\nSpecial relativity and electrodynamics\nEarly attempts to merge quantum mechanics with\nspecial relativity\ninvolved the replacement of the Schrödinger equation with a covariant equation such as the\nKlein–Gordon equation\nor the\nDirac equation\n. While these theories were successful in explaining many experimental results, they had certain unsatisfactory qualities stemming from their neglect of the relativistic creation and annihilation of particles. A fully relativistic quantum theory required the development of quantum field theory, which applies quantization to a field (rather than a fixed set of particles). The first complete quantum field theory,\nquantum electrodynamics\n, provides a fully quantum description of the\nelectromagnetic interaction\n. Quantum electrodynamics is, along with\ngeneral relativity\n, one of the most accurate physical theories ever devised.\n[\n45\n]\n[\n46\n]\nThe full apparatus of quantum field theory is often unnecessary for describing electrodynamic systems. A simpler approach, one that has been used since the inception of quantum mechanics, is to treat\ncharged\nparticles as quantum mechanical objects being acted on by a classical\nelectromagnetic field\n. For example, the elementary quantum model of the\nhydrogen atom\ndescribes the\nelectric field\nof the hydrogen atom using a classical\nCoulomb potential\n.\n[\n7\n]\n: 285 \nLikewise, in a\nStern–Gerlach experiment\n, a charged particle is modeled as a quantum system, while the background magnetic field is described classically.\n[\n42\n]\n: 26 \nThis \"semi-classical\" approach fails if quantum fluctuations in the electromagnetic field play an important role, such as in the emission of photons by\ncharged particles\n.\nQuantum field\ntheories for the\nstrong nuclear force\nand the\nweak nuclear force\nhave also been developed. The quantum field theory of the strong nuclear force is called\nquantum chromodynamics\n, and describes the interactions of subnuclear particles such as\nquarks\nand\ngluons\n. The weak nuclear force and the electromagnetic force were unified, in their quantized forms, into a single quantum field theory (known as\nelectroweak theory\n), by the physicists\nAbdus Salam\n,\nSheldon Glashow\nand\nSteven Weinberg\n.\n[\n47\n]\nRelation to general relativity\nEven though the predictions of both quantum theory and general relativity have been supported by rigorous and repeated\nempirical evidence\n, their abstract formalisms contradict each other and they have proven extremely difficult to incorporate into one consistent, cohesive model. Gravity is negligible in many areas of particle physics, so that unification between general relativity and quantum mechanics is not an urgent issue in those particular applications. However, the lack of a correct theory of\nquantum gravity\nis an important issue in\nphysical cosmology\nand the search by physicists for an elegant \"\nTheory of Everything\n\" (TOE). Consequently, resolving the inconsistencies between both theories has been a major goal of 20th- and 21st-century physics. This TOE would combine not only the models of subatomic physics but also derive the four fundamental forces of nature from a single force or phenomenon.\n[\n48\n]\nOne proposal for doing so is\nstring theory\n, which posits that the\npoint-like particles\nof\nparticle physics\nare replaced by\none-dimensional\nobjects called\nstrings\n. String theory describes how these strings propagate through space and interact with each other. On distance scales larger than the string scale, a string looks just like an ordinary particle, with its\nmass\n,\ncharge\n, and other properties determined by the\nvibrational\nstate of the string. In string theory, one of the many vibrational states of the string corresponds to the\ngraviton\n, a quantum mechanical particle that carries gravitational force.\n[\n49\n]\n[\n50\n]\nAnother popular theory is\nloop quantum gravity\n(LQG), which describes quantum properties of gravity and is thus a theory of\nquantum spacetime\n. LQG is an attempt to merge and adapt standard quantum mechanics and standard general relativity. This theory describes space as an extremely fine fabric \"woven\" of finite loops called\nspin networks\n. The evolution of a spin network over time is called a\nspin foam\n. The characteristic length scale of a spin foam is the\nPlanck length\n, approximately 1.616×10\n−35\nm, and so lengths shorter than the Planck length are not physically meaningful in LQG.\n[\n51\n]\nPhilosophical implications\nUnsolved problem in physics\nIs there a preferred interpretation of quantum mechanics? How does the quantum description of reality, which includes elements such as the \"\nsuperposition\nof states\" and \"\nwave function collapse\n\", give rise to the reality we perceive?\nSince its inception, the many counter-intuitive aspects and results of quantum mechanics have provoked strong\nphilosophical\ndebates and many\ninterpretations\n. The arguments centre on the probabilistic nature of quantum mechanics, the difficulties with\nwavefunction collapse\nand the related\nmeasurement problem\n, and\nquantum nonlocality\n. Perhaps the only consensus that exists about these issues is that there is no consensus.\nRichard Feynman\nonce said, \"I think I can safely say that nobody understands quantum mechanics.\"\n[\n52\n]\nAccording to\nSteven Weinberg\n, \"There is now in my opinion no entirely satisfactory interpretation of quantum mechanics.\"\n[\n53\n]\nThe views of\nNiels Bohr\n, Werner Heisenberg and other physicists are often grouped together as the \"\nCopenhagen interpretation\n\".\n[\n54\n]\n[\n55\n]\nAccording to these views, the probabilistic nature of quantum mechanics is not a\ntemporary\nfeature which will eventually be replaced by a deterministic theory, but is instead a\nfinal\nrenunciation of the classical idea of \"causality\". Bohr in particular emphasized that any well-defined application of the quantum mechanical formalism must always make reference to the experimental arrangement, due to the\ncomplementary\nnature of evidence obtained under different experimental situations. Copenhagen-type interpretations were adopted by Nobel laureates in quantum physics, including Bohr,\n[\n56\n]\nHeisenberg,\n[\n57\n]\nSchrödinger,\n[\n58\n]\nFeynman,\n[\n2\n]\nand\nZeilinger\n[\n59\n]\nas well as 21st-century researchers in quantum foundations.\n[\n60\n]\nAlbert Einstein\n, himself one of the founders of\nquantum theory\n, was troubled by its apparent failure to respect some cherished metaphysical principles, such as\ndeterminism\nand\nlocality\n. Einstein's long-running exchanges with Bohr about the meaning and status of quantum mechanics are now known as the\nBohr–Einstein debates\n. Einstein believed that underlying quantum mechanics must be a theory that explicitly forbids\naction at a distance\n. He argued that quantum mechanics was incomplete, a theory that was valid but not fundamental, analogous to how\nthermodynamics\nis valid, but the fundamental theory behind it is\nstatistical mechanics\n. In 1935, Einstein and his collaborators\nBoris Podolsky\nand\nNathan Rosen\npublished an argument that the principle of locality implies the incompleteness of quantum mechanics, a\nthought experiment\nlater termed the\nEinstein–Podolsky–Rosen paradox\n.\n[\nnote 4\n]\nIn 1964,\nJohn Bell\nshowed that EPR's principle of locality, together with determinism, was actually incompatible with quantum mechanics: they implied constraints on the correlations produced by distance systems, now known as\nBell inequalities\n, that can be violated by entangled particles.\n[\n65\n]\nSince then\nseveral experiments\nhave been performed to obtain these correlations, with the result that they do in fact violate Bell inequalities, and thus falsify the conjunction of locality with determinism.\n[\n16\n]\n[\n17\n]\nBohmian mechanics\nshows that it is possible to reformulate quantum mechanics to make it deterministic, at the price of making it explicitly nonlocal. It attributes not only a wave function to a physical system, but in addition a real position, that evolves deterministically under a nonlocal guiding equation. The evolution of a physical system is given at all times by the Schrödinger equation together with the guiding equation; there is never a collapse of the wave function. This solves the measurement problem.\n[\n66\n]\nThe\nSchrödinger's cat\nthought experiment can be used to visualize the many-worlds interpretation of quantum mechanics, where a branching of the universe occurs through a superposition of two quantum mechanical states.\nEverett's\nmany-worlds interpretation\n, formulated in 1956, holds that\nall\nthe possibilities described by quantum theory\nsimultaneously\noccur in a multiverse composed of mostly independent parallel universes.\n[\n67\n]\nThis is a consequence of removing the axiom of the collapse of the wave packet. All possible states of the measured system and the measuring apparatus, together with the observer, are present in a real physical quantum superposition. While the multiverse is deterministic, we perceive non-deterministic behavior governed by probabilities, because we do not observe the multiverse as a whole, but only one parallel universe at a time. Exactly how this is supposed to work has been the subject of much debate. Several attempts have been made to make sense of this and derive the Born rule,\n[\n68\n]\n[\n69\n]\nwith no consensus on whether they have been successful.\n[\n70\n]\n[\n71\n]\n[\n72\n]\nRelational quantum mechanics\nappeared in the late 1990s as a modern derivative of Copenhagen-type ideas,\n[\n73\n]\n[\n74\n]\nand\nQBism\nwas developed some years later.\n[\n75\n]\n[\n76\n]\nHistory\nQuantum mechanics was developed in the early decades of the 20th century, driven by the need to explain phenomena that, in some cases, had been observed in earlier times. Scientific inquiry into the wave nature of light began in the 17th and 18th centuries, when scientists such as\nRobert Hooke\n,\nChristiaan Huygens\nand\nLeonhard Euler\nproposed a wave theory of light based on experimental observations.\n[\n77\n]\nIn 1803 English\npolymath\nThomas Young\ndescribed the famous\ndouble-slit experiment\n.\n[\n78\n]\nThis experiment played a major role in the general acceptance of the\nwave theory of light\n.\nDuring the early 19th century,\nchemical\nresearch by\nJohn Dalton\nand\nAmedeo Avogadro\nlent weight to the\natomic theory\nof matter, an idea that\nJames Clerk Maxwell\n,\nLudwig Boltzmann\nand others built upon to establish the\nkinetic theory of gases\n. The successes of kinetic theory gave further credence to the idea that matter is composed of atoms, yet the theory also had shortcomings that would only be resolved by the development of quantum mechanics.\n[\n79\n]\nWhile the early conception of atoms from\nGreek philosophy\nhad been that they were indivisible units – the word \"atom\" deriving from the\nGreek\nfor 'uncuttable' – the 19th century saw the formulation of hypotheses about subatomic structure. One important discovery in that regard was\nMichael Faraday\n's 1838 observation of a glow caused by an electrical discharge inside a glass tube containing gas at low pressure.\nJulius Plücker\n,\nJohann Wilhelm Hittorf\nand\nEugen Goldstein\ncarried on and improved upon Faraday's work, leading to the identification of\ncathode rays\n, which\nJ. J. Thomson\nfound to consist of subatomic particles that would be called electrons.\n[\n80\n]\n[\n81\n]\nMax Planck\nis considered the father of the quantum theory.\nThe\nblack-body radiation\nproblem was discovered by\nGustav Kirchhoff\nin 1859. In 1900, Max Planck proposed the hypothesis that energy is radiated and absorbed in discrete \"quanta\" (or energy packets), yielding a calculation that precisely matched the observed patterns of black-body radiation.\n[\n82\n]\nThe word\nquantum\nderives from the\nLatin\n, meaning \"how great\" or \"how much\".\n[\n83\n]\nAccording to Planck, quantities of energy could be thought of as divided into \"elements\" whose size (\nE\n) would be proportional to their\nfrequency\n(\nν\n):\n,\nwhere\nh\nis the\nPlanck constant\n. Planck cautiously insisted that this was only an aspect of the processes of absorption and emission of radiation and was not the\nphysical reality\nof the radiation.\n[\n84\n]\nIn fact, he considered his quantum hypothesis a mathematical trick to get the right answer rather than a sizable discovery.\n[\n85\n]\nHowever, in 1905 Albert Einstein interpreted Planck's quantum hypothesis\nrealistically\nand used it to explain the\nphotoelectric effect\n, in which shining light on certain materials can eject electrons from the material. Niels Bohr then developed Planck's ideas about radiation into a\nmodel of the hydrogen atom\nthat successfully predicted the\nspectral lines\nof hydrogen.\n[\n86\n]\nEinstein further developed this idea to show that an\nelectromagnetic wave\nsuch as light could also be described as a particle (later called the photon), with a discrete amount of energy that depends on its frequency.\n[\n87\n]\nIn his paper \"On the Quantum Theory of Radiation\", Einstein expanded on the interaction between energy and matter to explain the absorption and emission of energy by atoms. Although overshadowed at the time by his general theory of relativity, this paper articulated the mechanism underlying the stimulated emission of radiation,\n[\n88\n]\nwhich became the basis of the laser.\n[\n89\n]\nThe 1927\nSolvay Conference\nin\nBrussels\nwas the fifth world physics conference.\nThis phase is known as the\nold quantum theory\n. Never complete or self-consistent, the old quantum theory was rather a set of\nheuristic\ncorrections to classical mechanics.\n[\n90\n]\n[\n91\n]\nThe theory is now understood as a\nsemi-classical approximation\nto modern quantum mechanics.\n[\n92\n]\n[\n93\n]\nNotable results from this period include, in addition to the work of Planck, Einstein and Bohr mentioned above, Einstein and\nPeter Debye\n's work on the\nspecific heat\nof solids, Bohr and\nHendrika Johanna van Leeuwen\n's\nproof\nthat classical physics cannot account for\ndiamagnetism\n, and\nArnold Sommerfeld\n's extension of the Bohr model to include special-relativistic effects.\n[\n90\n]\n[\n94\n]\nIn the mid-1920s quantum mechanics was developed to become the standard formulation for atomic physics. In 1923, the French physicist\nLouis de Broglie\nput forward his theory of matter waves by stating that particles can exhibit wave characteristics and vice versa. Building on de Broglie's approach, modern quantum mechanics was born in 1925, when the German physicists Werner Heisenberg, Max Born, and\nPascual Jordan\n[\n95\n]\n[\n96\n]\ndeveloped\nmatrix mechanics\nand the Austrian physicist Erwin Schrödinger invented\nwave mechanics\n. Born introduced the probabilistic interpretation of Schrödinger's wave function in July 1926.\n[\n97\n]\nThus, the entire field of quantum physics emerged, leading to its wider acceptance at the Fifth\nSolvay Conference\nin 1927.\n[\n98\n]\nBy 1930, quantum mechanics had been further unified and formalized by\nDavid Hilbert\n, Paul Dirac and\nJohn von Neumann\n[\n99\n]\nwith greater emphasis on\nmeasurement\n, the statistical nature of our knowledge of reality, and\nphilosophical speculation about the 'observer'\n. It has since permeated many disciplines, including quantum chemistry,\nquantum electronics\n,\nquantum optics\n, and\nquantum information science\n. It also provides a useful framework for many features of the modern\nperiodic table of elements\n, and describes the behaviors of\natoms\nduring\nchemical bonding\nand the flow of electrons in computer\nsemiconductors\n, and therefore plays a crucial role in many modern technologies. While quantum mechanics was constructed to describe the world of the very small, it is also needed to explain some\nmacroscopic\nphenomena such as\nsuperconductors\n[\n100\n]\nand\nsuperfluids\n.\n[\n101\n]\nSee also\nBra–ket notation\nEinstein's thought experiments\nList of textbooks on classical mechanics and quantum mechanics\nMacroscopic quantum phenomena\nPhase-space formulation\nRegularization (physics)\nTwo-state quantum system\nExplanatory notes\n^\nA momentum eigenstate would be a perfectly monochromatic wave of infinite extent, which is not square-integrable. Likewise, a position eigenstate would be a\nDirac delta distribution\n, not square-integrable and technically not a function at all. Consequently, neither can belong to the particle's Hilbert space. Physicists sometimes introduce fictitious \"bases\" for a Hilbert space comprising elements outside that space. These are invented for calculational convenience and do not represent physical states.\n[\n26\n]\n: 100–105 \n^\nSee, for example,\nthe Feynman Lectures on Physics\nfor some of the technological applications which use quantum mechanics, e.g.,\ntransistors\n(vol\nIII\n, pp. 14–11 ff),\nintegrated circuits\n, which are follow-on technology in solid-state physics (vol\nII\n, pp. 8–6), and\nlasers\n(vol\nIII\n, pp. 9–13).\n^\nSee\nMacroscopic quantum phenomena\n,\nBose–Einstein condensate\n, and\nQuantum machine\n^\nThe published form of the EPR argument was due to Podolsky, and Einstein himself was not satisfied with it. In his own publications and correspondence, Einstein used a different argument to insist that quantum mechanics is an incomplete theory.\n[\n61\n]\n[\n62\n]\n[\n63\n]\n[\n64\n]\nReferences\n^\nBorn, M.\n(1926). \"Zur Quantenmechanik der Stoßvorgänge\" [On the Quantum Mechanics of Collision Processes].\nZeitschrift für Physik\n(in German).\n37\n(12):\n863–\n867.\nBibcode\n:\n1926ZPhy...37..863B\n.\ndoi\n:\n10.1007\/BF01397477\n.\nISSN\n \n1434-6001\n.\nS2CID\n \n119896026\n.\n^\na\nb\nc\nd\nFeynman, Richard; Leighton, Robert; Sands, Matthew (1964).\nThe Feynman Lectures on Physics\n. Vol. 3. California Institute of Technology\n. Retrieved\n19 December\n2020\n.\nReprinted, Addison-Wesley, 1989,\nISBN\n \n978-0-201-50064-6\n^\nJaeger, Gregg (September 2014). \"What in the (quantum) world is macroscopic?\".\nAmerican Journal of Physics\n.\n82\n(9):\n896–\n905.\nBibcode\n:\n2014AmJPh..82..896J\n.\ndoi\n:\n10.1119\/1.4878358\n.\n^\nFein, Yaakov Y.; Geyer, Philipp; Zwick, Patrick; Kiałka, Filip; Pedalino, Sebastian; Mayor, Marcel; Gerlich, Stefan; Arndt, Markus (September 2019). \"Quantum superposition of molecules beyond 25 kDa\".\nNature Physics\n.\n15\n(12):\n1242–\n1245.\nBibcode\n:\n2019NatPh..15.1242F\n.\ndoi\n:\n10.1038\/s41567-019-0663-9\n.\nS2CID\n \n203638258\n.\n^\nBojowald, Martin (2015). \"Quantum cosmology: a review\".\nReports on Progress in Physics\n.\n78\n(2) 023901.\narXiv\n:\n1501.04899\n.\nBibcode\n:\n2015RPPh...78b3901B\n.\ndoi\n:\n10.1088\/0034-4885\/78\/2\/023901\n.\nPMID\n \n25582917\n.\nS2CID\n \n18463042\n.\n^\n^\na\nb\nc\nd\ne\nf\ng\nh\ni\nj\nZwiebach, Barton\n(2022).\nMastering Quantum Mechanics: Essentials, Theory, and Applications\n. MIT Press.\nISBN\n \n978-0-262-04613-8\n.\n^\na\nb\nc\nLederman, Leon M.; Hill, Christopher T. (2011).\nQuantum Physics for Poets\n. US: Prometheus Books.\nISBN\n \n978-1-61614-281-0\n.\n^\nMüller-Kirsten, H. J. W. (2006).\nIntroduction to Quantum Mechanics: Schrödinger Equation and Path Integral\n. US: World Scientific. p. 14.\nISBN\n \n978-981-256-691-1\n.\n^\nPlotnitsky, Arkady (2012).\nNiels Bohr and Complementarity: An Introduction\n. US: Springer. pp. \n75–\n76.\nISBN\n \n978-1-4614-4517-3\n.\n^\nGriffiths, David J.\n(1995).\nIntroduction to Quantum Mechanics\n. Prentice Hall.\nISBN\n \n0-13-124405-1\n.\n^\nTrixler, F. (2013).\n\"Quantum tunnelling to the origin and evolution of life\"\n.\nCurrent Organic Chemistry\n.\n17\n(16):\n1758–\n1770.\ndoi\n:\n10.2174\/13852728113179990083\n.\nPMC\n \n3768233\n.\nPMID\n \n24039543\n.\n^\nPhifer, Arnold (2012-03-27).\n\"Developing more energy-efficient transistors through quantum tunneling\"\n.\nNotre Dame News\n. Retrieved\n2024-06-07\n.\n^\nBub, Jeffrey\n(2019).\n\"Quantum entanglement\"\n. In Zalta, Edward N. (ed.).\nStanford Encyclopedia of Philosophy\n. Metaphysics Research Lab, Stanford University.\n^\na\nb\nCaves, Carlton M.\n(2015). \"Quantum Information Science: Emerging No More\". In Kelley, Paul; Agrawal, Govind; Bass, Mike; Hecht, Jeff; Stroud, Carlos (eds.).\nOSA Century of Optics\n.\nThe Optical Society\n. pp. \n320–\n323.\narXiv\n:\n1302.1864\n.\nBibcode\n:\n2013arXiv1302.1864C\n.\nISBN\n \n978-1-943580-04-0\n.\n^\na\nb\nWiseman, Howard\n(October 2015).\n\"Death by experiment for local realism\"\n.\nNature\n.\n526\n(7575):\n649–\n650.\ndoi\n:\n10.1038\/nature15631\n.\nISSN\n \n0028-0836\n.\nPMID\n \n26503054\n.\n^\na\nb\nWolchover, Natalie\n(7 February 2017).\n\"Experiment Reaffirms Quantum Weirdness\"\n.\nQuanta Magazine\n. Retrieved\n8 February\n2020\n.\n^\nBaez, John C.\n(20 March 2020).\n\"How to Learn Math and Physics\"\n.\nUniversity of California, Riverside\n. Retrieved\n19 December\n2020\n.\nthere's no way to understand the interpretation of quantum mechanics without also being able to\nsolve quantum mechanics problems\n – to understand the theory, you need to be able to use it (and vice versa)\n^\nSagan, Carl\n(1996).\nThe Demon-Haunted World: Science as a Candle in the Dark\n. Ballantine Books. p. 249.\nISBN\n \n0-345-40946-9\n.\n\"For most physics students, (the \"mathematical underpinning\" of quantum mechanics) might occupy them from, say, third grade to early graduate school – roughly 15 years. ... The job of the popularizer of science, trying to get across some idea of quantum mechanics to a general audience that has not gone through these initiation rites, is daunting. Indeed, there are no successful popularizations of quantum mechanics in my opinion – partly for this reason.\n^\na\nb\nc\nHolevo, Alexander S.\n(2001).\nStatistical Structure of Quantum Theory\n. Lecture Notes in Physics. Springer.\nISBN\n \n3-540-42082-7\n.\nOCLC\n \n318268606\n.\n^\nHall, Brian C. (2013).\nQuantum Theory for Mathematicians\n.\nGraduate Texts in Mathematics\n. Vol. 267. Springer. p. 125.\nISBN\n \n978-1-4614-7115-8\n.\n^\nWeinberg, Steven (2010).\nDreams Of A Final Theory: The Search for The Fundamental Laws of Nature\n. Random House. p. \n82\n.\nISBN\n \n978-1-4070-6396-6\n.\n^\nZhang, Ruiqin; Deng, Conghao (1993). \"Exact solutions of the Schrödinger equation for some quantum-mechanical many-body systems\".\nPhysical Review A\n.\n47\n(1):\n71–\n77.\nBibcode\n:\n1993PhRvA..47...71Z\n.\ndoi\n:\n10.1103\/PhysRevA.47.71\n.\nISSN\n \n1050-2947\n.\nPMID\n \n9908895\n.\n^\nLi, Jing; Drummond, N. D.; Schuck, Peter; Olevano, Valerio (2019-04-01).\n\"Comparing many-body approaches against the helium atom exact solution\"\n.\nSciPost Physics\n.\n6\n(4): 40.\narXiv\n:\n1801.09977\n.\nBibcode\n:\n2019ScPP....6...40L\n.\ndoi\n:\n10.21468\/SciPostPhys.6.4.040\n.\nISSN\n \n2542-4653\n.\n^\nDrake, Gordon W. F. (2023). \"High Precision Calculations for Helium\". In Drake, Gordon W. F. (ed.).\nSpringer Handbook of Atomic, Molecular, and Optical Physics\n. Springer Handbooks. Cham: Springer International Publishing. pp. \n199–\n216.\ndoi\n:\n10.1007\/978-3-030-73893-8_12\n.\nISBN\n \n978-3-030-73892-1\n.\n^\na\nb\nc\nd\nCohen-Tannoudji, Claude\n; Diu, Bernard; Laloë, Franck (2005).\nQuantum Mechanics\n. Translated by Hemley, Susan Reid; Ostrowsky, Nicole; Ostrowsky, Dan. John Wiley & Sons.\nISBN\n \n0-471-16433-X\n.\n^\nLandau, Lev D.\n;\nLifschitz, Evgeny M.\n(1977).\nQuantum Mechanics: Non-Relativistic Theory\n. Vol. 3 (3rd ed.).\nPergamon Press\n.\nISBN\n \n978-0-08-020940-1\n.\nOCLC\n \n2284121\n.\n^\nSection 3.2 of\nBallentine, Leslie E. (1970), \"The Statistical Interpretation of Quantum Mechanics\",\nReviews of Modern Physics\n,\n42\n(4):\n358–\n381,\nBibcode\n:\n1970RvMP...42..358B\n,\ndoi\n:\n10.1103\/RevModPhys.42.358\n,\nS2CID\n \n120024263\n. This fact is experimentally well-known for example in quantum optics; see e.g. chap. 2 and Fig. 2.1\nLeonhardt, Ulf (1997),\nMeasuring the Quantum State of Light\n, Cambridge: Cambridge University Press,\nBibcode\n:\n1997mqsl.book.....L\n,\nISBN\n \n0-521-49730-2\n.\n^\na\nb\nc\nNielsen, Michael A.\n;\nChuang, Isaac L.\n(2010).\nQuantum Computation and Quantum Information\n(2nd ed.). Cambridge University Press.\nISBN\n \n978-1-107-00217-3\n.\nOCLC\n \n844974180\n.\n^\na\nb\nRieffel, Eleanor G.\n; Polak, Wolfgang H. (2011).\nQuantum Computing: A Gentle Introduction\n. MIT Press.\nISBN\n \n978-0-262-01506-6\n.\n^\nWilde, Mark M. (2017).\nQuantum Information Theory\n(2nd ed.). Cambridge University Press.\narXiv\n:\n1106.1445\n.\ndoi\n:\n10.1017\/9781316809976.001\n.\nISBN\n \n978-1-107-17616-4\n.\nOCLC\n \n973404322\n.\nS2CID\n \n2515538\n.\n^\nSchlosshauer, Maximilian (October 2019). \"Quantum decoherence\".\nPhysics Reports\n.\n831\n:\n1–\n57.\narXiv\n:\n1911.06282\n.\nBibcode\n:\n2019PhR...831....1S\n.\ndoi\n:\n10.1016\/j.physrep.2019.10.001\n.\nS2CID\n \n208006050\n.\n^\nRechenberg, Helmut\n(1987).\n\"Erwin Schrödinger and the creation of wave mechanics\"\n(PDF)\n.\nActa Physica Polonica B\n.\n19\n(8):\n683–\n695\n. Retrieved\n13 June\n2016\n.\n^\nFeynman, Richard P.; Hibbs, Albert R. (2005). Steyer, Daniel F. (ed.).\nQuantum Mechanics and Path Integrals\n(Emended ed.). McGraw-Hill. pp. \nv–\nvii.\nISBN\n \n978-0-486-47722-0\n.\n^\nMathews, Piravonu Mathews; Venkatesan, K. (1976).\n\"The Schrödinger Equation and Stationary States\"\n.\nA Textbook of Quantum Mechanics\n. Tata McGraw-Hill. p. \n36\n.\nISBN\n \n978-0-07-096510-2\n.\n^\nParis, M. G. A. (1999). \"Entanglement and visibility at the output of a Mach–Zehnder interferometer\".\nPhysical Review A\n.\n59\n(2):\n1615–\n1621.\narXiv\n:\nquant-ph\/9811078\n.\nBibcode\n:\n1999PhRvA..59.1615P\n.\ndoi\n:\n10.1103\/PhysRevA.59.1615\n.\nS2CID\n \n13963928\n.\n^\nHaack, G. R.; Förster, H.; Büttiker, M. (2010). \"Parity detection and entanglement with a Mach-Zehnder interferometer\".\nPhysical Review B\n.\n82\n(15) 155303.\narXiv\n:\n1005.3976\n.\nBibcode\n:\n2010PhRvB..82o5303H\n.\ndoi\n:\n10.1103\/PhysRevB.82.155303\n.\nS2CID\n \n119261326\n.\n^\nVedral, Vlatko\n(2006).\nIntroduction to Quantum Information Science\n. Oxford University Press.\nISBN\n \n978-0-19-921570-6\n.\nOCLC\n \n442351498\n.\n^\nCohen, Marvin L. (2008).\n\"Essay: Fifty Years of Condensed Matter Physics\"\n.\nPhysical Review Letters\n.\n101\n(25) 250001.\nBibcode\n:\n2008PhRvL.101y0001C\n.\ndoi\n:\n10.1103\/PhysRevLett.101.250001\n.\nPMID\n \n19113681\n. Retrieved\n31 March\n2012\n.\n^\nMatson, John.\n\"What Is Quantum Mechanics Good for?\"\n.\nScientific American\n. Retrieved\n18 May\n2016\n.\n^\nTipler, Paul; Llewellyn, Ralph (2008).\nModern Physics\n(5th ed.). W. H. Freeman and Company. pp. \n160–\n161.\nISBN\n \n978-0-7167-7550-8\n.\n^\na\nb\nc\nPeres, Asher\n(1993).\nQuantum Theory: Concepts and Methods\n. Kluwer.\nISBN\n \n0-7923-2549-4\n.\n^\nBaez, John C.\n(2019-02-26).\n\"The Math That Takes Newton Into the Quantum World\"\n.\nNautilus Quarterly\n. Retrieved\n2024-03-23\n.\n^\n\"Atomic Properties\"\n. Academic.brooklyn.cuny.edu\n. Retrieved\n18 August\n2012\n.\n^\nHawking, Stephen; Penrose, Roger (2010).\nThe Nature of Space and Time\n. Princeton University Press.\nISBN\n \n978-1-4008-3474-7\n.\n^\nAoyama, Tatsumi; Hayakawa, Masashi; Kinoshita, Toichiro; Nio, Makiko (2012). \"Tenth-Order QED Contribution to the Electron g-2 and an Improved Value of the Fine Structure Constant\".\nPhysical Review Letters\n.\n109\n(11) 111807.\narXiv\n:\n1205.5368\n.\nBibcode\n:\n2012PhRvL.109k1807A\n.\ndoi\n:\n10.1103\/PhysRevLett.109.111807\n.\nPMID\n \n23005618\n.\nS2CID\n \n14712017\n.\n^\n\"The Nobel Prize in Physics 1979\"\n. Nobel Foundation\n. Retrieved\n16 December\n2020\n.\n^\nOverbye, Dennis\n(10 October 2022).\n\"Black Holes May Hide a Mind-Bending Secret About Our Universe – Take gravity, add quantum mechanics, stir. What do you get? Just maybe, a holographic cosmos\"\n.\nThe New York Times\n. Retrieved\n10 October\n2022\n.\n^\nBecker, Katrin;\nBecker, Melanie\n; Schwarz, John (2007).\nString theory and M-theory: A modern introduction\n. Cambridge University Press.\nISBN\n \n978-0-521-86069-7\n.\n^\nZwiebach, Barton\n(2009).\nA First Course in String Theory\n. Cambridge University Press.\nISBN\n \n978-0-521-88032-9\n.\n^\nRovelli, Carlo; Vidotto, Francesca (2014).\nCovariant Loop Quantum Gravity: An Elementary Introduction to Quantum Gravity and Spinfoam Theory\n. Cambridge University Press.\nISBN\n \n978-1-316-14811-2\n.\n^\nFeynman, Richard\n(1967).\nThe Character of Physical Law\n. MIT Press. p. 129.\nISBN\n \n0-262-56003-8\n.\n^\nWeinberg, Steven (2012). \"Collapse of the state vector\".\nPhysical Review A\n.\n85\n(6) 062116.\narXiv\n:\n1109.6462\n.\nBibcode\n:\n2012PhRvA..85f2116W\n.\ndoi\n:\n10.1103\/PhysRevA.85.062116\n.\nS2CID\n \n119273840\n.\n^\nHoward, Don (December 2004).\n\"Who Invented the 'Copenhagen Interpretation'? A Study in Mythology\"\n.\nPhilosophy of Science\n.\n71\n(5):\n669–\n682.\ndoi\n:\n10.1086\/425941\n.\nISSN\n \n0031-8248\n.\nS2CID\n \n9454552\n.\n^\nCamilleri, Kristian (May 2009).\n\"Constructing the Myth of the Copenhagen Interpretation\"\n.\nPerspectives on Science\n.\n17\n(1):\n26–\n57.\ndoi\n:\n10.1162\/posc.2009.17.1.26\n.\nISSN\n \n1063-6145\n.\nS2CID\n \n57559199\n.\n^\nBohr, Neils\n(1928).\n\"The Quantum Postulate and the Recent Development of Atomic Theory\"\n.\nNature\n.\n121\n(3050):\n580–\n590.\nBibcode\n:\n1928Natur.121..580B\n.\ndoi\n:\n10.1038\/121580a0\n.\n^\nHeisenberg, Werner\n(1971).\nPhysics and philosophy: the revolution in modern science\n. World perspectives (3 ed.). London: Allen & Unwin.\nISBN\n \n978-0-04-530016-7\n.\nOCLC\n \n743037461\n.\n^\nSchrödinger, Erwin (1980) [1935]. Trimmer, John (ed.). \"Die gegenwärtige Situation in der Quantenmechanik\" [The Present Situation in Quantum Mechanics].\nNaturwissenschaften\n(in German).\n23\n(50):\n844–\n849.\ndoi\n:\n10.1007\/BF01491987\n.\nJSTOR\n \n986572\n.\nS2CID\n \n22433857\n.\n^\nMa, Xiao-song; Kofler, Johannes; Zeilinger, Anton (2016-03-03). \"Delayed-choice gedanken experiments and their realizations\".\nReviews of Modern Physics\n.\n88\n(1) 015005.\narXiv\n:\n1407.2930\n.\nBibcode\n:\n2016RvMP...88a5005M\n.\ndoi\n:\n10.1103\/RevModPhys.88.015005\n.\nISSN\n \n0034-6861\n.\nS2CID\n \n34901303\n.\n^\nSchlosshauer, Maximilian; Kofler, Johannes; Zeilinger, Anton (1 August 2013). \"A snapshot of foundational attitudes toward quantum mechanics\".\nStudies in History and Philosophy of Science Part B\n.\n44\n(3):\n222–\n230.\narXiv\n:\n1301.1069\n.\nBibcode\n:\n2013SHPMP..44..222S\n.\ndoi\n:\n10.1016\/j.shpsb.2013.04.004\n.\nS2CID\n \n55537196\n.\n^\nHarrigan, Nicholas;\nSpekkens, Robert W.\n(2010). \"Einstein, incompleteness, and the epistemic view of quantum states\".\nFoundations of Physics\n.\n40\n(2): 125.\narXiv\n:\n0706.2661\n.\nBibcode\n:\n2010FoPh...40..125H\n.\ndoi\n:\n10.1007\/s10701-009-9347-0\n.\nS2CID\n \n32755624\n.\n^\nHoward, D. (1985). \"Einstein on locality and separability\".\nStudies in History and Philosophy of Science Part A\n.\n16\n(3):\n171–\n201.\nBibcode\n:\n1985SHPSA..16..171H\n.\ndoi\n:\n10.1016\/0039-3681(85)90001-9\n.\n^\nSauer, Tilman\n(1 December 2007).\n\"An Einstein manuscript on the EPR paradox for spin observables\"\n.\nStudies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics\n.\n38\n(4):\n879–\n887.\nBibcode\n:\n2007SHPMP..38..879S\n.\nCiteSeerX\n \n10.1.1.571.6089\n.\ndoi\n:\n10.1016\/j.shpsb.2007.03.002\n.\nISSN\n \n1355-2198\n.\n^\nEinstein, Albert (1949). \"Autobiographical Notes\". In Schilpp, Paul Arthur (ed.).\nAlbert Einstein: Philosopher-Scientist\n. Open Court Publishing Company.\n^\nBell, John Stewart\n(1 November 1964).\n\"On the Einstein Podolsky Rosen paradox\"\n.\nPhysics Physique Fizika\n.\n1\n(3):\n195–\n200.\nBibcode\n:\n1964PhyNY...1..195B\n.\ndoi\n:\n10.1103\/PhysicsPhysiqueFizika.1.195\n.\n^\nGoldstein, Sheldon (2017).\n\"Bohmian Mechanics\"\n.\nStanford Encyclopedia of Philosophy\n. Metaphysics Research Lab, Stanford University.\n^\nBarrett, Jeffrey (2018).\n\"Everett's Relative-State Formulation of Quantum Mechanics\"\n. In Zalta, Edward N. (ed.).\nStanford Encyclopedia of Philosophy\n. Metaphysics Research Lab, Stanford University.\n^\nEverett, Hugh\n;\nWheeler, J. A.\n;\nDeWitt, B. S.\n;\nCooper, L. N.\n; Van Vechten, D.; Graham, N. (1973).\nDeWitt, Bryce\n; Graham, R. Neill (eds.).\nThe Many-Worlds Interpretation of Quantum Mechanics\n. Princeton Series in Physics. Princeton, NJ:\nPrinceton University Press\n. p. v.\nISBN\n \n0-691-08131-X\n.\n^\nWallace, David (2003). \"Everettian Rationality: defending Deutsch's approach to probability in the Everett interpretation\".\nStud. Hist. Phil. Mod. Phys\n.\n34\n(3):\n415–\n438.\narXiv\n:\nquant-ph\/0303050\n.\nBibcode\n:\n2003SHPMP..34..415W\n.\ndoi\n:\n10.1016\/S1355-2198(03)00036-4\n.\nS2CID\n \n1921913\n.\n^\nBallentine, L. E. (1973). \"Can the statistical postulate of quantum theory be derived? – A critique of the many-universes interpretation\".\nFoundations of Physics\n.\n3\n(2):\n229–\n240.\nBibcode\n:\n1973FoPh....3..229B\n.\ndoi\n:\n10.1007\/BF00708440\n.\nS2CID\n \n121747282\n.\n^\nLandsman, N. P. (2008).\n\"The Born rule and its interpretation\"\n(PDF)\n. In Weinert, F.; Hentschel, K.; Greenberger, D.; Falkenburg, B. (eds.).\nCompendium of Quantum Physics\n. Springer.\nISBN\n \n978-3-540-70622-9\n.\nThe conclusion seems to be that no generally accepted derivation of the Born rule has been given to date, but this does not imply that such a derivation is impossible in principle.\n^\nKent, Adrian\n(2010). \"One world versus many: The inadequacy of Everettian accounts of evolution, probability, and scientific confirmation\". In S. Saunders; J. Barrett; A. Kent; D. Wallace (eds.).\nMany Worlds? Everett, Quantum Theory and Reality\n. Oxford University Press.\narXiv\n:\n0905.0624\n.\nBibcode\n:\n2009arXiv0905.0624K\n.\n^\nVan Fraassen, Bas C.\n(April 2010).\n\"Rovelli's World\"\n.\nFoundations of Physics\n.\n40\n(4):\n390–\n417.\nBibcode\n:\n2010FoPh...40..390V\n.\ndoi\n:\n10.1007\/s10701-009-9326-5\n.\nISSN\n \n0015-9018\n.\nS2CID\n \n17217776\n.\n^\nCalosi, Claudio\n; Riedel, Timotheus (2024).\n\"Relational Quantum Mechanics at the Crossroads\"\n.\nFoundations of Physics\n.\n54\n(6): 74.\nBibcode\n:\n2024FoPh...54...74C\n.\ndoi\n:\n10.1007\/s10701-024-00810-5\n.\nhdl\n:\n10278\/5097808\n.\n^\nHealey, Richard (2016).\n\"Quantum-Bayesian and Pragmatist Views of Quantum Theory\"\n. In Zalta, Edward N. (ed.).\nStanford Encyclopedia of Philosophy\n. Metaphysics Research Lab, Stanford University.\n^\nStacey, Blake C. (2023). \"Toward a World Game-Flavored as a Hawk's Wing\". In Berghofer, Philipp; Wiltsche, Harald A. (eds.).\nPhenomenology and QBism: New Approaches to Quantum Mechanics\n. Routledge. pp. \n49–\n77.\ndoi\n:\n10.4324\/9781003259008-3\n.\nISBN\n \n978-1-032-19181-2\n.\n^\nBorn, Max\n;\nWolf, Emil\n(1999).\nPrinciples of Optics\n. Cambridge University Press.\nISBN\n \n0-521-64222-1\n.\nOCLC\n \n1151058062\n.\n^\nScheider, Walter (April 1986).\n\"Bringing one of the great moments of science to the classroom\"\n.\nThe Physics Teacher\n.\n24\n(4):\n217–\n219.\nBibcode\n:\n1986PhTea..24..217S\n.\ndoi\n:\n10.1119\/1.2341987\n.\nISSN\n \n0031-921X\n.\n^\nFeynman, Richard; Leighton, Robert; Sands, Matthew (1964).\nThe Feynman Lectures on Physics\n. Vol. 1. California Institute of Technology\n. Retrieved\n30 September\n2021\n.\nReprinted, Addison-Wesley, 1989,\nISBN\n \n978-0-201-50064-6\n^\nMartin, Andre (1986), \"Cathode Ray Tubes for Industrial and Military Applications\", in Hawkes, Peter (ed.),\nAdvances in Electronics and Electron Physics, Volume 67\n, Academic Press, p. 183,\nISBN\n \n978-0-08-057733-3\n,\nEvidence for the existence of \"cathode-rays\" was first found by Plücker and Hittorf ...\n^\nDahl, Per F. (1997).\nFlash of the Cathode Rays: A History of J. J. Thomson's Electron\n. CRC Press. pp. \n47–\n57.\nISBN\n \n978-0-7503-0453-5\n.\n^\nMehra, J.\n; Rechenberg, H. (1982).\nThe Historical Development of Quantum Theory, Vol. 1: The Quantum Theory of Planck, Einstein, Bohr and Sommerfeld. Its Foundation and the Rise of Its Difficulties (1900–1925)\n. New York: Springer-Verlag.\nISBN\n \n978-0-387-90642-3\n.\n^\n\"Quantum\"\n.\nMerriam-Webster Dictionary\n.\nArchived\nfrom the original on Oct 26, 2012\n. Retrieved\n18 August\n2012\n.\n^\nKuhn, T. S.\n(1978).\nBlack-body theory and the quantum discontinuity 1894–1912\n. Oxford: Clarendon Press.\nISBN\n \n978-0-19-502383-1\n.\n^\nKragh, Helge\n(1 December 2000).\n\"Max Planck: the reluctant revolutionary\"\n.\nPhysics World\n. Retrieved\n12 December\n2020\n.\n^\nStachel, John\n(2009). \"Bohr and the Photon\".\nQuantum Reality, Relativistic Causality, and Closing the Epistemic Circle\n. The Western Ontario Series in Philosophy of Science. Vol. 73. Dordrecht: Springer. pp. \n69–\n83.\ndoi\n:\n10.1007\/978-1-4020-9107-0_5\n.\nISBN\n \n978-1-4020-9106-3\n.\n^\nEinstein, Albert (1905).\n\"Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt\"\n[On a heuristic point of view concerning the production and transformation of light].\nAnnalen der Physik\n(in German).\n17\n(6):\n132–\n148.\nBibcode\n:\n1905AnP...322..132E\n.\ndoi\n:\n10.1002\/andp.19053220607\n.\nReprinted in\nStachel, John\n, ed. (1989).\nThe Collected Papers of Albert Einstein\n(in German). Vol. 2. Princeton University Press. pp. \n149–\n166.\nSee also \"Einstein's early work on the quantum hypothesis\", ibid. pp. 134–148.\n^\nEinstein, Albert\n(1917). \"Zur Quantentheorie der Strahlung\" [On the Quantum Theory of Radiation].\nPhysikalische Zeitschrift\n(in German).\n18\n:\n121–\n128.\nBibcode\n:\n1917PhyZ...18..121E\n.\nTranslated in\nEinstein, A. (1967). \"On the Quantum Theory of Radiation\".\nThe Old Quantum Theory\n. Elsevier. pp. \n167–\n183.\ndoi\n:\n10.1016\/b978-0-08-012102-4.50018-8\n.\nISBN\n \n978-0-08-012102-4\n.\n^\nBall, Philip\n(2017-08-31).\n\"A century ago Einstein sparked the notion of the laser\"\n.\nPhysics World\n. Retrieved\n2024-03-23\n.\n^\na\nb\nter Haar, D. (1967).\nThe Old Quantum Theory\n. Pergamon Press. pp. \n3–\n75.\nISBN\n \n978-0-08-012101-7\n.\nLCCN\n \n66-29628\n.\n^\nBokulich, Alisa; Bokulich, Peter (2020-08-13).\n\"Bohr's Correspondence Principle\"\n. In\nZalta, Edward N.\n(ed.).\nStanford Encyclopedia of Philosophy\n.\nISSN\n \n1095-5054\n.\nOCLC\n \n429049174\n.\n^\n\"Semi-classical approximation\"\n.\nEncyclopedia of Mathematics\n. Archived from\nthe original\non 7 October 2022\n. Retrieved\n1 February\n2020\n.\n^\nSakurai, J. J.\n; Napolitano, J. (2014). \"Quantum Dynamics\".\nModern Quantum Mechanics\n. Pearson.\nISBN\n \n978-1-292-02410-3\n.\nOCLC\n \n929609283\n.\n^\nAharoni, Amikam\n(1996).\nIntroduction to the Theory of Ferromagnetism\n.\nClarendon Press\n. pp. \n6–7\n.\nISBN\n \n0-19-851791-2\n.\n^\nDavid Edwards, \"The Mathematical Foundations of Quantum Mechanics\",\nSynthese\n, Volume 42, Number 1\/September, 1979, pp. 1–70.\n^\nDavid Edwards, \"The Mathematical Foundations of Quantum Field Theory: Fermions, Gauge Fields, and Super-symmetry, Part I: Lattice Field Theories\",\nInternational Journal of Theoretical Physics\n, Vol. 20, No. 7 (1981).\n^\nBernstein, Jeremy\n(November 2005).\n\"Max Born and the quantum theory\"\n.\nAmerican Journal of Physics\n.\n73\n(11):\n999–\n1008.\nBibcode\n:\n2005AmJPh..73..999B\n.\ndoi\n:\n10.1119\/1.2060717\n.\nISSN\n \n0002-9505\n.\n^\nPais, Abraham\n(1997).\nA Tale of Two Continents: A Physicist's Life in a Turbulent World\n. Princeton, New Jersey: Princeton University Press.\nISBN\n \n0-691-01243-1\n.\n^\nVan Hove, Leon (1958).\n\"Von Neumann's contributions to quantum mechanics\"\n(PDF)\n.\nBulletin of the American Mathematical Society\n.\n64\n(3): Part 2:95–99.\ndoi\n:\n10.1090\/s0002-9904-1958-10206-2\n.\nArchived\n(PDF)\nfrom the original on Jan 20, 2024.\n^\nFeynman, Richard\n.\n\"The Feynman Lectures on Physics Vol. III Ch. 21: The Schrödinger Equation in a Classical Context: A Seminar on Superconductivity, 21-4\"\n.\nCalifornia Institute of Technology\n. Retrieved\n24 November\n2015\n.\n...it was long believed that the wave function of the Schrödinger equation would never have a macroscopic representation analogous to the macroscopic representation of the amplitude for photons. On the other hand, it is now realized that the phenomena of superconductivity presents us with just this situation.\n{{\ncite web\n}}\n:  CS1 maint: url-status (\nlink\n)\n^\nPackard, Richard (2006).\n\"Berkeley Experiments on Superfluid Macroscopic Quantum Effects\"\n(PDF)\n. Physics Department, University of California, Berkeley. Archived from\nthe original\n(PDF)\non 25 November 2015\n. Retrieved\n24 November\n2015\n.\nFurther reading\nThe following titles, all by working physicists, attempt to communicate quantum theory to lay people, using a minimum of technical apparatus:\nChester, Marvin\n(1987).\nPrimer of Quantum Mechanics\n. John Wiley.\nISBN\n \n0-486-42878-8\nCox, Brian\n;\nForshaw, Jeff\n(2011).\nThe Quantum Universe: Everything That Can Happen Does Happen\n. Allen Lane.\nISBN\n \n978-1-84614-432-5\n.\nRichard Feynman\n, 1985.\nQED: The Strange Theory of Light and Matter\n, Princeton University Press.\nISBN\n \n0-691-08388-6\n. Four elementary lectures on quantum electrodynamics and\nquantum field theory\n, yet containing many insights for the expert.\nGhirardi, GianCarlo\n, 2004.\nSneaking a Look at God's Cards\n, Gerald Malsbary, trans. Princeton Univ. Press. The most technical of the works cited here. Passages using\nalgebra\n,\ntrigonometry\n, and\nbra–ket notation\ncan be passed over on a first reading.\nN. David Mermin\n, 1990, \"Spooky actions at a distance: mysteries of the QT\" in his\nBoojums All the Way Through\n. Cambridge University Press: 110–76.\nVictor Stenger\n, 2000.\nTimeless Reality: Symmetry, Simplicity, and Multiple Universes\n. Buffalo, New York: Prometheus Books. Chpts. 5–8. Includes cosmological and philosophical considerations.\nMore technical:\nBernstein, Jeremy\n(2009).\nQuantum Leaps\n. Cambridge, Massachusetts: Belknap Press of Harvard University Press.\nISBN\n \n978-0-674-03541-6\n.\nBohm, David\n(1989).\nQuantum Theory\n. Dover Publications.\nISBN\n \n978-0-486-65969-5\n.\nBinney, James\n; Skinner, David (2008).\nThe Physics of Quantum Mechanics\n. Oxford University Press.\nISBN\n \n978-0-19-968857-9\n.\nEisberg, Robert;\nResnick, Robert\n(1985).\nQuantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles\n(2nd ed.). Wiley.\nISBN\n \n978-0-471-87373-0\n.\nBryce DeWitt\n, R. Neill Graham, eds., 1973.\nThe Many-Worlds Interpretation of Quantum Mechanics\n, Princeton Series in Physics, Princeton University Press.\nISBN\n \n0-691-08131-X\nEverett, Hugh\n(1957). \"Relative State Formulation of Quantum Mechanics\".\nReviews of Modern Physics\n.\n29\n(3):\n454–\n462.\nBibcode\n:\n1957RvMP...29..454E\n.\ndoi\n:\n10.1103\/RevModPhys.29.454\n.\nS2CID\n \n17178479\n.\nFeynman, Richard P.\n;\nLeighton, Robert B.\n; Sands, Matthew (1965).\nThe Feynman Lectures on Physics\n. Vol. \n1–\n3. Addison-Wesley.\nISBN\n \n978-0-7382-0008-8\n.\nD. Greenberger\n,\nK. Hentschel\n, F. Weinert, eds., 2009.\nCompendium of quantum physics, Concepts, experiments, history and philosophy\n, Springer-Verlag, Berlin, Heidelberg. Short articles on many QM topics.\nGriffiths, David J.\n(2004).\nIntroduction to Quantum Mechanics\n(2nd ed.). Prentice Hall.\nISBN\n \n978-0-13-111892-8\n.\nOCLC\n \n40251748\n.\nA standard undergraduate text.\nMax Jammer\n, 1966.\nThe Conceptual Development of Quantum Mechanics\n. McGraw Hill.\nHagen Kleinert\n, 2004.\nPath Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets\n, 3rd ed. Singapore: World Scientific.\nDraft of 4th edition.\nArchived\n2008-06-15 at the\nWayback Machine\nLandau, L. D.; Lifshitz, E. M. (1977).\nQuantum Mechanics: Non-Relativistic Theory\n. Vol. 3 (3rd ed.).\nPergamon Press\n.\nISBN\n \n978-0-08-020940-1\n.\nOnline copy\nLiboff, Richard L.\n(2002).\nIntroductory Quantum Mechanics\n. Addison-Wesley.\nISBN\n \n978-0-8053-8714-8\n.\nGunther Ludwig, 1968.\nWave Mechanics\n. London: Pergamon Press.\nISBN\n \n0-08-203204-1\nGeorge Mackey\n(2004).\nThe mathematical foundations of quantum mechanics\n. Dover Publications.\nISBN\n \n0-486-43517-2\n.\nMerzbacher, Eugen\n(1998).\nQuantum Mechanics\n. Wiley, John & Sons, Inc.\nISBN\n \n978-0-471-88702-7\n.\nAlbert Messiah\n, 1966.\nQuantum Mechanics\n(Vol. I), English translation from French by G. M. Temmer. North Holland, John Wiley & Sons. Cf. chpt. IV, section III.\nOmnès, Roland\n(1999).\nUnderstanding Quantum Mechanics\n. Princeton University Press.\nISBN\n \n978-0-691-00435-8\n.\nOCLC\n \n39849482\n.\nScerri, Eric. R.\n(2006).\nThe Periodic Table: Its Story and Its Significance\n. Oxford University Press.\nISBN\n \n0-19-530573-6\n.\nConsiders the extent to which chemistry and the periodic system have been reduced to quantum mechanics.\nSchiff, Leonard I.\n(1955).\nQuantum Mechanics\n. McGraw Hill.\nShankar, R.\n(1994).\nPrinciples of Quantum Mechanics\n. Springer.\nISBN\n \n978-0-306-44790-7\n.\nStone, A. Douglas\n(2013).\nEinstein and the Quantum\n. Princeton University Press.\nISBN\n \n978-0-691-13968-5\n.\nWhat is Quantum Mechanics? A Physics Adventure\n. Boston:\nTransnational College\n, Language Research Foundation. 1996.\nISBN\n \n978-0-9643504-1-0\n.\nOCLC\n \n34661512\n.\nVeltman, Martinus J. G.\n(2003),\nFacts and Mysteries in Elementary Particle Physics\n.\nExternal links\nIntroduction to quantum mechanics by Timon Idema\nQuantum Physics Made Relatively Simple\n: three video lectures by\nHans Bethe\n.\nCourse material\nQuantum Cook Book\nand\nPHYS 201: Fundamentals of Physics II\nby\nRamamurti Shankar\n, Yale OpenCourseware.\nModern Physics: With waves, thermodynamics, and optics\n– an online textbook.\nMIT OpenCourseWare\n:\nChemistry\nand\nPhysics\n. See\n8.04\n,\n8.05\nand\n8.06\n.\n⁠5\n+\n1\n\/\n2\n⁠\nExamples in Quantum Mechanics\n.\nPhilosophy\nIsmael, Jenann.\n\"Quantum Mechanics\"\n. In\nZalta, Edward N.\n(ed.).\nStanford Encyclopedia of Philosophy\n.\nISSN\n \n1095-5054\n.\nOCLC\n \n429049174\n.\nZalta, Edward N.\n(ed.).\n\"Philosophical Issues in Quantum Theory\"\n.\nStanford Encyclopedia of Philosophy\n.\nISSN\n \n1095-5054\n.\nOCLC\n \n429049174\n.","attrs_markdown":"[Jump to content](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#bodyContent)\n\nMain menu\n\nMain menu\n\nmove to sidebar\n\nhide\n\nNavigation\n\n- [Main page](https:\/\/en.wikipedia.org\/wiki\/Main_Page \"Visit the main page [z]\")\n- [Contents](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:Contents \"Guides to browsing Wikipedia\")\n- [Current events](https:\/\/en.wikipedia.org\/wiki\/Portal:Current_events \"Articles related to current events\")\n- [Random article](https:\/\/en.wikipedia.org\/wiki\/Special:Random \"Visit a randomly selected article [x]\")\n- [About Wikipedia](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:About \"Learn about Wikipedia and how it works\")\n- [Contact us](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:Contact_us \"How to contact Wikipedia\")\n\nContribute\n\n- [Help](https:\/\/en.wikipedia.org\/wiki\/Help:Contents \"Guidance on how to use and edit Wikipedia\")\n- [Learn to edit](https:\/\/en.wikipedia.org\/wiki\/Help:Introduction \"Learn how to edit Wikipedia\")\n- [Community portal](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:Community_portal \"The hub for editors\")\n- [Recent changes](https:\/\/en.wikipedia.org\/wiki\/Special:RecentChanges \"A list of recent changes to Wikipedia [r]\")\n- [Upload file](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:File_upload_wizard \"Add images or other media for use on Wikipedia\")\n- [Special pages](https:\/\/en.wikipedia.org\/wiki\/Special:SpecialPages \"A list of all special pages [q]\")\n\n[![](https:\/\/en.wikipedia.org\/static\/images\/icons\/enwiki-25.svg) ![Wikipedia](https:\/\/en.wikipedia.org\/static\/images\/mobile\/copyright\/wikipedia-wordmark-en-25.svg) ![The Free Encyclopedia](https:\/\/en.wikipedia.org\/static\/images\/mobile\/copyright\/wikipedia-tagline-en-25.svg)](https:\/\/en.wikipedia.org\/wiki\/Main_Page)\n\n[Search](https:\/\/en.wikipedia.org\/wiki\/Special:Search \"Search Wikipedia [f]\")\n\nAppearance\n\n- [Donate](https:\/\/donate.wikimedia.org\/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=en.wikipedia.org&uselang=en)\n- [Create account](https:\/\/en.wikipedia.org\/w\/index.php?title=Special:CreateAccount&returnto=Quantum+mechanics \"You are encouraged to create an account and log in; however, it is not mandatory\")\n- [Log in](https:\/\/en.wikipedia.org\/w\/index.php?title=Special:UserLogin&returnto=Quantum+mechanics \"You're encouraged to log in; however, it's not mandatory. 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[o]\")\n\n## Contents\nmove to sidebar\n\nhide\n\n- [(Top)](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics)\n- [1 Overview and fundamental concepts](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#Overview_and_fundamental_concepts)\n- [2 Mathematical formulation](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#Mathematical_formulation)\n  Toggle Mathematical formulation subsection\n  - [2\\.1 Time evolution of a quantum state](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#Time_evolution_of_a_quantum_state)\n  - [2\\.2 Uncertainty principle](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#Uncertainty_principle)\n  - [2\\.3 Composite systems and entanglement](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#Composite_systems_and_entanglement)\n  - [2\\.4 Equivalence between formulations](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#Equivalence_between_formulations)\n  - [2\\.5 Symmetries and conservation laws](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#Symmetries_and_conservation_laws)\n- [3 Examples](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#Examples)\n  Toggle Examples subsection\n  - [3\\.1 Free particle](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#Free_particle)\n  - [3\\.2 Particle in a box](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#Particle_in_a_box)\n  - [3\\.3 Harmonic oscillator](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#Harmonic_oscillator)\n  - [3\\.4 Mach–Zehnder interferometer](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#Mach%E2%80%93Zehnder_interferometer)\n- [4 Applications](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#Applications)\n- [5 Relation to other scientific theories](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#Relation_to_other_scientific_theories)\n  Toggle Relation to other scientific theories subsection\n  - [5\\.1 Classical mechanics](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#Classical_mechanics)\n  - [5\\.2 Special relativity and electrodynamics](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#Special_relativity_and_electrodynamics)\n  - [5\\.3 Relation to general relativity](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#Relation_to_general_relativity)\n- [6 Philosophical implications](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#Philosophical_implications)\n- [7 History](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#History)\n- [8 See also](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#See_also)\n- [9 Explanatory notes](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#Explanatory_notes)\n- [10 References](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#References)\n- [11 Further reading](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#Further_reading)\n- [12 External links](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#External_links)\n\nToggle the table of contents\n\n# Quantum mechanics\n142 languages\n\n- [Afrikaans](https:\/\/af.wikipedia.org\/wiki\/Kwantummeganika \"Kwantummeganika – Afrikaans\")\n- [Alemannisch](https:\/\/als.wikipedia.org\/wiki\/Quantenmechanik \"Quantenmechanik – Alemannic\")\n- [Aragonés](https:\/\/an.wikipedia.org\/wiki\/Mecanica_cuantica \"Mecanica cuantica – Aragonese\")\n- [अंगिका](https:\/\/anp.wikipedia.org\/wiki\/%E0%A4%95%E0%A5%8D%E0%A4%B5%E0%A4%BE%E0%A4%82%E0%A4%9F%E0%A4%AE_%E0%A4%B8%E0%A4%BF%E0%A4%A6%E0%A5%8D%E0%A4%A7%E0%A4%BE%E0%A4%82%E0%A4%A4 \"क्वांटम सिद्धांत – Angika\")\n- [العربية](https:\/\/ar.wikipedia.org\/wiki\/%D9%85%D9%8A%D9%83%D8%A7%D9%86%D9%8A%D9%83%D8%A7_%D8%A7%D9%84%D9%83%D9%85 \"ميكانيكا الكم – Arabic\")\n- [مصرى](https:\/\/arz.wikipedia.org\/wiki\/%D8%A7%D9%84%D9%85%D9%8A%D9%83%D8%A7%D9%86%D9%8A%D9%83%D8%A7_%D8%A7%D9%84%D9%83%D9%85%D9%8A%D9%87 \"الميكانيكا الكميه – Egyptian Arabic\")\n- [অসমীয়া](https:\/\/as.wikipedia.org\/wiki\/%E0%A6%95%E0%A7%8B%E0%A7%B1%E0%A6%BE%E0%A6%A3%E0%A7%8D%E0%A6%9F%E0%A6%BE%E0%A6%AE_%E0%A6%AC%E0%A6%B2%E0%A6%AC%E0%A6%BF%E0%A6%9C%E0%A7%8D%E0%A6%9E%E0%A6%BE%E0%A6%A8 \"কোৱাণ্টাম বলবিজ্ঞান – Assamese\")\n- [Asturianu](https:\/\/ast.wikipedia.org\/wiki\/Mec%C3%A1nica_cu%C3%A1ntica \"Mecánica cuántica – Asturian\")\n- [Azərbaycanca](https:\/\/az.wikipedia.org\/wiki\/Kvant_mexanikas%C4%B1 \"Kvant mexanikası – Azerbaijani\")\n- [تۆرکجه](https:\/\/azb.wikipedia.org\/wiki\/%DA%A9%D9%88%D8%A7%D9%86%D8%AA%D9%88%D9%85_%D9%85%DA%A9%D8%A7%D9%86%DB%8C%DA%A9%DB%8C \"کوانتوم مکانیکی – South Azerbaijani\")\n- [Башҡортса](https:\/\/ba.wikipedia.org\/wiki\/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0%D2%BB%D1%8B \"Квант механикаһы – Bashkir\")\n- [Basa Bali](https:\/\/ban.wikipedia.org\/wiki\/M%C3%A9kanika_kuantum \"Mékanika kuantum – Balinese\")\n- [Boarisch](https:\/\/bar.wikipedia.org\/wiki\/Fuzalmechanik \"Fuzalmechanik – Bavarian\")\n- [Žemaitėška](https:\/\/bat-smg.wikipedia.org\/wiki\/Kvant%C4%97n%C4%97_mekan%C4%97ka \"Kvantėnė mekanėka – Samogitian\")\n- [Bikol Central](https:\/\/bcl.wikipedia.org\/wiki\/Mekanikang_kwantum \"Mekanikang kwantum – Central Bikol\")\n- [Беларуская (тарашкевіца)](https:\/\/be-tarask.wikipedia.org\/wiki\/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D0%B0%D0%B2%D0%B0%D1%8F_%D0%BC%D1%8D%D1%85%D0%B0%D0%BD%D1%96%D0%BA%D0%B0 \"Квантавая мэханіка – Belarusian (Taraškievica orthography)\")\n- [Беларуская](https:\/\/be.wikipedia.org\/wiki\/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D0%B0%D0%B2%D0%B0%D1%8F_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D1%96%D0%BA%D0%B0 \"Квантавая механіка – Belarusian\")\n- [Български](https:\/\/bg.wikipedia.org\/wiki\/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D0%BE%D0%B2%D0%B0_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0 \"Квантова механика – Bulgarian\")\n- [भोजपुरी](https:\/\/bh.wikipedia.org\/wiki\/%E0%A4%95%E0%A5%8D%E0%A4%B5%E0%A4%BE%E0%A4%82%E0%A4%9F%E0%A4%AE_%E0%A4%AE%E0%A5%88%E0%A4%95%E0%A5%87%E0%A4%A8%E0%A4%BF%E0%A4%95%E0%A5%8D%E0%A4%B8 \"क्वांटम मैकेनिक्स – Bhojpuri\")\n- [ပအိုဝ်ႏဘာႏသာႏ](https:\/\/blk.wikipedia.org\/wiki\/%E1%80%80%E1%80%BD%E1%80%99%E1%80%BA%E1%80%90%E1%80%99%E1%80%BA%E1%80%99%E1%80%80%E1%80%B9%E1%80%80%E1%80%94%E1%80%85%E1%80%BA \"ကွမ်တမ်မက္ကနစ် – Pa'O\")\n- [বাংলা](https:\/\/bn.wikipedia.org\/wiki\/%E0%A6%95%E0%A7%8B%E0%A6%AF%E0%A6%BC%E0%A6%BE%E0%A6%A8%E0%A7%8D%E0%A6%9F%E0%A6%BE%E0%A6%AE_%E0%A6%AC%E0%A6%B2%E0%A6%AC%E0%A6%BF%E0%A6%9C%E0%A7%8D%E0%A6%9E%E0%A6%BE%E0%A6%A8 \"কোয়ান্টাম বলবিজ্ঞান – Bangla\")\n- [Brezhoneg](https:\/\/br.wikipedia.org\/wiki\/Mekanikerezh_kwantek \"Mekanikerezh kwantek – Breton\")\n- [Bosanski](https:\/\/bs.wikipedia.org\/wiki\/Kvantna_mehanika \"Kvantna mehanika – Bosnian\")\n- [Буряад](https:\/\/bxr.wikipedia.org\/wiki\/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D1%8B%D0%BD_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0 \"Квантын механика – Russia Buriat\")\n- [Català](https:\/\/ca.wikipedia.org\/wiki\/Mec%C3%A0nica_qu%C3%A0ntica \"Mecànica quàntica – Catalan\")\n- [閩東語 \/ Mìng-dĕ̤ng-ngṳ̄](https:\/\/cdo.wikipedia.org\/wiki\/Li%C3%B4ng-c%E1%B9%B3%CC%84_l%C4%ADk-h%C5%8Fk \"Liông-cṳ̄ lĭk-hŏk – Mindong\")\n- [کوردی](https:\/\/ckb.wikipedia.org\/wiki\/%D9%85%DB%8C%DA%A9%D8%A7%D9%86%DB%8C%DA%A9%DB%8C_%DA%A9%D9%88%D8%A7%D9%86%D8%AA%DB%86%D9%85 \"میکانیکی کوانتۆم – Central Kurdish\")\n- [Čeština](https:\/\/cs.wikipedia.org\/wiki\/Kvantov%C3%A1_mechanika \"Kvantová mechanika – Czech\")\n- [Чӑвашла](https:\/\/cv.wikipedia.org\/wiki\/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D0%BB%D0%B0_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0 \"Квантла механика – Chuvash\")\n- [Cymraeg](https:\/\/cy.wikipedia.org\/wiki\/Mecaneg_cwantwm \"Mecaneg cwantwm – Welsh\")\n- [Dansk](https:\/\/da.wikipedia.org\/wiki\/Kvantemekanik \"Kvantemekanik – Danish\")\n- [Deutsch](https:\/\/de.wikipedia.org\/wiki\/Quantenmechanik \"Quantenmechanik – German\")\n- [Ελληνικά](https:\/\/el.wikipedia.org\/wiki\/%CE%9A%CE%B2%CE%B1%CE%BD%CF%84%CE%B9%CE%BA%CE%AE_%CE%BC%CE%B7%CF%87%CE%B1%CE%BD%CE%B9%CE%BA%CE%AE \"Κβαντική μηχανική – Greek\")\n- [Esperanto](https:\/\/eo.wikipedia.org\/wiki\/Kvantuma_mekaniko \"Kvantuma mekaniko – Esperanto\")\n- [Español](https:\/\/es.wikipedia.org\/wiki\/Mec%C3%A1nica_cu%C3%A1ntica \"Mecánica cuántica – Spanish\")\n- [Eesti](https:\/\/et.wikipedia.org\/wiki\/Kvantmehaanika \"Kvantmehaanika – Estonian\")\n- [Euskara](https:\/\/eu.wikipedia.org\/wiki\/Mekanika_kuantiko \"Mekanika kuantiko – Basque\")\n- [Estremeñu](https:\/\/ext.wikipedia.org\/wiki\/Mec%C3%A1nica_cu%C3%A1ntica \"Mecánica cuántica – Extremaduran\")\n- [فارسی](https:\/\/fa.wikipedia.org\/wiki\/%D9%85%DA%A9%D8%A7%D9%86%DB%8C%DA%A9_%DA%A9%D9%88%D8%A7%D9%86%D8%AA%D9%88%D9%85%DB%8C \"مکانیک کوانتومی – Persian\")\n- [Suomi](https:\/\/fi.wikipedia.org\/wiki\/Kvanttimekaniikka \"Kvanttimekaniikka – Finnish\")\n- [Võro](https:\/\/fiu-vro.wikipedia.org\/wiki\/Kvantmekaaniga \"Kvantmekaaniga – Võro\")\n- [Français](https:\/\/fr.wikipedia.org\/wiki\/M%C3%A9canique_quantique \"Mécanique quantique – French\")\n- [Nordfriisk](https:\/\/frr.wikipedia.org\/wiki\/Kwantenmechaanik \"Kwantenmechaanik – Northern Frisian\")\n- [Gaeilge](https:\/\/ga.wikipedia.org\/wiki\/Meicnic_chandamach \"Meicnic chandamach – Irish\")\n- [Kriyòl gwiyannen](https:\/\/gcr.wikipedia.org\/wiki\/M%C3%A9kanik_kantik \"Mékanik kantik – Guianan Creole\")\n- [Gàidhlig](https:\/\/gd.wikipedia.org\/wiki\/Meacanaigs_quantumach \"Meacanaigs quantumach – Scottish Gaelic\")\n- [Galego](https:\/\/gl.wikipedia.org\/wiki\/Mec%C3%A1nica_cu%C3%A1ntica \"Mecánica cuántica – Galician\")\n- [Avañe'ẽ](https:\/\/gn.wikipedia.org\/wiki\/Mek%C3%A1nika_ku%C3%A1ntika \"Mekánika kuántika – Guarani\")\n- [Hausa](https:\/\/ha.wikipedia.org\/wiki\/Kimiyyar_kwantom \"Kimiyyar kwantom – Hausa\")\n- [עברית](https:\/\/he.wikipedia.org\/wiki\/%D7%9E%D7%9B%D7%A0%D7%99%D7%A7%D7%AA_%D7%94%D7%A7%D7%95%D7%95%D7%A0%D7%98%D7%99%D7%9D \"מכניקת הקוונטים – Hebrew\")\n- [हिन्दी](https:\/\/hi.wikipedia.org\/wiki\/%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%AE%E0%A4%BE%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%BE_%E0%A4%AF%E0%A4%BE%E0%A4%A8%E0%A5%8D%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%BF%E0%A4%95%E0%A5%80 \"प्रमात्रा यान्त्रिकी – Hindi\")\n- [Fiji Hindi](https:\/\/hif.wikipedia.org\/wiki\/Quantum_mechanics \"Quantum mechanics – Fiji Hindi\")\n- [Hrvatski](https:\/\/hr.wikipedia.org\/wiki\/Kvantna_mehanika \"Kvantna mehanika – Croatian\")\n- [Kreyòl ayisyen](https:\/\/ht.wikipedia.org\/wiki\/Mekanik_kantik \"Mekanik kantik – Haitian Creole\")\n- [Magyar](https:\/\/hu.wikipedia.org\/wiki\/Kvantummechanika \"Kvantummechanika – Hungarian\")\n- [Հայերեն](https:\/\/hy.wikipedia.org\/wiki\/%D5%94%D5%BE%D5%A1%D5%B6%D5%BF%D5%A1%D5%B5%D5%AB%D5%B6_%D5%B4%D5%A5%D5%AD%D5%A1%D5%B6%D5%AB%D5%AF%D5%A1 \"Քվանտային մեխանիկա – Armenian\")\n- [Արեւմտահայերէն](https:\/\/hyw.wikipedia.org\/wiki\/%D5%94%D5%B8%D6%82%D5%A1%D5%B6%D5%BF%D5%A1%D5%B5%D5%AB%D5%B6_%D5%B4%D5%A5%D5%A3%D5%A1%D5%B6%D5%AB%D5%AF \"Քուանտային մեգանիկ – Western Armenian\")\n- [Interlingua](https:\/\/ia.wikipedia.org\/wiki\/Mechanica_quantic \"Mechanica quantic – Interlingua\")\n- [Bahasa Indonesia](https:\/\/id.wikipedia.org\/wiki\/Mekanika_kuantum \"Mekanika kuantum – Indonesian\")\n- [Igbo](https:\/\/ig.wikipedia.org\/wiki\/Quantum_mechanics \"Quantum mechanics – Igbo\")\n- [Ido](https:\/\/io.wikipedia.org\/wiki\/Quantumala_mekaniko \"Quantumala mekaniko – Ido\")\n- [Íslenska](https:\/\/is.wikipedia.org\/wiki\/Skammtafr%C3%A6%C3%B0i \"Skammtafræði – Icelandic\")\n- [Italiano](https:\/\/it.wikipedia.org\/wiki\/Meccanica_quantistica \"Meccanica quantistica – Italian\")\n- [日本語](https:\/\/ja.wikipedia.org\/wiki\/%E9%87%8F%E5%AD%90%E5%8A%9B%E5%AD%A6 \"量子力学 – Japanese\")\n- [Patois](https:\/\/jam.wikipedia.org\/wiki\/Kuantom_mikianix \"Kuantom mikianix – Jamaican Creole English\")\n- [ქართული](https:\/\/ka.wikipedia.org\/wiki\/%E1%83%99%E1%83%95%E1%83%90%E1%83%9C%E1%83%A2%E1%83%A3%E1%83%A0%E1%83%98_%E1%83%9B%E1%83%94%E1%83%A5%E1%83%90%E1%83%9C%E1%83%98%E1%83%99%E1%83%90 \"კვანტური მექანიკა – Georgian\")\n- [Qaraqalpaqsha](https:\/\/kaa.wikipedia.org\/wiki\/Kvant_mexanika \"Kvant mexanika – Kara-Kalpak\")\n- [Kabɩyɛ](https:\/\/kbp.wikipedia.org\/wiki\/%C3%91%CA%8B%C5%8B_ho%C9%96e \"Ñʋŋ hoɖe – Kabiye\")\n- [Қазақша](https:\/\/kk.wikipedia.org\/wiki\/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D1%82%D1%8B%D2%9B_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0 \"Кванттық механика – Kazakh\")\n- [ಕನ್ನಡ](https:\/\/kn.wikipedia.org\/wiki\/%E0%B2%95%E0%B3%8D%E0%B2%B5%E0%B2%BE%E0%B2%82%E0%B2%9F%E0%B2%AE%E0%B3%8D_%E0%B2%AD%E0%B3%8C%E0%B2%A4%E0%B2%B6%E0%B2%BE%E0%B2%B8%E0%B3%8D%E0%B2%A4%E0%B3%8D%E0%B2%B0 \"ಕ್ವಾಂಟಮ್ ಭೌತಶಾಸ್ತ್ರ – Kannada\")\n- [한국어](https:\/\/ko.wikipedia.org\/wiki\/%EC%96%91%EC%9E%90%EC%97%AD%ED%95%99 \"양자역학 – Korean\")\n- [Кыргызча](https:\/\/ky.wikipedia.org\/wiki\/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D1%82%D1%8B%D0%BA_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0 \"Кванттык механика – Kyrgyz\")\n- [Latina](https:\/\/la.wikipedia.org\/wiki\/Mechanica_quantica \"Mechanica quantica – Latin\")\n- [Limburgs](https:\/\/li.wikipedia.org\/wiki\/Kwantummechanica \"Kwantummechanica – Limburgish\")\n- [Lombard](https:\/\/lmo.wikipedia.org\/wiki\/Mec%C3%A0nega_quant%C3%ACstega \"Mecànega quantìstega – Lombard\")\n- [Lietuvių](https:\/\/lt.wikipedia.org\/wiki\/Kvantin%C4%97_mechanika \"Kvantinė mechanika – Lithuanian\")\n- [Latviešu](https:\/\/lv.wikipedia.org\/wiki\/Kvantu_meh%C4%81nika \"Kvantu mehānika – Latvian\")\n- [Македонски](https:\/\/mk.wikipedia.org\/wiki\/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D0%BD%D0%B0_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0 \"Квантна механика – Macedonian\")\n- [മലയാളം](https:\/\/ml.wikipedia.org\/wiki\/%E0%B4%95%E0%B5%8D%E0%B4%B5%E0%B4%BE%E0%B4%A3%E0%B5%8D%E0%B4%9F%E0%B4%82_%E0%B4%AC%E0%B4%B2%E0%B4%A4%E0%B4%A8%E0%B5%8D%E0%B4%A4%E0%B5%8D%E0%B4%B0%E0%B4%82 \"ക്വാണ്ടം ബലതന്ത്രം – Malayalam\")\n- [Монгол](https:\/\/mn.wikipedia.org\/wiki\/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA \"Квант механик – Mongolian\")\n- [मराठी](https:\/\/mr.wikipedia.org\/wiki\/%E0%A4%AA%E0%A5%81%E0%A4%82%E0%A4%9C_%E0%A4%AF%E0%A4%BE%E0%A4%AE%E0%A4%BF%E0%A4%95%E0%A5%80 \"पुंज यामिकी – Marathi\")\n- [Bahasa Melayu](https:\/\/ms.wikipedia.org\/wiki\/Mekanik_kuantum \"Mekanik kuantum – Malay\")\n- [Malti](https:\/\/mt.wikipedia.org\/wiki\/Mekkanika_kwantistika \"Mekkanika kwantistika – Maltese\")\n- [မြန်မာဘာသာ](https:\/\/my.wikipedia.org\/wiki\/%E1%80%80%E1%80%BD%E1%80%99%E1%80%BA%E1%80%90%E1%80%99%E1%80%BA%E1%80%99%E1%80%80%E1%80%B9%E1%80%80%E1%80%84%E1%80%BA%E1%80%B8%E1%80%94%E1%80%85%E1%80%BA \"ကွမ်တမ်မက္ကင်းနစ် – Burmese\")\n- [مازِرونی](https:\/\/mzn.wikipedia.org\/wiki\/%DA%A9%D9%88%D8%A7%D9%86%D8%AA%D9%88%D9%85%DB%8C_%D9%81%DB%8C%D8%B2%DB%8C%DA%A9 \"کوانتومی فیزیک – Mazanderani\")\n- [Plattdüütsch](https:\/\/nds.wikipedia.org\/wiki\/Quantenmechanik \"Quantenmechanik – Low German\")\n- [नेपाली](https:\/\/ne.wikipedia.org\/wiki\/%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%AE%E0%A4%BE%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%BE_%E0%A4%AF%E0%A4%BE%E0%A4%A8%E0%A5%8D%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%BF%E0%A4%95%E0%A5%80 \"प्रमात्रा यान्त्रिकी – Nepali\")\n- [नेपाल भाषा](https:\/\/new.wikipedia.org\/wiki\/%E0%A4%95%E0%A5%8D%E0%A4%B5%E0%A4%BE%E0%A4%A8%E0%A5%8D%E0%A4%9F%E0%A4%AE_%E0%A4%AE%E0%A5%87%E0%A4%95%E0%A4%BE%E0%A4%A8%E0%A4%BF%E0%A4%95%E0%A5%8D%E0%A4%B8%E0%A5%8D \"क्वान्टम मेकानिक्स् – Newari\")\n- [Nederlands](https:\/\/nl.wikipedia.org\/wiki\/Kwantummechanica \"Kwantummechanica – Dutch\")\n- [Norsk nynorsk](https:\/\/nn.wikipedia.org\/wiki\/Kvantemekanikk \"Kvantemekanikk – Norwegian Nynorsk\")\n- [Norsk bokmål](https:\/\/no.wikipedia.org\/wiki\/Kvantemekanikk \"Kvantemekanikk – Norwegian Bokmål\")\n- [Occitan](https:\/\/oc.wikipedia.org\/wiki\/Mecanica_quantica \"Mecanica quantica – Occitan\")\n- [ਪੰਜਾਬੀ](https:\/\/pa.wikipedia.org\/wiki\/%E0%A8%95%E0%A9%81%E0%A8%86%E0%A8%82%E0%A8%9F%E0%A8%AE_%E0%A8%AE%E0%A8%95%E0%A9%88%E0%A8%A8%E0%A8%BF%E0%A8%95%E0%A8%B8 \"ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ – Punjabi\")\n- [Polski](https:\/\/pl.wikipedia.org\/wiki\/Mechanika_kwantowa \"Mechanika kwantowa – Polish\")\n- [Piemontèis](https:\/\/pms.wikipedia.org\/wiki\/Mec%C3%A0nica_qu%C3%A0ntica \"Mecànica quàntica – Piedmontese\")\n- [پنجابی](https:\/\/pnb.wikipedia.org\/wiki\/%DA%A9%D9%88%D8%A7%D9%86%D9%B9%D9%85_%D9%85%DA%A9%DB%8C%D9%86%DA%A9%D8%B3 \"کوانٹم مکینکس – Western Punjabi\")\n- [پښتو](https:\/\/ps.wikipedia.org\/wiki\/%DA%A9%D9%88%D8%A7%D9%86%D9%BC%D9%88%D9%85_%D9%85%DB%8C%D8%AE%D8%A7%D9%86%DB%8C%DA%A9 \"کوانټوم میخانیک – Pashto\")\n- [Português](https:\/\/pt.wikipedia.org\/wiki\/Mec%C3%A2nica_qu%C3%A2ntica \"Mecânica quântica – Portuguese\")\n- [ရခိုင်](https:\/\/rki.wikipedia.org\/wiki\/%E1%80%80%E1%80%BD%E1%80%99%E1%80%BA%E1%80%90%E1%80%99%E1%80%BA%E1%80%99%E1%80%80%E1%80%B9%E1%80%80%E1%80%84%E1%80%BA%E1%80%B8%E1%80%94%E1%80%85%E1%80%BA \"ကွမ်တမ်မက္ကင်းနစ် – Arakanese\")\n- [Română](https:\/\/ro.wikipedia.org\/wiki\/Mecanic%C4%83_cuantic%C4%83 \"Mecanică cuantică – Romanian\")\n- [Русский](https:\/\/ru.wikipedia.org\/wiki\/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D0%BE%D0%B2%D0%B0%D1%8F_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0 \"Квантовая механика – Russian\")\n- [Русиньскый](https:\/\/rue.wikipedia.org\/wiki\/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D0%BE%D0%B2%D0%B0_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D1%96%D0%BA%D0%B0 \"Квантова механіка – Rusyn\")\n- [संस्कृतम्](https:\/\/sa.wikipedia.org\/wiki\/%E0%A4%B8%E0%A5%82%E0%A4%95%E0%A5%8D%E0%A4%B7%E0%A5%8D%E0%A4%AE%E0%A4%AD%E0%A5%8C%E0%A4%A4%E0%A4%BF%E0%A4%95%E0%A4%B6%E0%A4%BE%E0%A4%B8%E0%A5%8D%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%AE%E0%A5%8D \"सूक्ष्मभौतिकशास्त्रम् – Sanskrit\")\n- [Саха тыла](https:\/\/sah.wikipedia.org\/wiki\/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D0%BE%D0%B2%D0%B0%D0%B9_%D1%84%D0%B8%D0%B7%D0%B8%D0%BA%D0%B0 \"Квантовай физика – Yakut\")\n- [Sicilianu](https:\/\/scn.wikipedia.org\/wiki\/Micc%C3%A0nica_quant%C3%ACstica \"Miccànica quantìstica – Sicilian\")\n- [Scots](https:\/\/sco.wikipedia.org\/wiki\/Quantum_mechanics \"Quantum mechanics – Scots\")\n- [سنڌي](https:\/\/sd.wikipedia.org\/wiki\/%DA%AA%D9%88%D8%A7%D9%86%D9%BD%D9%85_%D9%85%DA%AA%D9%8A%D9%86%DA%AA%D8%B3 \"ڪوانٽم مڪينڪس – Sindhi\")\n- [Srpskohrvatski \/ српскохрватски](https:\/\/sh.wikipedia.org\/wiki\/Kvantna_mehanika \"Kvantna mehanika – Serbo-Croatian\")\n- [Taclḥit](https:\/\/shi.wikipedia.org\/wiki\/Tamikanikt_tasmktant \"Tamikanikt tasmktant – Tachelhit\")\n- [සිංහල](https:\/\/si.wikipedia.org\/wiki\/%E0%B6%9A%E0%B7%8A%E0%B7%80%E0%B7%9C%E0%B6%B1%E0%B7%8A%E0%B6%A7%E0%B6%B8%E0%B7%8A_%E0%B6%BA%E0%B7%8F%E0%B6%B1%E0%B7%8A%E0%B6%AD%E0%B7%8A%E2%80%8D%E0%B6%BB_%E0%B7%80%E0%B7%92%E0%B6%AF%E0%B7%8A%E2%80%8D%E0%B6%BA%E0%B7%8F%E2%80%8D%E0%B7%80 \"ක්වොන්ටම් යාන්ත්‍ර විද්‍යා‍ව – Sinhala\")\n- [Simple English](https:\/\/simple.wikipedia.org\/wiki\/Quantum_mechanics \"Quantum mechanics – Simple English\")\n- [Slovenčina](https:\/\/sk.wikipedia.org\/wiki\/Kvantov%C3%A1_mechanika \"Kvantová mechanika – Slovak\")\n- [Slovenščina](https:\/\/sl.wikipedia.org\/wiki\/Kvantna_mehanika \"Kvantna mehanika – Slovenian\")\n- [Shqip](https:\/\/sq.wikipedia.org\/wiki\/Mekanika_kuantike \"Mekanika kuantike – Albanian\")\n- [Српски \/ srpski](https:\/\/sr.wikipedia.org\/wiki\/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D0%BD%D0%B0_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0 \"Квантна механика – Serbian\")\n- [Sunda](https:\/\/su.wikipedia.org\/wiki\/M%C3%A9kanika_kuantum \"Mékanika kuantum – Sundanese\")\n- [Svenska](https:\/\/sv.wikipedia.org\/wiki\/Kvantmekanik \"Kvantmekanik – Swedish\")\n- [Kiswahili](https:\/\/sw.wikipedia.org\/wiki\/Umakanika_kwanta \"Umakanika kwanta – Swahili\")\n- [Ślůnski](https:\/\/szl.wikipedia.org\/wiki\/Kwantow%C5%8F_mechanika \"Kwantowŏ mechanika – Silesian\")\n- [தமிழ்](https:\/\/ta.wikipedia.org\/wiki\/%E0%AE%95%E0%AF%81%E0%AE%B5%E0%AE%BE%E0%AE%A3%E0%AF%8D%E0%AE%9F%E0%AE%AE%E0%AF%8D_%E0%AE%87%E0%AE%AF%E0%AE%99%E0%AF%8D%E0%AE%95%E0%AE%BF%E0%AE%AF%E0%AE%B2%E0%AF%8D \"குவாண்டம் இயங்கியல் – Tamil\")\n- [తెలుగు](https:\/\/te.wikipedia.org\/wiki\/%E0%B0%95%E0%B1%8D%E0%B0%B5%E0%B0%BE%E0%B0%82%E0%B0%9F%E0%B0%82_%E0%B0%AF%E0%B0%BE%E0%B0%82%E0%B0%A4%E0%B1%8D%E0%B0%B0%E0%B0%BF%E0%B0%95_%E0%B0%B6%E0%B0%BE%E0%B0%B8%E0%B1%8D%E0%B0%A4%E0%B1%8D%E0%B0%B0%E0%B0%82 \"క్వాంటం యాంత్రిక శాస్త్రం – Telugu\")\n- [Тоҷикӣ](https:\/\/tg.wikipedia.org\/wiki\/%D0%9C%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0%D0%B8_%D0%BA%D0%B2%D0%B0%D0%BD%D1%82%D3%A3 \"Механикаи квантӣ – Tajik\")\n- [ไทย](https:\/\/th.wikipedia.org\/wiki\/%E0%B8%81%E0%B8%A5%E0%B8%A8%E0%B8%B2%E0%B8%AA%E0%B8%95%E0%B8%A3%E0%B9%8C%E0%B8%84%E0%B8%A7%E0%B8%AD%E0%B8%99%E0%B8%95%E0%B8%B1%E0%B8%A1 \"กลศาสตร์ควอนตัม – Thai\")\n- [Tagalog](https:\/\/tl.wikipedia.org\/wiki\/Mekanikang_quantum \"Mekanikang quantum – Tagalog\")\n- [Toki pona](https:\/\/tok.wikipedia.org\/wiki\/sona_tawa_pi_wan_lili \"sona tawa pi wan lili – Toki Pona\")\n- [Türkçe](https:\/\/tr.wikipedia.org\/wiki\/Kuantum_mekani%C4%9Fi \"Kuantum mekaniği – Turkish\")\n- 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[Wikiversity](https:\/\/en.wikiversity.org\/wiki\/Quantum_mechanics)\n- [Wikidata item](https:\/\/www.wikidata.org\/wiki\/Special:EntityPage\/Q944 \"Structured data on this page hosted by Wikidata [g]\")\n\nAppearance\n\nmove to sidebar\n\nhide\n\n[![This is a good article. Click here for more information.](https:\/\/upload.wikimedia.org\/wikipedia\/en\/thumb\/9\/94\/Symbol_support_vote.svg\/20px-Symbol_support_vote.svg.png)](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:Good_articles* \"This is a good article. Click here for more information.\")\n\n[![Page semi-protected](https:\/\/upload.wikimedia.org\/wikipedia\/en\/thumb\/1\/1b\/Semi-protection-shackle.svg\/20px-Semi-protection-shackle.svg.png)](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:Protection_policy#semi \"This article is semi-protected.\")\n\nFrom Wikipedia, the free encyclopedia\n\nDescription of physical properties at the atomic and subatomic scale\n\n\"Quantum systems\" redirects here. For the company, see [Quantum-Systems](https:\/\/en.wikipedia.org\/wiki\/Quantum-Systems \"Quantum-Systems\").\n\nFor a more accessible and less technical introduction to this topic, see [Introduction to quantum mechanics](https:\/\/en.wikipedia.org\/wiki\/Introduction_to_quantum_mechanics \"Introduction to quantum mechanics\").\n\n[![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/e\/e7\/Hydrogen_Density_Plots.png\/500px-Hydrogen_Density_Plots.png)](https:\/\/en.wikipedia.org\/wiki\/File:Hydrogen_Density_Plots.png)\n\n[Wave functions](https:\/\/en.wikipedia.org\/wiki\/Wave_function \"Wave function\") of the [electron](https:\/\/en.wikipedia.org\/wiki\/Electron \"Electron\") in a hydrogen atom at different energy levels. Quantum mechanics cannot predict the exact location of a particle in space, only the probability of finding it at different locations.[\\[1\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Born1926-1) The brighter areas represent a higher probability of finding the electron.\n\n|   |\n|---|\n| Part of a series of articles about |\n| [Quantum mechanics]() |\n| i ℏ d d t \\| Ψ ⟩ \\= H ^ \\| Ψ ⟩ {\\\\displaystyle i\\\\hbar {\\\\frac {d}{dt}}\\|\\\\Psi \\\\rangle ={\\\\hat {H}}\\|\\\\Psi \\\\rangle } ![{\\\\displaystyle i\\\\hbar {\\\\frac {d}{dt}}\\|\\\\Psi \\\\rangle ={\\\\hat {H}}\\|\\\\Psi \\\\rangle }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/1799e4a910c7d26396922a20ef5ceec25ca1871c)[Schrödinger equation](https:\/\/en.wikipedia.org\/wiki\/Schr%C3%B6dinger_equation \"Schrödinger equation\") |\n\n**Quantum mechanics** is the fundamental physical [theory](https:\/\/en.wikipedia.org\/wiki\/Scientific_theory \"Scientific theory\") that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of [atoms](https:\/\/en.wikipedia.org\/wiki\/Atom \"Atom\").[\\[2\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Feynman-2): 1.1  It is the foundation of all **quantum physics**, which includes [quantum chemistry](https:\/\/en.wikipedia.org\/wiki\/Quantum_chemistry \"Quantum chemistry\"), [quantum biology](https:\/\/en.wikipedia.org\/wiki\/Quantum_biology \"Quantum biology\"), [quantum field theory](https:\/\/en.wikipedia.org\/wiki\/Quantum_field_theory \"Quantum field theory\"), [quantum technology](https:\/\/en.wikipedia.org\/wiki\/Quantum_technology \"Quantum technology\"), and [quantum information science](https:\/\/en.wikipedia.org\/wiki\/Quantum_information_science \"Quantum information science\").\n\nQuantum mechanics can describe many systems that [classical physics](https:\/\/en.wikipedia.org\/wiki\/Classical_physics \"Classical physics\") cannot. Classical physics can describe many aspects of nature at an ordinary ([macroscopic](https:\/\/en.wikipedia.org\/wiki\/Macroscopic \"Macroscopic\") and [(optical) microscopic](https:\/\/en.wikipedia.org\/wiki\/Microscopic_scale \"Microscopic scale\")) scale, however is insufficient for describing them at very small [submicroscopic](https:\/\/en.wikipedia.org\/wiki\/Submicroscopic \"Submicroscopic\") (atomic and [subatomic](https:\/\/en.wikipedia.org\/wiki\/Subatomic \"Subatomic\")) scales. Classical mechanics can be derived from quantum mechanics as an approximation that is valid at ordinary scales.[\\[3\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-3)\n\n**Quantum systems** have [bound](https:\/\/en.wikipedia.org\/wiki\/Bound_state \"Bound state\") states that are [quantized](https:\/\/en.wikipedia.org\/wiki\/Quantization_\\(physics\\) \"Quantization (physics)\") to [discrete values](https:\/\/en.wikipedia.org\/wiki\/Discrete_mathematics \"Discrete mathematics\") of [energy](https:\/\/en.wikipedia.org\/wiki\/Energy \"Energy\"), [momentum](https:\/\/en.wikipedia.org\/wiki\/Momentum \"Momentum\"), [angular momentum](https:\/\/en.wikipedia.org\/wiki\/Angular_momentum \"Angular momentum\"), and other quantities, in contrast to classical systems where these quantities can be measured continuously. Measurements of quantum systems show characteristics of both [particles](https:\/\/en.wikipedia.org\/wiki\/Particle \"Particle\") and [waves](https:\/\/en.wikipedia.org\/wiki\/Wave \"Wave\") ([wave–particle duality](https:\/\/en.wikipedia.org\/wiki\/Wave%E2%80%93particle_duality \"Wave–particle duality\")), and there are limits to how accurately the value of a physical quantity can be predicted prior to its measurement, given a complete set of initial conditions (the [uncertainty principle](https:\/\/en.wikipedia.org\/wiki\/Uncertainty_principle \"Uncertainty principle\")).\n\nQuantum mechanics arose gradually from theories to explain observations that could not be reconciled with [classical physics](https:\/\/en.wikipedia.org\/wiki\/Classical_physics \"Classical physics\"), such as [Max Planck](https:\/\/en.wikipedia.org\/wiki\/Max_Planck \"Max Planck\")'s solution in 1900 to the [black-body radiation](https:\/\/en.wikipedia.org\/wiki\/Black-body_radiation \"Black-body radiation\") problem, and the correspondence between energy and frequency in [Albert Einstein](https:\/\/en.wikipedia.org\/wiki\/Albert_Einstein \"Albert Einstein\")'s [1905 paper](https:\/\/en.wikipedia.org\/wiki\/Annus_Mirabilis_papers#Photoelectric_effect \"Annus Mirabilis papers\"), which explained the [photoelectric effect](https:\/\/en.wikipedia.org\/wiki\/Photoelectric_effect \"Photoelectric effect\"). These early attempts to understand microscopic phenomena, now known as the \"[old quantum theory](https:\/\/en.wikipedia.org\/wiki\/Old_quantum_theory \"Old quantum theory\")\", led to the full development of quantum mechanics in the mid-1920s by [Niels Bohr](https:\/\/en.wikipedia.org\/wiki\/Niels_Bohr \"Niels Bohr\"), [Erwin Schrödinger](https:\/\/en.wikipedia.org\/wiki\/Erwin_Schr%C3%B6dinger \"Erwin Schrödinger\"), [Werner Heisenberg](https:\/\/en.wikipedia.org\/wiki\/Werner_Heisenberg \"Werner Heisenberg\"), [Max Born](https:\/\/en.wikipedia.org\/wiki\/Max_Born \"Max Born\"), [Paul Dirac](https:\/\/en.wikipedia.org\/wiki\/Paul_Dirac \"Paul Dirac\") and others. The modern theory is formulated in various [specially developed mathematical formalisms](https:\/\/en.wikipedia.org\/wiki\/Mathematical_formulations_of_quantum_mechanics \"Mathematical formulations of quantum mechanics\"). In one of them, a mathematical entity called the [wave function](https:\/\/en.wikipedia.org\/wiki\/Wave_function \"Wave function\") provides information, in the form of [probability amplitudes](https:\/\/en.wikipedia.org\/wiki\/Probability_amplitude \"Probability amplitude\"), about what measurements of a particle's energy, momentum, and other physical properties may yield.\n\n## Overview and fundamental concepts\nQuantum mechanics allows the calculation of properties and behaviour of [physical systems](https:\/\/en.wikipedia.org\/wiki\/Physical_systems \"Physical systems\"). It is typically applied to microscopic systems: [molecules](https:\/\/en.wikipedia.org\/wiki\/Molecules \"Molecules\"), [atoms](https:\/\/en.wikipedia.org\/wiki\/Atoms \"Atoms\") and [subatomic particles](https:\/\/en.wikipedia.org\/wiki\/Subatomic_particle \"Subatomic particle\"). It has been demonstrated to hold for complex molecules with thousands of atoms,[\\[4\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-4) but its application to human beings raises philosophical problems, such as [Wigner's friend](https:\/\/en.wikipedia.org\/wiki\/Wigner%27s_friend \"Wigner's friend\"), and its application to the universe as a whole remains speculative.[\\[5\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-5) Predictions of quantum mechanics have been verified experimentally to an extremely high degree of [accuracy](https:\/\/en.wikipedia.org\/wiki\/Accuracy \"Accuracy\"). For example, the refinement of quantum mechanics for the interaction of light and matter, known as [quantum electrodynamics](https:\/\/en.wikipedia.org\/wiki\/Quantum_electrodynamics \"Quantum electrodynamics\") (QED), has been [shown to agree with experiment](https:\/\/en.wikipedia.org\/wiki\/Precision_tests_of_QED \"Precision tests of QED\") to within 1 part in 1012 when predicting the magnetic properties of an electron.[\\[6\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-6)\n\nA fundamental feature of the theory is that it usually cannot predict with certainty what will happen, but only gives probabilities. Mathematically, a probability is found by taking the square of the absolute value of a [complex number](https:\/\/en.wikipedia.org\/wiki\/Complex_number \"Complex number\"), known as a probability amplitude. This is known as the [Born rule](https:\/\/en.wikipedia.org\/wiki\/Born_rule \"Born rule\"), named after physicist [Max Born](https:\/\/en.wikipedia.org\/wiki\/Max_Born \"Max Born\"). For example, a quantum particle like an [electron](https:\/\/en.wikipedia.org\/wiki\/Electron \"Electron\") can be described by a wave function, which associates to each point in space a probability amplitude. Applying the Born rule to these amplitudes gives a [probability density function](https:\/\/en.wikipedia.org\/wiki\/Probability_density_function \"Probability density function\") for the position that the electron will be found to have when an experiment is performed to measure it. This is the best the theory can do; it cannot say for certain where the electron will be found. The [Schrödinger equation](https:\/\/en.wikipedia.org\/wiki\/Schr%C3%B6dinger_equation \"Schrödinger equation\") relates the collection of probability amplitudes that pertain to one moment of time to the collection of probability amplitudes that pertain to another.[\\[7\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Zwiebach2022-7): 67–87\n\nOne consequence of the mathematical rules of quantum mechanics is a tradeoff in predictability between measurable quantities. The most famous form of this [uncertainty principle](https:\/\/en.wikipedia.org\/wiki\/Uncertainty_principle \"Uncertainty principle\") says that no matter how a quantum particle is prepared or how carefully experiments upon it are arranged, it is impossible to have a precise prediction for a measurement of its position and also at the same time for a measurement of its [momentum](https:\/\/en.wikipedia.org\/wiki\/Momentum \"Momentum\").[\\[7\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Zwiebach2022-7): 427–435\n\n[![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/c\/cd\/Double-slit.svg\/330px-Double-slit.svg.png)](https:\/\/en.wikipedia.org\/wiki\/File:Double-slit.svg)\n\nAn illustration of the [double-slit experiment](https:\/\/en.wikipedia.org\/wiki\/Double-slit_experiment \"Double-slit experiment\")\n\nAnother consequence of the mathematical rules of quantum mechanics is the phenomenon of [quantum interference](https:\/\/en.wikipedia.org\/wiki\/Quantum_interference \"Quantum interference\"), which is often illustrated with the [double-slit experiment](https:\/\/en.wikipedia.org\/wiki\/Double-slit_experiment \"Double-slit experiment\"). In the basic version of this experiment, a [coherent light source](https:\/\/en.wikipedia.org\/wiki\/Coherence_\\(physics\\) \"Coherence (physics)\"), such as a [laser](https:\/\/en.wikipedia.org\/wiki\/Laser \"Laser\") beam, illuminates a plate pierced by two parallel slits, and the light passing through the slits is observed on a screen behind the plate.[\\[8\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Lederman-8): 102–111 [\\[2\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Feynman-2): 1.1–1.8  The wave nature of light causes the light waves passing through the two slits to [interfere](https:\/\/en.wikipedia.org\/wiki\/Interference_\\(wave_propagation\\) \"Interference (wave propagation)\"), producing bright and dark bands on the screen – a result that would not be expected if light consisted of classical particles.[\\[8\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Lederman-8) However, the light is always found to be absorbed at the screen at discrete points, as individual particles rather than waves; the interference pattern appears via the varying density of these particle hits on the screen. Furthermore, versions of the experiment that include detectors at the slits find that each detected [photon](https:\/\/en.wikipedia.org\/wiki\/Photon \"Photon\") passes through one slit (as would a classical particle), and not through both slits (as would a wave).[\\[8\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Lederman-8): 109 [\\[9\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-M%C3%BCller-Kirsten-9)[\\[10\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Plotnitsky-10) However, [such experiments](https:\/\/en.wikipedia.org\/wiki\/Double-slit_experiment#Which_way \"Double-slit experiment\") demonstrate that particles do not form the interference pattern if one detects which slit they pass through. This behavior is known as [wave–particle duality](https:\/\/en.wikipedia.org\/wiki\/Wave%E2%80%93particle_duality \"Wave–particle duality\"). In addition to light, [electrons](https:\/\/en.wikipedia.org\/wiki\/Electrons \"Electrons\"), [atoms](https:\/\/en.wikipedia.org\/wiki\/Atoms \"Atoms\"), and [molecules](https:\/\/en.wikipedia.org\/wiki\/Molecules \"Molecules\") are all found to exhibit the same dual behavior when fired towards a double slit.[\\[2\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Feynman-2)\n\n[![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/5\/5f\/QuantumTunnel.jpg\/330px-QuantumTunnel.jpg)](https:\/\/en.wikipedia.org\/wiki\/File:QuantumTunnel.jpg)\n\nA simplified diagram of [quantum tunneling](https:\/\/en.wikipedia.org\/wiki\/Quantum_tunneling \"Quantum tunneling\"), a phenomenon by which a particle may move through a barrier which would be impossible under classical mechanics\n\nAnother non-classical phenomenon predicted by quantum mechanics is [quantum tunnelling](https:\/\/en.wikipedia.org\/wiki\/Quantum_tunnelling \"Quantum tunnelling\"): a particle that goes up against a [potential barrier](https:\/\/en.wikipedia.org\/wiki\/Potential_barrier \"Potential barrier\") can cross it, even if its kinetic energy is smaller than the maximum of the potential.[\\[11\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-11) In classical mechanics this particle would be trapped. Quantum tunnelling has several important consequences, enabling [radioactive decay](https:\/\/en.wikipedia.org\/wiki\/Radioactive_decay \"Radioactive decay\"), [nuclear fusion](https:\/\/en.wikipedia.org\/wiki\/Nuclear_fusion \"Nuclear fusion\") in stars, and applications such as [scanning tunnelling microscopy](https:\/\/en.wikipedia.org\/wiki\/Scanning_tunnelling_microscopy \"Scanning tunnelling microscopy\"), [tunnel diode](https:\/\/en.wikipedia.org\/wiki\/Tunnel_diode \"Tunnel diode\") and [tunnel field-effect transistor](https:\/\/en.wikipedia.org\/wiki\/Tunnel_field-effect_transistor \"Tunnel field-effect transistor\").[\\[12\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Trixler2013-12)[\\[13\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-13)\n\nWhen quantum systems interact, the result can be the creation of [quantum entanglement](https:\/\/en.wikipedia.org\/wiki\/Quantum_entanglement \"Quantum entanglement\"): their properties become so intertwined that a description of the whole solely in terms of the individual parts is no longer possible. Erwin Schrödinger called entanglement \"...*the* characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought\".[\\[14\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-14) Quantum entanglement enables [quantum computing](https:\/\/en.wikipedia.org\/wiki\/Quantum_computing \"Quantum computing\") and is part of quantum communication protocols, such as [quantum key distribution](https:\/\/en.wikipedia.org\/wiki\/Quantum_key_distribution \"Quantum key distribution\") and [superdense coding](https:\/\/en.wikipedia.org\/wiki\/Superdense_coding \"Superdense coding\").[\\[15\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Caves-15) Contrary to popular misconception, entanglement does not allow sending signals [faster than light](https:\/\/en.wikipedia.org\/wiki\/Faster_than_light \"Faster than light\"), as demonstrated by the [no-communication theorem](https:\/\/en.wikipedia.org\/wiki\/No-communication_theorem \"No-communication theorem\").[\\[15\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Caves-15)\n\nAnother possibility opened by entanglement is testing for \"[hidden variables](https:\/\/en.wikipedia.org\/wiki\/Hidden_variable_theory \"Hidden variable theory\")\", hypothetical properties more fundamental than the quantities addressed in quantum theory itself, knowledge of which would allow more exact predictions than quantum theory provides. A collection of results, most significantly [Bell's theorem](https:\/\/en.wikipedia.org\/wiki\/Bell%27s_theorem \"Bell's theorem\"), have demonstrated that broad classes of such hidden-variable theories are in fact incompatible with quantum physics. According to Bell's theorem, if nature actually operates in accord with any theory of *local* hidden variables, then the results of a [Bell test](https:\/\/en.wikipedia.org\/wiki\/Bell_test \"Bell test\") will be constrained in a particular, quantifiable way. Many Bell tests have been performed and they have shown results incompatible with the constraints imposed by local hidden variables.[\\[16\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-wiseman15-16)[\\[17\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-wolchover17-17)\n\nIt is not possible to present these concepts in more than a superficial way without introducing the mathematics involved; understanding quantum mechanics requires not only manipulating complex numbers, but also [linear algebra](https:\/\/en.wikipedia.org\/wiki\/Linear_algebra \"Linear algebra\"), [differential equations](https:\/\/en.wikipedia.org\/wiki\/Differential_equation \"Differential equation\"), [group theory](https:\/\/en.wikipedia.org\/wiki\/Group_theory \"Group theory\"), and other more advanced subjects.[\\[18\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-18)[\\[19\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-19) Accordingly, this article will present a mathematical formulation of quantum mechanics and survey its application to some useful and oft-studied examples.\n\n## Mathematical formulation\nMain article: [Mathematical formulation of quantum mechanics](https:\/\/en.wikipedia.org\/wiki\/Mathematical_formulation_of_quantum_mechanics \"Mathematical formulation of quantum mechanics\")\n\nIn the mathematically rigorous formulation of quantum mechanics, the state of a quantum mechanical system is a vector ψ {\\\\displaystyle \\\\psi } ![{\\\\displaystyle \\\\psi }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a) belonging to a ([separable](https:\/\/en.wikipedia.org\/wiki\/Separable_space \"Separable space\")) complex [Hilbert space](https:\/\/en.wikipedia.org\/wiki\/Hilbert_space \"Hilbert space\") H {\\\\displaystyle {\\\\mathcal {H}}} ![{\\\\displaystyle {\\\\mathcal {H}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/19ef4c7b923a5125ac91aa491838a95ee15b804f). This vector is postulated to be normalized under the Hilbert space inner product, that is, it obeys ⟨ ψ , ψ ⟩ \\= 1 {\\\\displaystyle \\\\langle \\\\psi ,\\\\psi \\\\rangle =1} ![{\\\\displaystyle \\\\langle \\\\psi ,\\\\psi \\\\rangle =1}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/7d5758e7a60b4e54bc46e01b0618919c65b787a0), and it is well-defined up to a complex number of modulus 1 (the global phase), that is, ψ {\\\\displaystyle \\\\psi } ![{\\\\displaystyle \\\\psi }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a) and e i α ψ {\\\\displaystyle e^{i\\\\alpha }\\\\psi } ![{\\\\displaystyle e^{i\\\\alpha }\\\\psi }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/7193e9b0479da798b9cf2dcc492f49166d0d7103) represent the same physical system. In other words, the possible states are points in the [projective space](https:\/\/en.wikipedia.org\/wiki\/Projective_space \"Projective space\") of a Hilbert space, usually called the [complex projective space](https:\/\/en.wikipedia.org\/wiki\/Complex_projective_space \"Complex projective space\"). The exact nature of this Hilbert space is dependent on the system – for example, for describing position and momentum the Hilbert space is the space of complex [square-integrable](https:\/\/en.wikipedia.org\/wiki\/Square-integrable \"Square-integrable\") functions L 2 ( C ) {\\\\displaystyle L^{2}(\\\\mathbb {C} )} ![{\\\\displaystyle L^{2}(\\\\mathbb {C} )}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/3a3649087af50f42a95da2163656076529ca3139),[\\[20\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Holevo2001-20): 13  while the Hilbert space for the [spin](https:\/\/en.wikipedia.org\/wiki\/Spin_\\(physics\\) \"Spin (physics)\") of a single proton is simply the space of two-dimensional complex vectors C 2 {\\\\displaystyle \\\\mathbb {C} ^{2}} ![{\\\\displaystyle \\\\mathbb {C} ^{2}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/6f43d6ec8a1e1fe5a85aec0dd9bdcd45ae09b06b) with the usual inner product.[\\[20\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Holevo2001-20): 20\n\nPhysical quantities of interest – position, momentum, energy, spin – are represented by observables, which are [Hermitian](https:\/\/en.wikipedia.org\/wiki\/Hermitian_adjoint#Hermitian_operators \"Hermitian adjoint\") (more precisely, [self-adjoint](https:\/\/en.wikipedia.org\/wiki\/Self-adjoint_operator \"Self-adjoint operator\")) linear [operators](https:\/\/en.wikipedia.org\/wiki\/Operator_\\(physics\\) \"Operator (physics)\") acting on the Hilbert space.[\\[20\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Holevo2001-20): 17  A quantum state can be an [eigenvector](https:\/\/en.wikipedia.org\/wiki\/Eigenvector \"Eigenvector\") of an observable, in which case it is called an [eigenstate](https:\/\/en.wikipedia.org\/wiki\/Eigenstate \"Eigenstate\"), and the associated [eigenvalue](https:\/\/en.wikipedia.org\/wiki\/Eigenvalue \"Eigenvalue\") corresponds to the value of the observable in that eigenstate. More generally, a quantum state will be a linear combination of the eigenstates, known as a [quantum superposition](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition \"Quantum superposition\"). When an observable is measured, the result will be one of its eigenvalues with probability given by the [Born rule](https:\/\/en.wikipedia.org\/wiki\/Born_rule \"Born rule\"): in the simplest case the eigenvalue λ {\\\\displaystyle \\\\lambda } ![{\\\\displaystyle \\\\lambda }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/b43d0ea3c9c025af1be9128e62a18fa74bedda2a) is non-degenerate and the probability is given by \\| ⟨ λ → , ψ ⟩ \\| 2 {\\\\displaystyle \\|\\\\langle {\\\\vec {\\\\lambda }},\\\\psi \\\\rangle \\|^{2}} ![{\\\\displaystyle \\|\\\\langle {\\\\vec {\\\\lambda }},\\\\psi \\\\rangle \\|^{2}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/de4b465224a805b558697f44dcf3824fb2fbfe20), where λ → {\\\\displaystyle {\\\\vec {\\\\lambda }}} ![{\\\\displaystyle {\\\\vec {\\\\lambda }}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/d8c69573b9c55ec8ea726662102b8e693bf32f01) is its associated unit-length eigenvector. More generally, the eigenvalue is degenerate and the probability is given by ⟨ ψ , P λ ψ ⟩ {\\\\displaystyle \\\\langle \\\\psi ,P\\_{\\\\lambda }\\\\psi \\\\rangle } ![{\\\\displaystyle \\\\langle \\\\psi ,P\\_{\\\\lambda }\\\\psi \\\\rangle }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/6ba3648378b52e2e710937b36948cd908b2d6531), where P λ {\\\\displaystyle P\\_{\\\\lambda }} ![{\\\\displaystyle P\\_{\\\\lambda }}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/330591f9b6fffc93ca78514576fd0d8cfac6f0c7) is the projector onto its associated eigenspace.[\\[21\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-21) In the continuous case, these formulas give instead the [probability density](https:\/\/en.wikipedia.org\/wiki\/Probability_density \"Probability density\").\n\nAfter the [measurement](https:\/\/en.wikipedia.org\/wiki\/Measurement_in_quantum_mechanics \"Measurement in quantum mechanics\"), if result λ {\\\\displaystyle \\\\lambda } ![{\\\\displaystyle \\\\lambda }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/b43d0ea3c9c025af1be9128e62a18fa74bedda2a) was obtained, the quantum state is postulated to [collapse](https:\/\/en.wikipedia.org\/wiki\/Collapse_of_the_wavefunction \"Collapse of the wavefunction\") to λ → {\\\\displaystyle {\\\\vec {\\\\lambda }}} ![{\\\\displaystyle {\\\\vec {\\\\lambda }}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/d8c69573b9c55ec8ea726662102b8e693bf32f01), in the non-degenerate case, or to P λ ψ \/ ⟨ ψ , P λ ψ ⟩ {\\\\textstyle P\\_{\\\\lambda }\\\\psi {\\\\big \/}\\\\!{\\\\sqrt {\\\\langle \\\\psi ,P\\_{\\\\lambda }\\\\psi \\\\rangle }}} ![{\\\\textstyle P\\_{\\\\lambda }\\\\psi {\\\\big \/}\\\\!{\\\\sqrt {\\\\langle \\\\psi ,P\\_{\\\\lambda }\\\\psi \\\\rangle }}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/e38d8eb9122b686c4d8f937fe59e2ec511be43ed), in the general case. The [probabilistic](https:\/\/en.wikipedia.org\/wiki\/Probabilistic \"Probabilistic\") nature of quantum mechanics thus stems from the act of measurement. This is one of the most debated aspects of quantum theory, with different [interpretations of quantum mechanics](https:\/\/en.wikipedia.org\/wiki\/Interpretations_of_quantum_mechanics \"Interpretations of quantum mechanics\") giving radically different answers to questions regarding quantum-state collapse, as discussed [below](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#Philosophical_implications).\n\n### Time evolution of a quantum state\nThe time evolution of a quantum state is described by the Schrödinger equation: i ℏ ∂ ∂ t ψ ( t ) \\= H ψ ( t ) . {\\\\displaystyle i\\\\hbar {\\\\frac {\\\\partial }{\\\\partial t}}\\\\psi (t)=H\\\\psi (t).} ![{\\\\displaystyle i\\\\hbar {\\\\frac {\\\\partial }{\\\\partial t}}\\\\psi (t)=H\\\\psi (t).}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/5c41b5a62afa3aa83e6ea98e69d692ac0f77c2c6) Here H {\\\\displaystyle H} ![{\\\\displaystyle H}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/75a9edddcca2f782014371f75dca39d7e13a9c1b) denotes the [Hamiltonian](https:\/\/en.wikipedia.org\/wiki\/Hamiltonian_\\(quantum_mechanics\\) \"Hamiltonian (quantum mechanics)\"), the observable corresponding to the [total energy](https:\/\/en.wikipedia.org\/wiki\/Total_energy \"Total energy\") of the system, and ℏ {\\\\displaystyle \\\\hbar } ![{\\\\displaystyle \\\\hbar }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/de68de3a92517953436c93b5a76461d49160cc41) is the reduced [Planck constant](https:\/\/en.wikipedia.org\/wiki\/Planck_constant \"Planck constant\"). The constant i ℏ {\\\\displaystyle i\\\\hbar } ![{\\\\displaystyle i\\\\hbar }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/3d724d600903f701570a9f97c2a8378a6f3a80e5) is introduced so that the Hamiltonian is reduced to the [classical Hamiltonian](https:\/\/en.wikipedia.org\/wiki\/Hamiltonian_mechanics \"Hamiltonian mechanics\") in cases where the quantum system can be approximated by a classical system; the ability to make such an approximation in certain limits is called the [correspondence principle](https:\/\/en.wikipedia.org\/wiki\/Correspondence_principle \"Correspondence principle\").\n\nThe solution of this differential equation is given by ψ ( t ) \\= e − i H t \/ ℏ ψ ( 0 ) . {\\\\displaystyle \\\\psi (t)=e^{-iHt\/\\\\hbar }\\\\psi (0).} ![{\\\\displaystyle \\\\psi (t)=e^{-iHt\/\\\\hbar }\\\\psi (0).}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/3b0a822b842f8d952c4e5827f312597e72a45f4d) The operator U ( t ) \\= e − i H t \/ ℏ {\\\\displaystyle U(t)=e^{-iHt\/\\\\hbar }} ![{\\\\displaystyle U(t)=e^{-iHt\/\\\\hbar }}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/17a17e78953fa7d61788cc8bc41a25ef254fc8e0) is known as the time-evolution operator, and has the crucial property that it is [unitary](https:\/\/en.wikipedia.org\/wiki\/Unitarity_\\(physics\\) \"Unitarity (physics)\"). This time evolution is [deterministic](https:\/\/en.wikipedia.org\/wiki\/Deterministic \"Deterministic\") in the sense that – given an initial quantum state ψ ( 0 ) {\\\\displaystyle \\\\psi (0)} ![{\\\\displaystyle \\\\psi (0)}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/fcfdf3fefa6feee8b2192815ac15a2650d8a8db2) – it makes a definite prediction of what the quantum state ψ ( t ) {\\\\displaystyle \\\\psi (t)} ![{\\\\displaystyle \\\\psi (t)}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/6cf4a36b5f945be90a527b3dbe3d55d3f0439cdb) will be at any later time.[\\[22\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-22)\n\n[![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/3\/35\/Atomic-orbital-clouds_spd_m0.png\/330px-Atomic-orbital-clouds_spd_m0.png)](https:\/\/en.wikipedia.org\/wiki\/File:Atomic-orbital-clouds_spd_m0.png)\n\nFig. 1: [Probability densities](https:\/\/en.wikipedia.org\/wiki\/Probability_density_function \"Probability density function\") corresponding to the wave functions of an electron in a hydrogen atom possessing definite energy levels (increasing from the top of the image to the bottom: *n* = 1, 2, 3, ...) and angular momenta (increasing across from left to right: *s*, *p*, *d*, ...). Denser areas correspond to higher probability density in a position measurement.\n\nSuch wave functions are directly comparable to [Chladni's figures](https:\/\/en.wikipedia.org\/wiki\/Chladni%27s_figures \"Chladni's figures\") of [acoustic](https:\/\/en.wikipedia.org\/wiki\/Acoustics \"Acoustics\") modes of vibration in classical physics and are modes of oscillation as well, possessing a sharp energy and thus, a definite frequency. The [angular momentum](https:\/\/en.wikipedia.org\/wiki\/Angular_momentum \"Angular momentum\") and energy are [quantized](https:\/\/en.wikipedia.org\/wiki\/Quantization_\\(physics\\) \"Quantization (physics)\") and take *only* discrete values like those shown – as is the case for [resonant frequencies](https:\/\/en.wikipedia.org\/wiki\/Resonant_frequencies \"Resonant frequencies\") in acoustics.\n\nSome wave functions produce probability distributions that are independent of time, such as [eigenstates](https:\/\/en.wikipedia.org\/wiki\/Eigenstate \"Eigenstate\") of the Hamiltonian.[\\[7\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Zwiebach2022-7): 133–137  Many systems that are treated dynamically in classical mechanics are described by such \"static\" wave functions. For example, a single electron in an unexcited atom is pictured classically as a particle moving in a circular trajectory around the [atomic nucleus](https:\/\/en.wikipedia.org\/wiki\/Atomic_nucleus \"Atomic nucleus\"), whereas in quantum mechanics, it is described by a static wave function surrounding the nucleus. For example, the electron wave function for an unexcited hydrogen atom is a spherically symmetric function known as an [*s* orbital](https:\/\/en.wikipedia.org\/wiki\/Atomic_orbital \"Atomic orbital\") ([Fig. 1](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#fig1)).\n\nAnalytic solutions of the Schrödinger equation are known for [very few relatively simple model Hamiltonians](https:\/\/en.wikipedia.org\/wiki\/List_of_quantum-mechanical_systems_with_analytical_solutions \"List of quantum-mechanical systems with analytical solutions\") including the [quantum harmonic oscillator](https:\/\/en.wikipedia.org\/wiki\/Quantum_harmonic_oscillator \"Quantum harmonic oscillator\"), the [particle in a box](https:\/\/en.wikipedia.org\/wiki\/Particle_in_a_box \"Particle in a box\"), the [dihydrogen cation](https:\/\/en.wikipedia.org\/wiki\/Dihydrogen_cation \"Dihydrogen cation\"), and the [hydrogen atom](https:\/\/en.wikipedia.org\/wiki\/Hydrogen_atom \"Hydrogen atom\"). Even the [helium](https:\/\/en.wikipedia.org\/wiki\/Helium \"Helium\") atom – which contains just two electrons – has defied all attempts at a fully analytic treatment, admitting no solution in [closed form](https:\/\/en.wikipedia.org\/wiki\/Closed-form_expression \"Closed-form expression\").[\\[23\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-23)[\\[24\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-24)[\\[25\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-25)\n\nHowever, there are techniques for finding approximate solutions. One method, called [perturbation theory](https:\/\/en.wikipedia.org\/wiki\/Perturbation_theory_\\(quantum_mechanics\\) \"Perturbation theory (quantum mechanics)\"), uses the analytic result for a simple quantum mechanical model to create a result for a related but more complicated model by (for example) the addition of a weak [potential energy](https:\/\/en.wikipedia.org\/wiki\/Potential_energy \"Potential energy\").[\\[7\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Zwiebach2022-7): 793  Another approximation method applies to systems for which quantum mechanics produces only small deviations from classical behavior. These deviations can then be computed based on the classical motion.[\\[7\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Zwiebach2022-7): 849\n\n### Uncertainty principle\nOne consequence of the basic quantum formalism is the uncertainty principle. In its most familiar form, this states that no preparation of a quantum particle can imply simultaneously precise predictions both for a measurement of its position and for a measurement of its momentum.[\\[26\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Cohen-Tannoudji-26)[\\[27\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-L&L-27) Both position and momentum are observables, meaning that they are represented by [Hermitian operators](https:\/\/en.wikipedia.org\/wiki\/Hermitian_operators \"Hermitian operators\"). The position operator X ^ {\\\\displaystyle {\\\\hat {X}}} ![{\\\\displaystyle {\\\\hat {X}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/acc59ad6d9a06d55b96b65beb0fdfc89acc1e40e) and momentum operator P ^ {\\\\displaystyle {\\\\hat {P}}} ![{\\\\displaystyle {\\\\hat {P}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/7a46a8cf7bc789e8c4f8de7ca0d9946a46fb8842) do not commute, but rather satisfy the [canonical commutation relation](https:\/\/en.wikipedia.org\/wiki\/Canonical_commutation_relation \"Canonical commutation relation\"): \\[ X ^ , P ^ \\] \\= i ℏ . {\\\\displaystyle \\[{\\\\hat {X}},{\\\\hat {P}}\\]=i\\\\hbar .} ![{\\\\displaystyle \\[{\\\\hat {X}},{\\\\hat {P}}\\]=i\\\\hbar .}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/803fe39b0eeaff8d1570df480e738cf5a968cc71) Given a quantum state, the Born rule lets us compute expectation values for both X {\\\\displaystyle X} ![{\\\\displaystyle X}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/68baa052181f707c662844a465bfeeb135e82bab) and P {\\\\displaystyle P} ![{\\\\displaystyle P}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/b4dc73bf40314945ff376bd363916a738548d40a), and moreover for powers of them. Defining the uncertainty for an observable by a [standard deviation](https:\/\/en.wikipedia.org\/wiki\/Standard_deviation \"Standard deviation\"), we have σ X \\= ⟨ X 2 ⟩ − ⟨ X ⟩ 2 , {\\\\displaystyle \\\\sigma \\_{X}={\\\\textstyle {\\\\sqrt {\\\\left\\\\langle X^{2}\\\\right\\\\rangle -\\\\left\\\\langle X\\\\right\\\\rangle ^{2}}}},} ![{\\\\displaystyle \\\\sigma \\_{X}={\\\\textstyle {\\\\sqrt {\\\\left\\\\langle X^{2}\\\\right\\\\rangle -\\\\left\\\\langle X\\\\right\\\\rangle ^{2}}}},}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/457ec20972d63dfb1ecc9087e18d1f949f908c8f) and likewise for the momentum: σ P \\= ⟨ P 2 ⟩ − ⟨ P ⟩ 2 . {\\\\displaystyle \\\\sigma \\_{P}={\\\\sqrt {\\\\left\\\\langle P^{2}\\\\right\\\\rangle -\\\\left\\\\langle P\\\\right\\\\rangle ^{2}}}.} ![{\\\\displaystyle \\\\sigma \\_{P}={\\\\sqrt {\\\\left\\\\langle P^{2}\\\\right\\\\rangle -\\\\left\\\\langle P\\\\right\\\\rangle ^{2}}}.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/63ec8f5f7b9e5957ea6d06c56068b06244acc184) The uncertainty principle states that σ X σ P ≥ ℏ 2 . {\\\\displaystyle \\\\sigma \\_{X}\\\\sigma \\_{P}\\\\geq {\\\\frac {\\\\hbar }{2}}.} ![{\\\\displaystyle \\\\sigma \\_{X}\\\\sigma \\_{P}\\\\geq {\\\\frac {\\\\hbar }{2}}.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/538cada7fa57155ece387e5d53d90ca366e323fe) Either standard deviation can in principle be made arbitrarily small, but not both simultaneously.[\\[28\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-ballentine1970-28) This inequality generalizes to arbitrary pairs of self-adjoint operators A {\\\\displaystyle A} ![{\\\\displaystyle A}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/7daff47fa58cdfd29dc333def748ff5fa4c923e3) and B {\\\\displaystyle B} ![{\\\\displaystyle B}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/47136aad860d145f75f3eed3022df827cee94d7a). The [commutator](https:\/\/en.wikipedia.org\/wiki\/Commutator \"Commutator\") of these two operators is \\[ A , B \\] \\= A B − B A , {\\\\displaystyle \\[A,B\\]=AB-BA,} ![{\\\\displaystyle \\[A,B\\]=AB-BA,}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/b2a47259b42e63c048c65f67d304404867841951) and this provides the lower bound on the product of standard deviations: σ A σ B ≥ 1 2 \\| ⟨ \\[ A , B \\] ⟩ \\| . {\\\\displaystyle \\\\sigma \\_{A}\\\\sigma \\_{B}\\\\geq {\\\\tfrac {1}{2}}\\\\left\\|{\\\\bigl \\\\langle }\\[A,B\\]{\\\\bigr \\\\rangle }\\\\right\\|.} ![{\\\\displaystyle \\\\sigma \\_{A}\\\\sigma \\_{B}\\\\geq {\\\\tfrac {1}{2}}\\\\left\\|{\\\\bigl \\\\langle }\\[A,B\\]{\\\\bigr \\\\rangle }\\\\right\\|.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/d0fd768b447334e150b8b98181f74b475e41ee52)\n\nAnother consequence of the canonical commutation relation is that the position and momentum operators are [Fourier transforms](https:\/\/en.wikipedia.org\/wiki\/Fourier_transform#Uncertainty_principle \"Fourier transform\") of each other, so that a description of an object according to its momentum is the Fourier transform of its description according to its position. The fact that dependence in momentum is the Fourier transform of the dependence in position means that the momentum operator is equivalent (up to an i \/ ℏ {\\\\displaystyle i\/\\\\hbar } ![{\\\\displaystyle i\/\\\\hbar }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/44a04e1f19b5e7bea2bfa8002a841bf8d1b4e66a) factor) to taking the derivative according to the position, since in Fourier analysis [differentiation corresponds to multiplication in the dual space](https:\/\/en.wikipedia.org\/wiki\/Fourier_transform#Differentiation \"Fourier transform\"). This is why in quantum equations in position space, the momentum p i {\\\\displaystyle p\\_{i}} ![{\\\\displaystyle p\\_{i}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/5bab39399bf5424f25d957cdc57c84a0622626d2) is replaced by − i ℏ ∂ ∂ x {\\\\displaystyle -i\\\\hbar {\\\\frac {\\\\partial }{\\\\partial x}}} ![{\\\\displaystyle -i\\\\hbar {\\\\frac {\\\\partial }{\\\\partial x}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/ab7fffcee704fc55eb36b137e0cc25132b5dc1bf), and in particular in the [non-relativistic Schrödinger equation in position space](https:\/\/en.wikipedia.org\/wiki\/Schr%C3%B6dinger_equation#Equation \"Schrödinger equation\") the momentum-squared term is replaced with a Laplacian times − ℏ 2 {\\\\displaystyle -\\\\hbar ^{2}} ![{\\\\displaystyle -\\\\hbar ^{2}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/185934c9e69984a0905432abe63d3b41741342b4).[\\[26\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Cohen-Tannoudji-26)\n\n### Composite systems and entanglement\nWhen two different quantum systems are considered together, the Hilbert space of the combined system is the [tensor product](https:\/\/en.wikipedia.org\/wiki\/Tensor_product \"Tensor product\") of the Hilbert spaces of the two components. For example, let A and B be two quantum systems, with Hilbert spaces H A {\\\\displaystyle {\\\\mathcal {H}}\\_{A}} ![{\\\\displaystyle {\\\\mathcal {H}}\\_{A}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/4611f8542d6c72789b7e82bfcf29014d44c13aa3) and H B {\\\\displaystyle {\\\\mathcal {H}}\\_{B}} ![{\\\\displaystyle {\\\\mathcal {H}}\\_{B}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/bfea1102a35aff8c6435147dc9dffc21b4cfe4ef), respectively. The Hilbert space of the composite system is then H A B \\= H A ⊗ H B . {\\\\displaystyle {\\\\mathcal {H}}\\_{AB}={\\\\mathcal {H}}\\_{A}\\\\otimes {\\\\mathcal {H}}\\_{B}.} ![{\\\\displaystyle {\\\\mathcal {H}}\\_{AB}={\\\\mathcal {H}}\\_{A}\\\\otimes {\\\\mathcal {H}}\\_{B}.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/54e4f6aff2be5ec9f3eb4657a133d1be17f6e2b3) If the state for the first system is the vector ψ A {\\\\displaystyle \\\\psi \\_{A}} ![{\\\\displaystyle \\\\psi \\_{A}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/a04e49109107e7765357dd92e975895840fa7700) and the state for the second system is ψ B {\\\\displaystyle \\\\psi \\_{B}} ![{\\\\displaystyle \\\\psi \\_{B}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/7ca3b90bba42c451129c4969322e175f5a1d3368), then the state of the composite system is ψ A ⊗ ψ B . {\\\\displaystyle \\\\psi \\_{A}\\\\otimes \\\\psi \\_{B}.} ![{\\\\displaystyle \\\\psi \\_{A}\\\\otimes \\\\psi \\_{B}.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/852c1e09787e91af3af1a10d1eee4eadc232d48c) Not all states in the joint Hilbert space H A B {\\\\displaystyle {\\\\mathcal {H}}\\_{AB}} ![{\\\\displaystyle {\\\\mathcal {H}}\\_{AB}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/951b54ea87d5bd62464abaf1be530e6a37a86f1b) can be written in this form, however, because the superposition principle implies that linear combinations of these \"separable\" or \"product states\" are also valid. For example, if ψ A {\\\\displaystyle \\\\psi \\_{A}} ![{\\\\displaystyle \\\\psi \\_{A}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/a04e49109107e7765357dd92e975895840fa7700) and ϕ A {\\\\displaystyle \\\\phi \\_{A}} ![{\\\\displaystyle \\\\phi \\_{A}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/5e38d84dcc21ea769a51ab502691f566c830b430) are both possible states for system A {\\\\displaystyle A} ![{\\\\displaystyle A}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/7daff47fa58cdfd29dc333def748ff5fa4c923e3), and likewise ψ B {\\\\displaystyle \\\\psi \\_{B}} ![{\\\\displaystyle \\\\psi \\_{B}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/7ca3b90bba42c451129c4969322e175f5a1d3368) and ϕ B {\\\\displaystyle \\\\phi \\_{B}} ![{\\\\displaystyle \\\\phi \\_{B}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/a829a7a85bb29904dec5e93605d0b976baf45755) are both possible states for system B {\\\\displaystyle B} ![{\\\\displaystyle B}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/47136aad860d145f75f3eed3022df827cee94d7a), then 1 2 ( ψ A ⊗ ψ B \\+ ϕ A ⊗ ϕ B ) {\\\\displaystyle {\\\\tfrac {1}{\\\\sqrt {2}}}\\\\left(\\\\psi \\_{A}\\\\otimes \\\\psi \\_{B}+\\\\phi \\_{A}\\\\otimes \\\\phi \\_{B}\\\\right)} ![{\\\\displaystyle {\\\\tfrac {1}{\\\\sqrt {2}}}\\\\left(\\\\psi \\_{A}\\\\otimes \\\\psi \\_{B}+\\\\phi \\_{A}\\\\otimes \\\\phi \\_{B}\\\\right)}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/53d226ba585a99942ec855197c23ed885b635258) is a valid joint state that is not separable. States that are not separable are called [entangled](https:\/\/en.wikipedia.org\/wiki\/Quantum_entanglement \"Quantum entanglement\").[\\[29\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-:0-29)[\\[30\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-:1-30)\n\nIf the state for a composite system is entangled, it is impossible to describe either component system A or system B by a state vector. One can instead define [reduced density matrices](https:\/\/en.wikipedia.org\/wiki\/Reduced_density_matrix \"Reduced density matrix\") that describe the statistics that can be obtained by making measurements on either component system alone. This necessarily causes a loss of information, though: knowing the reduced density matrices of the individual systems is not enough to reconstruct the state of the composite system.[\\[29\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-:0-29)[\\[30\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-:1-30) Just as density matrices specify the state of a subsystem of a larger system, analogously, [positive operator-valued measures](https:\/\/en.wikipedia.org\/wiki\/POVM \"POVM\") (POVMs) describe the effect on a subsystem of a measurement performed on a larger system. POVMs are extensively used in quantum information theory.[\\[29\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-:0-29)[\\[31\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-wilde-31)\n\nAs described above, entanglement is a key feature of models of measurement processes in which an apparatus becomes entangled with the system being measured. Systems interacting with the environment in which they reside generally become entangled with that environment, a phenomenon known as [quantum decoherence](https:\/\/en.wikipedia.org\/wiki\/Quantum_decoherence \"Quantum decoherence\"). This can explain why, in practice, quantum effects are difficult to observe in systems larger than microscopic.[\\[32\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-32)\n\n### Equivalence between formulations\nThere are many mathematically equivalent formulations of quantum mechanics. One of the oldest and most common is the \"[transformation theory](https:\/\/en.wikipedia.org\/wiki\/Transformation_theory_\\(quantum_mechanics\\) \"Transformation theory (quantum mechanics)\")\" proposed by [Paul Dirac](https:\/\/en.wikipedia.org\/wiki\/Paul_Dirac \"Paul Dirac\"), which unifies and generalizes the two earliest formulations of quantum mechanics – [matrix mechanics](https:\/\/en.wikipedia.org\/wiki\/Matrix_mechanics \"Matrix mechanics\") (invented by [Werner Heisenberg](https:\/\/en.wikipedia.org\/wiki\/Werner_Heisenberg \"Werner Heisenberg\")) and wave mechanics (invented by [Erwin Schrödinger](https:\/\/en.wikipedia.org\/wiki\/Erwin_Schr%C3%B6dinger \"Erwin Schrödinger\")).[\\[33\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-33) An alternative formulation of quantum mechanics is [Feynman](https:\/\/en.wikipedia.org\/wiki\/Feynman \"Feynman\")'s [path integral formulation](https:\/\/en.wikipedia.org\/wiki\/Path_integral_formulation \"Path integral formulation\"), in which a quantum-mechanical amplitude is considered as a sum over all possible classical and non-classical paths between the initial and final states. This is the quantum-mechanical counterpart of the [action principle](https:\/\/en.wikipedia.org\/wiki\/Action_principle \"Action principle\") in classical mechanics.[\\[34\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-34)\n\n### Symmetries and conservation laws\nMain article: [Noether's theorem](https:\/\/en.wikipedia.org\/wiki\/Noether%27s_theorem \"Noether's theorem\")\n\nThe Hamiltonian H {\\\\displaystyle H} ![{\\\\displaystyle H}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/75a9edddcca2f782014371f75dca39d7e13a9c1b) is known as the *generator* of time evolution, since it defines a unitary time-evolution operator U ( t ) \\= e − i H t \/ ℏ {\\\\displaystyle U(t)=e^{-iHt\/\\\\hbar }} ![{\\\\displaystyle U(t)=e^{-iHt\/\\\\hbar }}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/17a17e78953fa7d61788cc8bc41a25ef254fc8e0) for each value of t {\\\\displaystyle t} ![{\\\\displaystyle t}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/65658b7b223af9e1acc877d848888ecdb4466560). From this relation between U ( t ) {\\\\displaystyle U(t)} ![{\\\\displaystyle U(t)}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/666c639df532e88616357c4991cabce9a57b5611) and H {\\\\displaystyle H} ![{\\\\displaystyle H}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/75a9edddcca2f782014371f75dca39d7e13a9c1b), it follows that any observable A {\\\\displaystyle A} ![{\\\\displaystyle A}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/7daff47fa58cdfd29dc333def748ff5fa4c923e3) that commutes with H {\\\\displaystyle H} ![{\\\\displaystyle H}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/75a9edddcca2f782014371f75dca39d7e13a9c1b) will be *conserved*: its expectation value will not change over time.[\\[7\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Zwiebach2022-7): 471  This statement generalizes, as mathematically, any Hermitian operator A {\\\\displaystyle A} ![{\\\\displaystyle A}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/7daff47fa58cdfd29dc333def748ff5fa4c923e3) can generate a family of unitary operators parameterized by a variable t {\\\\displaystyle t} ![{\\\\displaystyle t}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/65658b7b223af9e1acc877d848888ecdb4466560). Under the evolution generated by A {\\\\displaystyle A} ![{\\\\displaystyle A}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/7daff47fa58cdfd29dc333def748ff5fa4c923e3), any observable B {\\\\displaystyle B} ![{\\\\displaystyle B}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/47136aad860d145f75f3eed3022df827cee94d7a) that commutes with A {\\\\displaystyle A} ![{\\\\displaystyle A}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/7daff47fa58cdfd29dc333def748ff5fa4c923e3) will be conserved. Moreover, if B {\\\\displaystyle B} ![{\\\\displaystyle B}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/47136aad860d145f75f3eed3022df827cee94d7a) is conserved by evolution under A {\\\\displaystyle A} ![{\\\\displaystyle A}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/7daff47fa58cdfd29dc333def748ff5fa4c923e3), then A {\\\\displaystyle A} ![{\\\\displaystyle A}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/7daff47fa58cdfd29dc333def748ff5fa4c923e3) is conserved under the evolution generated by B {\\\\displaystyle B} ![{\\\\displaystyle B}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/47136aad860d145f75f3eed3022df827cee94d7a). This implies a quantum version of the result proven by [Emmy Noether](https:\/\/en.wikipedia.org\/wiki\/Emmy_Noether \"Emmy Noether\") in classical ([Lagrangian](https:\/\/en.wikipedia.org\/wiki\/Lagrangian_mechanics \"Lagrangian mechanics\")) mechanics: for every [differentiable](https:\/\/en.wikipedia.org\/wiki\/Differentiable \"Differentiable\") [symmetry](https:\/\/en.wikipedia.org\/wiki\/Symmetry_\\(physics\\) \"Symmetry (physics)\") of a Hamiltonian, there exists a corresponding [conservation law](https:\/\/en.wikipedia.org\/wiki\/Conservation_law \"Conservation law\").\n\n## Examples\n### Free particle\nMain article: [Free particle](https:\/\/en.wikipedia.org\/wiki\/Free_particle \"Free particle\")\n\n[![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/5\/56\/Guassian_Dispersion.gif)](https:\/\/en.wikipedia.org\/wiki\/File:Guassian_Dispersion.gif)\n\nPosition space probability density of a Gaussian [wave packet](https:\/\/en.wikipedia.org\/wiki\/Wave_packet \"Wave packet\") moving in one dimension in free space\n\nThe simplest example of a quantum system with a position degree of freedom is a free particle in a single spatial dimension. A free particle is one which is not subject to external influences, so that its Hamiltonian consists only of its kinetic energy: H \\= 1 2 m P 2 \\= − ℏ 2 2 m d 2 d x 2 . {\\\\displaystyle H={\\\\frac {1}{2m}}P^{2}=-{\\\\frac {\\\\hbar ^{2}}{2m}}{\\\\frac {d^{2}}{dx^{2}}}.} ![{\\\\displaystyle H={\\\\frac {1}{2m}}P^{2}=-{\\\\frac {\\\\hbar ^{2}}{2m}}{\\\\frac {d^{2}}{dx^{2}}}.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/084f585ee5c6cd8a34e323a6de7943227128afc3) The general solution of the Schrödinger equation is given by ψ ( x , t ) \\= 1 2 π ∫ − ∞ ∞ ψ ^ ( k , 0 ) e i ( k x − ℏ k 2 2 m t ) d k , {\\\\displaystyle \\\\psi (x,t)={\\\\frac {1}{\\\\sqrt {2\\\\pi }}}\\\\int \\_{-\\\\infty }^{\\\\infty }{\\\\hat {\\\\psi }}(k,0)e^{i(kx-{\\\\frac {\\\\hbar k^{2}}{2m}}t)}\\\\mathrm {d} k,} ![{\\\\displaystyle \\\\psi (x,t)={\\\\frac {1}{\\\\sqrt {2\\\\pi }}}\\\\int \\_{-\\\\infty }^{\\\\infty }{\\\\hat {\\\\psi }}(k,0)e^{i(kx-{\\\\frac {\\\\hbar k^{2}}{2m}}t)}\\\\mathrm {d} k,}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/ef4f021ba945856e3676808b11724109a8a74dad) which is a superposition of all possible [plane waves](https:\/\/en.wikipedia.org\/wiki\/Plane_wave \"Plane wave\") e i ( k x − ℏ k 2 2 m t ) {\\\\displaystyle e^{i(kx-{\\\\frac {\\\\hbar k^{2}}{2m}}t)}} ![{\\\\displaystyle e^{i(kx-{\\\\frac {\\\\hbar k^{2}}{2m}}t)}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/fb4cd9a9984c84a493ce547babcf58e31b04f7e7), which are eigenstates of the momentum operator with momentum p \\= ℏ k {\\\\displaystyle p=\\\\hbar k} ![{\\\\displaystyle p=\\\\hbar k}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/24fee69175538303b28ac54e907baf53d0a58dbf). The coefficients of the superposition are ψ ^ ( k , 0 ) {\\\\displaystyle {\\\\hat {\\\\psi }}(k,0)} ![{\\\\displaystyle {\\\\hat {\\\\psi }}(k,0)}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/8b8323c08418da8bc376c6d78b578d4729b927ea), which is the Fourier transform of the initial quantum state ψ ( x , 0 ) {\\\\displaystyle \\\\psi (x,0)} ![{\\\\displaystyle \\\\psi (x,0)}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/55ad442e07ca2d7986ef0787f9129fc325cde137).\n\nIt is not possible for the solution to be a single momentum eigenstate, or a single position eigenstate, as these are not normalizable quantum states.[\\[note 1\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-35) Instead, we can consider a Gaussian [wave packet](https:\/\/en.wikipedia.org\/wiki\/Wave_packet \"Wave packet\"): ψ ( x , 0 ) \\= 1 π a 4 e − x 2 2 a {\\\\displaystyle \\\\psi (x,0)={\\\\frac {1}{\\\\sqrt\\[{4}\\]{\\\\pi a}}}e^{-{\\\\frac {x^{2}}{2a}}}} ![{\\\\displaystyle \\\\psi (x,0)={\\\\frac {1}{\\\\sqrt\\[{4}\\]{\\\\pi a}}}e^{-{\\\\frac {x^{2}}{2a}}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/a4c2dae82312897d5fd4c58986c426a6009e6840) which has Fourier transform, and therefore momentum distribution ψ ^ ( k , 0 ) \\= a π 4 e − a k 2 2 . {\\\\displaystyle {\\\\hat {\\\\psi }}(k,0)={\\\\sqrt\\[{4}\\]{\\\\frac {a}{\\\\pi }}}e^{-{\\\\frac {ak^{2}}{2}}}.} ![{\\\\displaystyle {\\\\hat {\\\\psi }}(k,0)={\\\\sqrt\\[{4}\\]{\\\\frac {a}{\\\\pi }}}e^{-{\\\\frac {ak^{2}}{2}}}.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/c4991535bba434314af8c27c16fff74f49ce367e) We see that as we make a {\\\\displaystyle a} ![{\\\\displaystyle a}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/ffd2487510aa438433a2579450ab2b3d557e5edc) smaller the spread in position gets smaller, but the spread in momentum gets larger. Conversely, by making a {\\\\displaystyle a} ![{\\\\displaystyle a}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/ffd2487510aa438433a2579450ab2b3d557e5edc) larger we make the spread in momentum smaller, but the spread in position gets larger. This illustrates the uncertainty principle.\n\nAs we let the Gaussian wave packet evolve in time, we see that its center moves through space at a constant velocity (like a classical particle with no forces acting on it). However, the wave packet will also spread out as time progresses, which means that the position becomes more and more uncertain. The uncertainty in momentum, however, stays constant.[\\[35\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-36)\n\n### Particle in a box\n[![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/2\/27\/Infinite_potential_well.svg\/250px-Infinite_potential_well.svg.png)](https:\/\/en.wikipedia.org\/wiki\/File:Infinite_potential_well.svg)\n\n1-dimensional potential energy box (or infinite potential well)\n\nMain article: [Particle in a box](https:\/\/en.wikipedia.org\/wiki\/Particle_in_a_box \"Particle in a box\")\n\nThe particle in a one-dimensional potential energy box is the most mathematically simple example where restraints lead to the quantization of energy levels. The box is defined as having zero potential energy everywhere *inside* a certain region, and therefore infinite potential energy everywhere *outside* that region.[\\[26\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Cohen-Tannoudji-26): 77–78  For the one-dimensional case in the x {\\\\displaystyle x} ![{\\\\displaystyle x}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/87f9e315fd7e2ba406057a97300593c4802b53e4) direction, the time-independent Schrödinger equation may be written − ℏ 2 2 m d 2 ψ d x 2 \\= E ψ . {\\\\displaystyle -{\\\\frac {\\\\hbar ^{2}}{2m}}{\\\\frac {d^{2}\\\\psi }{dx^{2}}}=E\\\\psi .} ![{\\\\displaystyle -{\\\\frac {\\\\hbar ^{2}}{2m}}{\\\\frac {d^{2}\\\\psi }{dx^{2}}}=E\\\\psi .}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/cc73e9216faf1390c3ed550b72be21fc068ec747)\n\nWith the differential operator defined by p ^ x \\= − i ℏ d d x {\\\\displaystyle {\\\\hat {p}}\\_{x}=-i\\\\hbar {\\\\frac {d}{dx}}} ![{\\\\displaystyle {\\\\hat {p}}\\_{x}=-i\\\\hbar {\\\\frac {d}{dx}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/2511885975007002c7582a9c8175689076df210a)the previous equation is evocative of the [classic kinetic energy analogue](https:\/\/en.wikipedia.org\/wiki\/Kinetic_energy#Kinetic_energy_of_rigid_bodies \"Kinetic energy\"), 1 2 m p ^ x 2 \\= E , {\\\\displaystyle {\\\\frac {1}{2m}}{\\\\hat {p}}\\_{x}^{2}=E,} ![{\\\\displaystyle {\\\\frac {1}{2m}}{\\\\hat {p}}\\_{x}^{2}=E,}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/422f57813ede91b8e348e786616c6297742b4fd7) with state ψ {\\\\displaystyle \\\\psi } ![{\\\\displaystyle \\\\psi }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a) in this case having energy E {\\\\displaystyle E} ![{\\\\displaystyle E}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b) coincident with the kinetic energy of the particle.\n\nThe general solutions of the Schrödinger equation for the particle in a box are ψ ( x ) \\= A e i k x \\+ B e − i k x E \\= ℏ 2 k 2 2 m {\\\\displaystyle \\\\psi (x)=Ae^{ikx}+Be^{-ikx}\\\\qquad \\\\qquad E={\\\\frac {\\\\hbar ^{2}k^{2}}{2m}}} ![{\\\\displaystyle \\\\psi (x)=Ae^{ikx}+Be^{-ikx}\\\\qquad \\\\qquad E={\\\\frac {\\\\hbar ^{2}k^{2}}{2m}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/b5fb1b2f1d5afb42edb4eb98bf89791d283c1e53) or, from [Euler's formula](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_formula \"Euler's formula\"), ψ ( x ) \\= C sin ⁡ ( k x ) \\+ D cos ⁡ ( k x ) . {\\\\displaystyle \\\\psi (x)=C\\\\sin(kx)+D\\\\cos(kx).\\\\!} ![{\\\\displaystyle \\\\psi (x)=C\\\\sin(kx)+D\\\\cos(kx).\\\\!}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/229ebbbe587e0c678318cb85089a74ed289c971a)\n\nThe infinite potential walls of the box determine the values of C , D , {\\\\displaystyle C,D,} ![{\\\\displaystyle C,D,}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/6e153d2c75e3492a32d4fafefec88846862c3b9b) and k {\\\\displaystyle k} ![{\\\\displaystyle k}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/c3c9a2c7b599b37105512c5d570edc034056dd40) at x \\= 0 {\\\\displaystyle x=0} ![{\\\\displaystyle x=0}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/953917eaf52f2e1baad54c8c9e3d6f9bb3710cdc) and x \\= L {\\\\displaystyle x=L} ![{\\\\displaystyle x=L}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/e5fe40c588800aaab69041986b49a59664cd767a) where ψ {\\\\displaystyle \\\\psi } ![{\\\\displaystyle \\\\psi }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a) must be zero. Thus, at x \\= 0 {\\\\displaystyle x=0} ![{\\\\displaystyle x=0}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/953917eaf52f2e1baad54c8c9e3d6f9bb3710cdc), ψ ( 0 ) \\= 0 \\= C sin ⁡ ( 0 ) \\+ D cos ⁡ ( 0 ) \\= D {\\\\displaystyle \\\\psi (0)=0=C\\\\sin(0)+D\\\\cos(0)=D} ![{\\\\displaystyle \\\\psi (0)=0=C\\\\sin(0)+D\\\\cos(0)=D}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/34188e64cbf80486b3ad311b53b512fa36de59c6) and D \\= 0 {\\\\displaystyle D=0} ![{\\\\displaystyle D=0}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/d375dfda80ee8df1d1d7aa8b962114044e464305). At x \\= L {\\\\displaystyle x=L} ![{\\\\displaystyle x=L}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/e5fe40c588800aaab69041986b49a59664cd767a), ψ ( L ) \\= 0 \\= C sin ⁡ ( k L ) , {\\\\displaystyle \\\\psi (L)=0=C\\\\sin(kL),} ![{\\\\displaystyle \\\\psi (L)=0=C\\\\sin(kL),}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/c7a5bc9ef330efb5b6eba5d89c2ee026bbeea6a8) in which C {\\\\displaystyle C} ![{\\\\displaystyle C}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/4fc55753007cd3c18576f7933f6f089196732029) cannot be zero as this would conflict with the postulate that ψ {\\\\displaystyle \\\\psi } ![{\\\\displaystyle \\\\psi }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a) has norm 1. Therefore, since sin ⁡ ( k L ) \\= 0 {\\\\displaystyle \\\\sin(kL)=0} ![{\\\\displaystyle \\\\sin(kL)=0}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/e5177ccdb2057c5c1be728af20b8ef3d61f79999), k L {\\\\displaystyle kL} ![{\\\\displaystyle kL}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/d8865c1184b2c1dff6226dae50d3be91f4f01cfe) must be an integer multiple of π {\\\\displaystyle \\\\pi } ![{\\\\displaystyle \\\\pi }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/9be4ba0bb8df3af72e90a0535fabcc17431e540a), k \\= n π L n \\= 1 , 2 , 3 , … . {\\\\displaystyle k={\\\\frac {n\\\\pi }{L}}\\\\qquad \\\\qquad n=1,2,3,\\\\ldots .} ![{\\\\displaystyle k={\\\\frac {n\\\\pi }{L}}\\\\qquad \\\\qquad n=1,2,3,\\\\ldots .}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/3fd91af3024e1f59cf57e04884494fd2c55664f8)\n\nThis constraint on k {\\\\displaystyle k} ![{\\\\displaystyle k}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/c3c9a2c7b599b37105512c5d570edc034056dd40) implies a constraint on the energy levels, yielding E n \\= ℏ 2 π 2 n 2 2 m L 2 \\= n 2 h 2 8 m L 2 . {\\\\displaystyle E\\_{n}={\\\\frac {\\\\hbar ^{2}\\\\pi ^{2}n^{2}}{2mL^{2}}}={\\\\frac {n^{2}h^{2}}{8mL^{2}}}.} ![{\\\\displaystyle E\\_{n}={\\\\frac {\\\\hbar ^{2}\\\\pi ^{2}n^{2}}{2mL^{2}}}={\\\\frac {n^{2}h^{2}}{8mL^{2}}}.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/fe507fd1f92ebfe133aeb4f1da46ea27f569f38b)\n\nA [finite potential well](https:\/\/en.wikipedia.org\/wiki\/Finite_potential_well \"Finite potential well\") is the generalization of the infinite potential well problem to potential wells having finite depth. The finite potential well problem is mathematically more complicated than the infinite particle-in-a-box problem as the wave function is not pinned to zero at the walls of the well. Instead, the wave function must satisfy more complicated mathematical boundary conditions as it is nonzero in regions outside the well. Another related problem is that of the [rectangular potential barrier](https:\/\/en.wikipedia.org\/wiki\/Rectangular_potential_barrier \"Rectangular potential barrier\"), which furnishes a model for the [quantum tunneling](https:\/\/en.wikipedia.org\/wiki\/Quantum_tunneling \"Quantum tunneling\") effect that plays an important role in the performance of modern technologies such as [flash memory](https:\/\/en.wikipedia.org\/wiki\/Flash_memory \"Flash memory\") and [scanning tunneling microscopy](https:\/\/en.wikipedia.org\/wiki\/Scanning_tunneling_microscopy \"Scanning tunneling microscopy\").\n\n### Harmonic oscillator\nMain article: [Quantum harmonic oscillator](https:\/\/en.wikipedia.org\/wiki\/Quantum_harmonic_oscillator \"Quantum harmonic oscillator\")\n\n[![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/9\/90\/QuantumHarmonicOscillatorAnimation.gif)](https:\/\/en.wikipedia.org\/wiki\/File:QuantumHarmonicOscillatorAnimation.gif)\n\nSome trajectories of a [harmonic oscillator](https:\/\/en.wikipedia.org\/wiki\/Harmonic_oscillator \"Harmonic oscillator\") (i.e. a ball attached to a [spring](https:\/\/en.wikipedia.org\/wiki\/Hooke%27s_law \"Hooke's law\")) in [classical mechanics](https:\/\/en.wikipedia.org\/wiki\/Classical_mechanics \"Classical mechanics\") (A-B) and quantum mechanics (C-H). In quantum mechanics, the position of the ball is represented by a [wave](https:\/\/en.wikipedia.org\/wiki\/Wave \"Wave\") (called the wave function), with the [real part](https:\/\/en.wikipedia.org\/wiki\/Real_part \"Real part\") shown in blue and the [imaginary part](https:\/\/en.wikipedia.org\/wiki\/Imaginary_part \"Imaginary part\") shown in red. Some of the trajectories (such as C, D, E, and F) are [standing waves](https:\/\/en.wikipedia.org\/wiki\/Standing_wave \"Standing wave\") (or \"[stationary states](https:\/\/en.wikipedia.org\/wiki\/Stationary_state \"Stationary state\")\"). Each standing-wave frequency is proportional to a possible [energy level](https:\/\/en.wikipedia.org\/wiki\/Energy_level \"Energy level\") of the oscillator. This \"energy quantization\" does not occur in classical physics, where the oscillator can have *any* energy.\n\nAs in the classical case, the potential for the quantum harmonic oscillator is given by[\\[7\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Zwiebach2022-7): 234  V ( x ) \\= 1 2 m ω 2 x 2 . {\\\\displaystyle V(x)={\\\\frac {1}{2}}m\\\\omega ^{2}x^{2}.} ![{\\\\displaystyle V(x)={\\\\frac {1}{2}}m\\\\omega ^{2}x^{2}.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/ac9d8735e10198c090b576c765dc2778fe458b9c)\n\nThis problem can either be treated by directly solving the Schrödinger equation, which is not trivial, or by using the more elegant \"ladder method\" first proposed by Paul Dirac. The [eigenstates](https:\/\/en.wikipedia.org\/wiki\/Eigenstate \"Eigenstate\") are given by ψ n ( x ) \\= 1 2 n n \\! ⋅ ( m ω π ℏ ) 1 \/ 4 ⋅ e − m ω x 2 2 ℏ ⋅ H n ( m ω ℏ x ) , {\\\\displaystyle \\\\psi \\_{n}(x)={\\\\sqrt {\\\\frac {1}{2^{n}\\\\,n!}}}\\\\cdot \\\\left({\\\\frac {m\\\\omega }{\\\\pi \\\\hbar }}\\\\right)^{1\/4}\\\\cdot e^{-{\\\\frac {m\\\\omega x^{2}}{2\\\\hbar }}}\\\\cdot H\\_{n}\\\\left({\\\\sqrt {\\\\frac {m\\\\omega }{\\\\hbar }}}x\\\\right),\\\\qquad } ![{\\\\displaystyle \\\\psi \\_{n}(x)={\\\\sqrt {\\\\frac {1}{2^{n}\\\\,n!}}}\\\\cdot \\\\left({\\\\frac {m\\\\omega }{\\\\pi \\\\hbar }}\\\\right)^{1\/4}\\\\cdot e^{-{\\\\frac {m\\\\omega x^{2}}{2\\\\hbar }}}\\\\cdot H\\_{n}\\\\left({\\\\sqrt {\\\\frac {m\\\\omega }{\\\\hbar }}}x\\\\right),\\\\qquad }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/8a0031b1b0c9fa4a9e4e6957718d509846522c4d) n \\= 0 , 1 , 2 , … . {\\\\displaystyle n=0,1,2,\\\\ldots .} ![{\\\\displaystyle n=0,1,2,\\\\ldots .}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/d652f9756f144ea772e4e632cb1398652d36372c) where *Hn* are the [Hermite polynomials](https:\/\/en.wikipedia.org\/wiki\/Hermite_polynomials \"Hermite polynomials\") H n ( x ) \\= ( − 1 ) n e x 2 d n d x n ( e − x 2 ) , {\\\\displaystyle H\\_{n}(x)=(-1)^{n}e^{x^{2}}{\\\\frac {d^{n}}{dx^{n}}}\\\\left(e^{-x^{2}}\\\\right),} ![{\\\\displaystyle H\\_{n}(x)=(-1)^{n}e^{x^{2}}{\\\\frac {d^{n}}{dx^{n}}}\\\\left(e^{-x^{2}}\\\\right),}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/0da8b4a24291ce8b32bcc0665f490f3c76afc7d3) and the corresponding energy levels are E n \\= ℏ ω ( n \\+ 1 2 ) . {\\\\displaystyle E\\_{n}=\\\\hbar \\\\omega \\\\left(n+{1 \\\\over 2}\\\\right).} ![{\\\\displaystyle E\\_{n}=\\\\hbar \\\\omega \\\\left(n+{1 \\\\over 2}\\\\right).}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/635fb242f4c34cd5ba947f4c8132ddd47f2872c0)\n\nThis is another example illustrating the discretization of energy for [bound states](https:\/\/en.wikipedia.org\/wiki\/Bound_state \"Bound state\").\n\n### Mach–Zehnder interferometer\n[![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/c\/c9\/Mach-Zehnder_interferometer.svg\/330px-Mach-Zehnder_interferometer.svg.png)](https:\/\/en.wikipedia.org\/wiki\/File:Mach-Zehnder_interferometer.svg)\n\nSchematic of a Mach–Zehnder interferometer\n\nThe [Mach–Zehnder interferometer](https:\/\/en.wikipedia.org\/wiki\/Mach%E2%80%93Zehnder_interferometer \"Mach–Zehnder interferometer\") (MZI) illustrates the concepts of superposition and interference with linear algebra in dimension 2, rather than differential equations. It can be seen as a simplified version of the double-slit experiment, but it is of interest in its own right, for example in the [delayed choice quantum eraser](https:\/\/en.wikipedia.org\/wiki\/Delayed_choice_quantum_eraser \"Delayed choice quantum eraser\"), the [Elitzur–Vaidman bomb tester](https:\/\/en.wikipedia.org\/wiki\/Elitzur%E2%80%93Vaidman_bomb_tester \"Elitzur–Vaidman bomb tester\"), and in studies of quantum entanglement.[\\[36\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Paris1999-37)[\\[37\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Haack2010-38)\n\nWe can model a photon going through the interferometer by considering that at each point it can be in a superposition of only two paths: the \"lower\" path which starts from the left, goes straight through both beam splitters, and ends at the top, and the \"upper\" path which starts from the bottom, goes straight through both beam splitters, and ends at the right. The quantum state of the photon is therefore a vector ψ ∈ C 2 {\\\\displaystyle \\\\psi \\\\in \\\\mathbb {C} ^{2}} ![{\\\\displaystyle \\\\psi \\\\in \\\\mathbb {C} ^{2}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/a7a51a8280039fc22cd88d90915a952f8e020f47) that is a superposition of the \"lower\" path ψ l \\= ( 1 0 ) {\\\\displaystyle \\\\psi \\_{l}={\\\\begin{pmatrix}1\\\\\\\\0\\\\end{pmatrix}}} ![{\\\\displaystyle \\\\psi \\_{l}={\\\\begin{pmatrix}1\\\\\\\\0\\\\end{pmatrix}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/feca39f1f03b4dc63be6f7a7c2060430b1217e2f) and the \"upper\" path ψ u \\= ( 0 1 ) {\\\\displaystyle \\\\psi \\_{u}={\\\\begin{pmatrix}0\\\\\\\\1\\\\end{pmatrix}}} ![{\\\\displaystyle \\\\psi \\_{u}={\\\\begin{pmatrix}0\\\\\\\\1\\\\end{pmatrix}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/400e9751fcd7816718398d3892bec2ad26bb4713), that is, ψ \\= α ψ l \\+ β ψ u {\\\\displaystyle \\\\psi =\\\\alpha \\\\psi \\_{l}+\\\\beta \\\\psi \\_{u}} ![{\\\\displaystyle \\\\psi =\\\\alpha \\\\psi \\_{l}+\\\\beta \\\\psi \\_{u}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/fb88364b79f3c611ca2ca1edb5356139bf4e1085) for complex α , β {\\\\displaystyle \\\\alpha ,\\\\beta } ![{\\\\displaystyle \\\\alpha ,\\\\beta }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/e4b46b57cfa0011b643037751809904d915c1b48). In order to respect the postulate that ⟨ ψ , ψ ⟩ \\= 1 {\\\\displaystyle \\\\langle \\\\psi ,\\\\psi \\\\rangle =1} ![{\\\\displaystyle \\\\langle \\\\psi ,\\\\psi \\\\rangle =1}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/7d5758e7a60b4e54bc46e01b0618919c65b787a0) we require that \\| α \\| 2 \\+ \\| β \\| 2 \\= 1 {\\\\displaystyle \\|\\\\alpha \\|^{2}+\\|\\\\beta \\|^{2}=1} ![{\\\\displaystyle \\|\\\\alpha \\|^{2}+\\|\\\\beta \\|^{2}=1}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/18cd7473cdb894839d10852890517b1fb687c73b).\n\nBoth [beam splitters](https:\/\/en.wikipedia.org\/wiki\/Beam_splitter \"Beam splitter\") are modelled as the unitary matrix B \\= 1 2 ( 1 i i 1 ) {\\\\displaystyle B={\\\\frac {1}{\\\\sqrt {2}}}{\\\\begin{pmatrix}1\\&i\\\\\\\\i&1\\\\end{pmatrix}}} ![{\\\\displaystyle B={\\\\frac {1}{\\\\sqrt {2}}}{\\\\begin{pmatrix}1\\&i\\\\\\\\i&1\\\\end{pmatrix}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/ddf502efcb65d0cbac5bb8ef1a6f163ac9cf2145), which means that when a photon meets the beam splitter it will either stay on the same path with a probability amplitude of 1 \/ 2 {\\\\displaystyle 1\/{\\\\sqrt {2}}} ![{\\\\displaystyle 1\/{\\\\sqrt {2}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/75a0bbdb60fcb73ac67d9970a5eb0808b87fd37d), or be reflected to the other path with a probability amplitude of i \/ 2 {\\\\displaystyle i\/{\\\\sqrt {2}}} ![{\\\\displaystyle i\/{\\\\sqrt {2}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/e846b6a5731f3799960a4968399d85bc0b7fb9fd). The phase shifter on the upper arm is modelled as the unitary matrix P \\= ( 1 0 0 e i Δ Φ ) {\\\\displaystyle P={\\\\begin{pmatrix}1&0\\\\\\\\0\\&e^{i\\\\Delta \\\\Phi }\\\\end{pmatrix}}} ![{\\\\displaystyle P={\\\\begin{pmatrix}1&0\\\\\\\\0\\&e^{i\\\\Delta \\\\Phi }\\\\end{pmatrix}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/3df9457946dd8035c51a39e0926be9f07c7f0a3e), which means that if the photon is on the \"upper\" path it will gain a relative phase of Δ Φ {\\\\displaystyle \\\\Delta \\\\Phi } ![{\\\\displaystyle \\\\Delta \\\\Phi }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/20cedb08e6edea3cad9b2829ef67311bbe518dd2), and it will stay unchanged if it is in the lower path.\n\nA photon that enters the interferometer from the left will then be acted upon with a beam splitter B {\\\\displaystyle B} ![{\\\\displaystyle B}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/47136aad860d145f75f3eed3022df827cee94d7a), a phase shifter P {\\\\displaystyle P} ![{\\\\displaystyle P}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/b4dc73bf40314945ff376bd363916a738548d40a), and another beam splitter B {\\\\displaystyle B} ![{\\\\displaystyle B}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/47136aad860d145f75f3eed3022df827cee94d7a), and so end up in the state B P B ψ l \\= i e i Δ Φ \/ 2 ( − sin ⁡ ( Δ Φ \/ 2 ) cos ⁡ ( Δ Φ \/ 2 ) ) , {\\\\displaystyle BPB\\\\psi \\_{l}=ie^{i\\\\Delta \\\\Phi \/2}{\\\\begin{pmatrix}-\\\\sin(\\\\Delta \\\\Phi \/2)\\\\\\\\\\\\cos(\\\\Delta \\\\Phi \/2)\\\\end{pmatrix}},} ![{\\\\displaystyle BPB\\\\psi \\_{l}=ie^{i\\\\Delta \\\\Phi \/2}{\\\\begin{pmatrix}-\\\\sin(\\\\Delta \\\\Phi \/2)\\\\\\\\\\\\cos(\\\\Delta \\\\Phi \/2)\\\\end{pmatrix}},}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/7927a94da54f5d57b8accffdb9ad456a1e3b5033) and the probabilities that it will be detected at the right or at the top are given respectively by p ( u ) \\= \\| ⟨ ψ u , B P B ψ l ⟩ \\| 2 \\= cos 2 ⁡ Δ Φ 2 , {\\\\displaystyle p(u)=\\|\\\\langle \\\\psi \\_{u},BPB\\\\psi \\_{l}\\\\rangle \\|^{2}=\\\\cos ^{2}{\\\\frac {\\\\Delta \\\\Phi }{2}},} ![{\\\\displaystyle p(u)=\\|\\\\langle \\\\psi \\_{u},BPB\\\\psi \\_{l}\\\\rangle \\|^{2}=\\\\cos ^{2}{\\\\frac {\\\\Delta \\\\Phi }{2}},}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/110acb8ee7dc4e309de846470778f4767fe97f8f) p ( l ) \\= \\| ⟨ ψ l , B P B ψ l ⟩ \\| 2 \\= sin 2 ⁡ Δ Φ 2 . {\\\\displaystyle p(l)=\\|\\\\langle \\\\psi \\_{l},BPB\\\\psi \\_{l}\\\\rangle \\|^{2}=\\\\sin ^{2}{\\\\frac {\\\\Delta \\\\Phi }{2}}.} ![{\\\\displaystyle p(l)=\\|\\\\langle \\\\psi \\_{l},BPB\\\\psi \\_{l}\\\\rangle \\|^{2}=\\\\sin ^{2}{\\\\frac {\\\\Delta \\\\Phi }{2}}.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/86880ce53051688a4f591ebfb5183fbae013deee) One can therefore use the Mach–Zehnder interferometer to estimate the [phase shift](https:\/\/en.wikipedia.org\/wiki\/Phase_\\(waves\\) \"Phase (waves)\") by estimating these probabilities.\n\nIt is interesting to consider what would happen if the photon were definitely in either the \"lower\" or \"upper\" paths between the beam splitters. This can be accomplished by blocking one of the paths, or equivalently by removing the first beam splitter (and feeding the photon from the left or the bottom, as desired). In both cases, there will be no interference between the paths anymore, and the probabilities are given by p ( u ) \\= p ( l ) \\= 1 \/ 2 {\\\\displaystyle p(u)=p(l)=1\/2} ![{\\\\displaystyle p(u)=p(l)=1\/2}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/d2a9b3426984de1a64c07261c3e14b485320fee4), independently of the phase Δ Φ {\\\\displaystyle \\\\Delta \\\\Phi } ![{\\\\displaystyle \\\\Delta \\\\Phi }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/20cedb08e6edea3cad9b2829ef67311bbe518dd2). From this we can conclude that the photon does not take one path or another after the first beam splitter, but rather that it is in a genuine quantum superposition of the two paths.[\\[38\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-vedral-39)\n\n## Applications\nMain article: [Applications of quantum mechanics](https:\/\/en.wikipedia.org\/wiki\/Applications_of_quantum_mechanics \"Applications of quantum mechanics\")\n\nQuantum mechanics has had enormous success in explaining many of the features of our universe, with regard to small-scale and discrete quantities and interactions which cannot be explained by [classical methods](https:\/\/en.wikipedia.org\/wiki\/Classical_physics \"Classical physics\").[\\[note 2\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-feynmanIII-40) Quantum mechanics is often the only theory that can reveal the individual behaviors of the subatomic particles that make up all forms of matter (electrons, [protons](https:\/\/en.wikipedia.org\/wiki\/Proton \"Proton\"), [neutrons](https:\/\/en.wikipedia.org\/wiki\/Neutron \"Neutron\"), [photons](https:\/\/en.wikipedia.org\/wiki\/Photon \"Photon\"), and others). [Solid-state physics](https:\/\/en.wikipedia.org\/wiki\/Solid-state_physics \"Solid-state physics\") and [materials science](https:\/\/en.wikipedia.org\/wiki\/Materials_science \"Materials science\") are dependent upon quantum mechanics.[\\[39\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-marvincohen2008-41)\n\nIn many aspects, modern technology operates at a scale where quantum effects are significant. Important applications of quantum theory include [quantum chemistry](https:\/\/en.wikipedia.org\/wiki\/Quantum_chemistry \"Quantum chemistry\"), [quantum optics](https:\/\/en.wikipedia.org\/wiki\/Quantum_optics \"Quantum optics\"), [quantum computing](https:\/\/en.wikipedia.org\/wiki\/Quantum_computing \"Quantum computing\"), [superconducting magnets](https:\/\/en.wikipedia.org\/wiki\/Superconducting_magnet \"Superconducting magnet\"), [light-emitting diodes](https:\/\/en.wikipedia.org\/wiki\/Light-emitting_diode \"Light-emitting diode\"), the [optical amplifier](https:\/\/en.wikipedia.org\/wiki\/Optical_amplifier \"Optical amplifier\") and the laser, the [transistor](https:\/\/en.wikipedia.org\/wiki\/Transistor \"Transistor\") and [semiconductors](https:\/\/en.wikipedia.org\/wiki\/Semiconductor \"Semiconductor\") such as the [microprocessor](https:\/\/en.wikipedia.org\/wiki\/Microprocessor \"Microprocessor\"), [medical and research imaging](https:\/\/en.wikipedia.org\/wiki\/Medical_imaging \"Medical imaging\") such as [magnetic resonance imaging](https:\/\/en.wikipedia.org\/wiki\/Magnetic_resonance_imaging \"Magnetic resonance imaging\") and [electron microscopy](https:\/\/en.wikipedia.org\/wiki\/Electron_microscopy \"Electron microscopy\").[\\[40\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-42) Explanations for many biological and physical phenomena are rooted in the nature of the chemical bond, most notably the macro-molecule [DNA](https:\/\/en.wikipedia.org\/wiki\/DNA \"DNA\").\n\n## Relation to other scientific theories\n| [Modern physics](https:\/\/en.wikipedia.org\/wiki\/Modern_physics \"Modern physics\") |\n|---|\n| H ^ \\| ψ n ( t ) ⟩ \\= i ℏ d d t \\| ψ n ( t ) ⟩ {\\\\displaystyle {\\\\hat {H}}\\|\\\\psi \\_{n}(t)\\\\rangle =i\\\\hbar {\\\\frac {d}{dt}}\\|\\\\psi \\_{n}(t)\\\\rangle } ![{\\\\displaystyle {\\\\hat {H}}\\|\\\\psi \\_{n}(t)\\\\rangle =i\\\\hbar {\\\\frac {d}{dt}}\\|\\\\psi \\_{n}(t)\\\\rangle }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/6227214310d46ffac904ca257878eee1fb6ce726)  G μ ν \\+ Λ g μ ν \\= κ T μ ν {\\\\displaystyle G\\_{\\\\mu \\\\nu }+\\\\Lambda g\\_{\\\\mu \\\\nu }={\\\\kappa }T\\_{\\\\mu \\\\nu }} ![{\\\\displaystyle G\\_{\\\\mu \\\\nu }+\\\\Lambda g\\_{\\\\mu \\\\nu }={\\\\kappa }T\\_{\\\\mu \\\\nu }}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/124ab80fcb17e2733cc17ff6f93da5e52f355c77)[Schrödinger](https:\/\/en.wikipedia.org\/wiki\/Schr%C3%B6dinger_equation \"Schrödinger equation\") and [Einstein field equations](https:\/\/en.wikipedia.org\/wiki\/Einstein_field_equations \"Einstein field equations\") |\n### Classical mechanics\nThe rules of quantum mechanics assert that the state space of a system is a Hilbert space and that observables of the system are Hermitian operators acting on vectors in that space – although they do not tell us which Hilbert space or which operators. These can be chosen appropriately in order to obtain a quantitative description of a quantum system, a necessary step in making physical predictions. An important guide for making these choices is the [correspondence principle](https:\/\/en.wikipedia.org\/wiki\/Correspondence_principle \"Correspondence principle\"), a heuristic which states that the predictions of quantum mechanics reduce to those of [classical mechanics](https:\/\/en.wikipedia.org\/wiki\/Classical_mechanics \"Classical mechanics\") in the regime of large [quantum numbers](https:\/\/en.wikipedia.org\/wiki\/Quantum_number \"Quantum number\").[\\[41\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Tipler-43) One can also start from an established classical model of a particular system, and then try to guess the underlying quantum model that would give rise to the classical model in the correspondence limit. This approach is known as [quantization](https:\/\/en.wikipedia.org\/wiki\/Canonical_quantization \"Canonical quantization\").[\\[42\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Peres1993-44): 299 [\\[43\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-45)\n\nWhen quantum mechanics was originally formulated, it was applied to models whose correspondence limit was [non-relativistic](https:\/\/en.wikipedia.org\/wiki\/Theory_of_relativity \"Theory of relativity\") classical mechanics. For instance, the well-known model of the [quantum harmonic oscillator](https:\/\/en.wikipedia.org\/wiki\/Quantum_harmonic_oscillator \"Quantum harmonic oscillator\") uses an explicitly non-relativistic expression for the [kinetic energy](https:\/\/en.wikipedia.org\/wiki\/Kinetic_energy \"Kinetic energy\") of the oscillator, and is thus a quantum version of the [classical harmonic oscillator](https:\/\/en.wikipedia.org\/wiki\/Harmonic_oscillator \"Harmonic oscillator\").[\\[7\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Zwiebach2022-7): 234\n\nComplications arise with [chaotic systems](https:\/\/en.wikipedia.org\/wiki\/Chaotic_systems \"Chaotic systems\"), which do not have good quantum numbers, and [quantum chaos](https:\/\/en.wikipedia.org\/wiki\/Quantum_chaos \"Quantum chaos\") studies the relationship between classical and quantum descriptions in these systems.[\\[42\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Peres1993-44): 353\n\n[Quantum decoherence](https:\/\/en.wikipedia.org\/wiki\/Quantum_decoherence \"Quantum decoherence\") is a mechanism through which quantum systems lose [coherence](https:\/\/en.wikipedia.org\/wiki\/Quantum_coherence \"Quantum coherence\"), and thus become incapable of displaying many typically quantum effects: [quantum superpositions](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition \"Quantum superposition\") become simply probabilistic mixtures, and quantum entanglement becomes simply classical correlations.[\\[7\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Zwiebach2022-7): 687–730  Quantum coherence is not typically evident at macroscopic scales, though at temperatures approaching [absolute zero](https:\/\/en.wikipedia.org\/wiki\/Absolute_zero \"Absolute zero\") quantum behavior may manifest macroscopically.[\\[note 3\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-46)\n\nMany macroscopic properties of a classical system are a direct consequence of the quantum behavior of its parts. For example, the stability of bulk matter (consisting of atoms and [molecules](https:\/\/en.wikipedia.org\/wiki\/Molecule \"Molecule\") which would quickly collapse under electric forces alone), the rigidity of solids, and the mechanical, thermal, chemical, optical and magnetic properties of matter are all results of the interaction of [electric charges](https:\/\/en.wikipedia.org\/wiki\/Electric_charge \"Electric charge\") under the rules of quantum mechanics.[\\[44\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-47)\n\n### Special relativity and electrodynamics\nEarly attempts to merge quantum mechanics with [special relativity](https:\/\/en.wikipedia.org\/wiki\/Special_relativity \"Special relativity\") involved the replacement of the Schrödinger equation with a covariant equation such as the [Klein–Gordon equation](https:\/\/en.wikipedia.org\/wiki\/Klein%E2%80%93Gordon_equation \"Klein–Gordon equation\") or the [Dirac equation](https:\/\/en.wikipedia.org\/wiki\/Dirac_equation \"Dirac equation\"). While these theories were successful in explaining many experimental results, they had certain unsatisfactory qualities stemming from their neglect of the relativistic creation and annihilation of particles. A fully relativistic quantum theory required the development of quantum field theory, which applies quantization to a field (rather than a fixed set of particles). The first complete quantum field theory, [quantum electrodynamics](https:\/\/en.wikipedia.org\/wiki\/Quantum_electrodynamics \"Quantum electrodynamics\"), provides a fully quantum description of the [electromagnetic interaction](https:\/\/en.wikipedia.org\/wiki\/Electromagnetic_interaction \"Electromagnetic interaction\"). Quantum electrodynamics is, along with [general relativity](https:\/\/en.wikipedia.org\/wiki\/General_relativity \"General relativity\"), one of the most accurate physical theories ever devised.[\\[45\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-48)[\\[46\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-49)\n\nThe full apparatus of quantum field theory is often unnecessary for describing electrodynamic systems. A simpler approach, one that has been used since the inception of quantum mechanics, is to treat [charged](https:\/\/en.wikipedia.org\/wiki\/Electric_charge \"Electric charge\") particles as quantum mechanical objects being acted on by a classical [electromagnetic field](https:\/\/en.wikipedia.org\/wiki\/Electromagnetic_field \"Electromagnetic field\"). For example, the elementary quantum model of the [hydrogen atom](https:\/\/en.wikipedia.org\/wiki\/Hydrogen_atom \"Hydrogen atom\") describes the [electric field](https:\/\/en.wikipedia.org\/wiki\/Electric_field \"Electric field\") of the hydrogen atom using a classical − e 2 \/ ( 4 π ϵ 0 r ) {\\\\displaystyle \\\\textstyle -e^{2}\/(4\\\\pi \\\\epsilon \\_{\\_{0}}r)} ![{\\\\displaystyle \\\\textstyle -e^{2}\/(4\\\\pi \\\\epsilon \\_{\\_{0}}r)}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/7c4876353362da0a234af137590c6379d8f8c273) [Coulomb potential](https:\/\/en.wikipedia.org\/wiki\/Electric_potential \"Electric potential\").[\\[7\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Zwiebach2022-7): 285  Likewise, in a [Stern–Gerlach experiment](https:\/\/en.wikipedia.org\/wiki\/Stern%E2%80%93Gerlach_experiment \"Stern–Gerlach experiment\"), a charged particle is modeled as a quantum system, while the background magnetic field is described classically.[\\[42\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Peres1993-44): 26  This \"semi-classical\" approach fails if quantum fluctuations in the electromagnetic field play an important role, such as in the emission of photons by [charged particles](https:\/\/en.wikipedia.org\/wiki\/Charged_particle \"Charged particle\").\n\n[Quantum field](https:\/\/en.wikipedia.org\/wiki\/Field_\\(physics\\) \"Field (physics)\") theories for the [strong nuclear force](https:\/\/en.wikipedia.org\/wiki\/Strong_nuclear_force \"Strong nuclear force\") and the [weak nuclear force](https:\/\/en.wikipedia.org\/wiki\/Weak_nuclear_force \"Weak nuclear force\") have also been developed. The quantum field theory of the strong nuclear force is called [quantum chromodynamics](https:\/\/en.wikipedia.org\/wiki\/Quantum_chromodynamics \"Quantum chromodynamics\"), and describes the interactions of subnuclear particles such as [quarks](https:\/\/en.wikipedia.org\/wiki\/Quark \"Quark\") and [gluons](https:\/\/en.wikipedia.org\/wiki\/Gluon \"Gluon\"). The weak nuclear force and the electromagnetic force were unified, in their quantized forms, into a single quantum field theory (known as [electroweak theory](https:\/\/en.wikipedia.org\/wiki\/Electroweak_theory \"Electroweak theory\")), by the physicists [Abdus Salam](https:\/\/en.wikipedia.org\/wiki\/Abdus_Salam \"Abdus Salam\"), [Sheldon Glashow](https:\/\/en.wikipedia.org\/wiki\/Sheldon_Glashow \"Sheldon Glashow\") and [Steven Weinberg](https:\/\/en.wikipedia.org\/wiki\/Steven_Weinberg \"Steven Weinberg\").[\\[47\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-50)\n\n### Relation to general relativity\nEven though the predictions of both quantum theory and general relativity have been supported by rigorous and repeated [empirical evidence](https:\/\/en.wikipedia.org\/wiki\/Empirical_evidence \"Empirical evidence\"), their abstract formalisms contradict each other and they have proven extremely difficult to incorporate into one consistent, cohesive model. Gravity is negligible in many areas of particle physics, so that unification between general relativity and quantum mechanics is not an urgent issue in those particular applications. However, the lack of a correct theory of [quantum gravity](https:\/\/en.wikipedia.org\/wiki\/Quantum_gravity \"Quantum gravity\") is an important issue in [physical cosmology](https:\/\/en.wikipedia.org\/wiki\/Physical_cosmology \"Physical cosmology\") and the search by physicists for an elegant \"[Theory of Everything](https:\/\/en.wikipedia.org\/wiki\/Theory_of_Everything \"Theory of Everything\")\" (TOE). Consequently, resolving the inconsistencies between both theories has been a major goal of 20th- and 21st-century physics. This TOE would combine not only the models of subatomic physics but also derive the four fundamental forces of nature from a single force or phenomenon.[\\[48\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-NYT-20221010-51)\n\nOne proposal for doing so is [string theory](https:\/\/en.wikipedia.org\/wiki\/String_theory \"String theory\"), which posits that the [point-like particles](https:\/\/en.wikipedia.org\/wiki\/Point_particle \"Point particle\") of [particle physics](https:\/\/en.wikipedia.org\/wiki\/Particle_physics \"Particle physics\") are replaced by [one-dimensional](https:\/\/en.wikipedia.org\/wiki\/Dimension_\\(mathematics_and_physics\\) \"Dimension (mathematics and physics)\") objects called [strings](https:\/\/en.wikipedia.org\/wiki\/String_\\(physics\\) \"String (physics)\"). String theory describes how these strings propagate through space and interact with each other. On distance scales larger than the string scale, a string looks just like an ordinary particle, with its [mass](https:\/\/en.wikipedia.org\/wiki\/Mass \"Mass\"), [charge](https:\/\/en.wikipedia.org\/wiki\/Charge_\\(physics\\) \"Charge (physics)\"), and other properties determined by the [vibrational](https:\/\/en.wikipedia.org\/wiki\/Vibration \"Vibration\") state of the string. In string theory, one of the many vibrational states of the string corresponds to the [graviton](https:\/\/en.wikipedia.org\/wiki\/Graviton \"Graviton\"), a quantum mechanical particle that carries gravitational force.[\\[49\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-52)[\\[50\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-53)\n\nAnother popular theory is [loop quantum gravity](https:\/\/en.wikipedia.org\/wiki\/Loop_quantum_gravity \"Loop quantum gravity\") (LQG), which describes quantum properties of gravity and is thus a theory of [quantum spacetime](https:\/\/en.wikipedia.org\/wiki\/Quantum_spacetime \"Quantum spacetime\"). LQG is an attempt to merge and adapt standard quantum mechanics and standard general relativity. This theory describes space as an extremely fine fabric \"woven\" of finite loops called [spin networks](https:\/\/en.wikipedia.org\/wiki\/Spin_network \"Spin network\"). The evolution of a spin network over time is called a [spin foam](https:\/\/en.wikipedia.org\/wiki\/Spin_foam \"Spin foam\"). The characteristic length scale of a spin foam is the [Planck length](https:\/\/en.wikipedia.org\/wiki\/Planck_length \"Planck length\"), approximately 1.616×10−35 m, and so lengths shorter than the Planck length are not physically meaningful in LQG.[\\[51\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-54)\n\n## Philosophical implications\nMain article: [Interpretations of quantum mechanics](https:\/\/en.wikipedia.org\/wiki\/Interpretations_of_quantum_mechanics \"Interpretations of quantum mechanics\")\n\nUnsolved problem in physics\n\nIs there a preferred interpretation of quantum mechanics? How does the quantum description of reality, which includes elements such as the \"[superposition](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle \"Superposition principle\") of states\" and \"[wave function collapse](https:\/\/en.wikipedia.org\/wiki\/Wave_function_collapse \"Wave function collapse\")\", give rise to the reality we perceive?\n\n[More unsolved problems in physics](https:\/\/en.wikipedia.org\/wiki\/List_of_unsolved_problems_in_physics \"List of unsolved problems in physics\")\n\nSince its inception, the many counter-intuitive aspects and results of quantum mechanics have provoked strong [philosophical](https:\/\/en.wikipedia.org\/wiki\/Philosophical \"Philosophical\") debates and many [interpretations](https:\/\/en.wikipedia.org\/wiki\/Interpretations_of_quantum_mechanics \"Interpretations of quantum mechanics\"). The arguments centre on the probabilistic nature of quantum mechanics, the difficulties with [wavefunction collapse](https:\/\/en.wikipedia.org\/wiki\/Wavefunction_collapse \"Wavefunction collapse\") and the related [measurement problem](https:\/\/en.wikipedia.org\/wiki\/Measurement_problem \"Measurement problem\"), and [quantum nonlocality](https:\/\/en.wikipedia.org\/wiki\/Quantum_nonlocality \"Quantum nonlocality\"). Perhaps the only consensus that exists about these issues is that there is no consensus. [Richard Feynman](https:\/\/en.wikipedia.org\/wiki\/Richard_Feynman \"Richard Feynman\") once said, \"I think I can safely say that nobody understands quantum mechanics.\"[\\[52\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-55) According to [Steven Weinberg](https:\/\/en.wikipedia.org\/wiki\/Steven_Weinberg \"Steven Weinberg\"), \"There is now in my opinion no entirely satisfactory interpretation of quantum mechanics.\"[\\[53\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-56)\n\nThe views of [Niels Bohr](https:\/\/en.wikipedia.org\/wiki\/Niels_Bohr \"Niels Bohr\"), Werner Heisenberg and other physicists are often grouped together as the \"[Copenhagen interpretation](https:\/\/en.wikipedia.org\/wiki\/Copenhagen_interpretation \"Copenhagen interpretation\")\".[\\[54\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-57)[\\[55\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-58) According to these views, the probabilistic nature of quantum mechanics is not a *temporary* feature which will eventually be replaced by a deterministic theory, but is instead a *final* renunciation of the classical idea of \"causality\". Bohr in particular emphasized that any well-defined application of the quantum mechanical formalism must always make reference to the experimental arrangement, due to the [complementary](https:\/\/en.wikipedia.org\/wiki\/Complementarity_\\(physics\\) \"Complementarity (physics)\") nature of evidence obtained under different experimental situations. Copenhagen-type interpretations were adopted by Nobel laureates in quantum physics, including Bohr,[\\[56\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-BohrComo-59) Heisenberg,[\\[57\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-60) Schrödinger,[\\[58\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-61) Feynman,[\\[2\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Feynman-2) and [Zeilinger](https:\/\/en.wikipedia.org\/wiki\/Anton_Zeilinger \"Anton Zeilinger\")[\\[59\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-MaKoflerZeilinger-62) as well as 21st-century researchers in quantum foundations.[\\[60\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-:25-63)\n\n[Albert Einstein](https:\/\/en.wikipedia.org\/wiki\/Albert_Einstein \"Albert Einstein\"), himself one of the founders of [quantum theory](https:\/\/en.wikipedia.org\/wiki\/Old_quantum_theory \"Old quantum theory\"), was troubled by its apparent failure to respect some cherished metaphysical principles, such as [determinism](https:\/\/en.wikipedia.org\/wiki\/Determinism \"Determinism\") and [locality](https:\/\/en.wikipedia.org\/wiki\/Principle_of_locality \"Principle of locality\"). Einstein's long-running exchanges with Bohr about the meaning and status of quantum mechanics are now known as the [Bohr–Einstein debates](https:\/\/en.wikipedia.org\/wiki\/Bohr%E2%80%93Einstein_debates \"Bohr–Einstein debates\"). Einstein believed that underlying quantum mechanics must be a theory that explicitly forbids [action at a distance](https:\/\/en.wikipedia.org\/wiki\/Action_at_a_distance \"Action at a distance\"). He argued that quantum mechanics was incomplete, a theory that was valid but not fundamental, analogous to how [thermodynamics](https:\/\/en.wikipedia.org\/wiki\/Thermodynamics \"Thermodynamics\") is valid, but the fundamental theory behind it is [statistical mechanics](https:\/\/en.wikipedia.org\/wiki\/Statistical_mechanics \"Statistical mechanics\"). In 1935, Einstein and his collaborators [Boris Podolsky](https:\/\/en.wikipedia.org\/wiki\/Boris_Podolsky \"Boris Podolsky\") and [Nathan Rosen](https:\/\/en.wikipedia.org\/wiki\/Nathan_Rosen \"Nathan Rosen\") published an argument that the principle of locality implies the incompleteness of quantum mechanics, a [thought experiment](https:\/\/en.wikipedia.org\/wiki\/Thought_experiment \"Thought experiment\") later termed the [Einstein–Podolsky–Rosen paradox](https:\/\/en.wikipedia.org\/wiki\/Einstein%E2%80%93Podolsky%E2%80%93Rosen_paradox \"Einstein–Podolsky–Rosen paradox\").[\\[note 4\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-68) In 1964, [John Bell](https:\/\/en.wikipedia.org\/wiki\/John_Stewart_Bell \"John Stewart Bell\") showed that EPR's principle of locality, together with determinism, was actually incompatible with quantum mechanics: they implied constraints on the correlations produced by distance systems, now known as [Bell inequalities](https:\/\/en.wikipedia.org\/wiki\/Bell_inequalities \"Bell inequalities\"), that can be violated by entangled particles.[\\[65\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-69) Since then [several experiments](https:\/\/en.wikipedia.org\/wiki\/Bell_test \"Bell test\") have been performed to obtain these correlations, with the result that they do in fact violate Bell inequalities, and thus falsify the conjunction of locality with determinism.[\\[16\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-wiseman15-16)[\\[17\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-wolchover17-17)\n\n[Bohmian mechanics](https:\/\/en.wikipedia.org\/wiki\/Bohmian_mechanics \"Bohmian mechanics\") shows that it is possible to reformulate quantum mechanics to make it deterministic, at the price of making it explicitly nonlocal. It attributes not only a wave function to a physical system, but in addition a real position, that evolves deterministically under a nonlocal guiding equation. The evolution of a physical system is given at all times by the Schrödinger equation together with the guiding equation; there is never a collapse of the wave function. This solves the measurement problem.[\\[66\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-70)\n\n[![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/c\/c8\/Schroedingers_cat_film.svg\/250px-Schroedingers_cat_film.svg.png)](https:\/\/en.wikipedia.org\/wiki\/File:Schroedingers_cat_film.svg)\n\nThe [Schrödinger's cat](https:\/\/en.wikipedia.org\/wiki\/Schr%C3%B6dinger%27s_cat \"Schrödinger's cat\") thought experiment can be used to visualize the many-worlds interpretation of quantum mechanics, where a branching of the universe occurs through a superposition of two quantum mechanical states.\n\nEverett's [many-worlds interpretation](https:\/\/en.wikipedia.org\/wiki\/Many-worlds_interpretation \"Many-worlds interpretation\"), formulated in 1956, holds that *all* the possibilities described by quantum theory *simultaneously* occur in a multiverse composed of mostly independent parallel universes.[\\[67\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-71) This is a consequence of removing the axiom of the collapse of the wave packet. All possible states of the measured system and the measuring apparatus, together with the observer, are present in a real physical quantum superposition. While the multiverse is deterministic, we perceive non-deterministic behavior governed by probabilities, because we do not observe the multiverse as a whole, but only one parallel universe at a time. Exactly how this is supposed to work has been the subject of much debate. Several attempts have been made to make sense of this and derive the Born rule,[\\[68\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-dewitt73-72)[\\[69\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-wallace2003-73) with no consensus on whether they have been successful.[\\[70\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-ballentine1973-74)[\\[71\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-75)[\\[72\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-kent2009-76)\n\n[Relational quantum mechanics](https:\/\/en.wikipedia.org\/wiki\/Relational_quantum_mechanics \"Relational quantum mechanics\") appeared in the late 1990s as a modern derivative of Copenhagen-type ideas,[\\[73\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-77)[\\[74\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-78) and [QBism](https:\/\/en.wikipedia.org\/wiki\/QBism \"QBism\") was developed some years later.[\\[75\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-:23-79)[\\[76\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-80)\n\n## History\nMain articles: [History of quantum mechanics](https:\/\/en.wikipedia.org\/wiki\/History_of_quantum_mechanics \"History of quantum mechanics\") and [Atomic theory](https:\/\/en.wikipedia.org\/wiki\/Atomic_theory \"Atomic theory\")\n\nQuantum mechanics was developed in the early decades of the 20th century, driven by the need to explain phenomena that, in some cases, had been observed in earlier times. Scientific inquiry into the wave nature of light began in the 17th and 18th centuries, when scientists such as [Robert Hooke](https:\/\/en.wikipedia.org\/wiki\/Robert_Hooke \"Robert Hooke\"), [Christiaan Huygens](https:\/\/en.wikipedia.org\/wiki\/Christiaan_Huygens \"Christiaan Huygens\") and [Leonhard Euler](https:\/\/en.wikipedia.org\/wiki\/Leonhard_Euler \"Leonhard Euler\") proposed a wave theory of light based on experimental observations.[\\[77\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Born_&_Wolf-81) In 1803 English [polymath](https:\/\/en.wikipedia.org\/wiki\/Polymath \"Polymath\") [Thomas Young](https:\/\/en.wikipedia.org\/wiki\/Thomas_Young_\\(scientist\\) \"Thomas Young (scientist)\") described the famous [double-slit experiment](https:\/\/en.wikipedia.org\/wiki\/Young%27s_interference_experiment \"Young's interference experiment\").[\\[78\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-82) This experiment played a major role in the general acceptance of the [wave theory of light](https:\/\/en.wikipedia.org\/wiki\/Wave_theory_of_light \"Wave theory of light\").\n\nDuring the early 19th century, [chemical](https:\/\/en.wikipedia.org\/wiki\/Chemistry \"Chemistry\") research by [John Dalton](https:\/\/en.wikipedia.org\/wiki\/John_Dalton \"John Dalton\") and [Amedeo Avogadro](https:\/\/en.wikipedia.org\/wiki\/Amedeo_Avogadro \"Amedeo Avogadro\") lent weight to the [atomic theory](https:\/\/en.wikipedia.org\/wiki\/Atomic_theory \"Atomic theory\") of matter, an idea that [James Clerk Maxwell](https:\/\/en.wikipedia.org\/wiki\/James_Clerk_Maxwell \"James Clerk Maxwell\"), [Ludwig Boltzmann](https:\/\/en.wikipedia.org\/wiki\/Ludwig_Boltzmann \"Ludwig Boltzmann\") and others built upon to establish the [kinetic theory of gases](https:\/\/en.wikipedia.org\/wiki\/Kinetic_theory_of_gases \"Kinetic theory of gases\"). The successes of kinetic theory gave further credence to the idea that matter is composed of atoms, yet the theory also had shortcomings that would only be resolved by the development of quantum mechanics.[\\[79\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Feynman-kinetic-theory-83) While the early conception of atoms from [Greek philosophy](https:\/\/en.wikipedia.org\/wiki\/Greek_philosophy \"Greek philosophy\") had been that they were indivisible units – the word \"atom\" deriving from the [Greek](https:\/\/en.wikipedia.org\/wiki\/Greek_language \"Greek language\") for 'uncuttable' – the 19th century saw the formulation of hypotheses about subatomic structure. One important discovery in that regard was [Michael Faraday](https:\/\/en.wikipedia.org\/wiki\/Michael_Faraday \"Michael Faraday\")'s 1838 observation of a glow caused by an electrical discharge inside a glass tube containing gas at low pressure. [Julius Plücker](https:\/\/en.wikipedia.org\/wiki\/Julius_Pl%C3%BCcker \"Julius Plücker\"), [Johann Wilhelm Hittorf](https:\/\/en.wikipedia.org\/wiki\/Johann_Wilhelm_Hittorf \"Johann Wilhelm Hittorf\") and [Eugen Goldstein](https:\/\/en.wikipedia.org\/wiki\/Eugen_Goldstein \"Eugen Goldstein\") carried on and improved upon Faraday's work, leading to the identification of [cathode rays](https:\/\/en.wikipedia.org\/wiki\/Cathode_rays \"Cathode rays\"), which [J. J. Thomson](https:\/\/en.wikipedia.org\/wiki\/J._J._Thomson \"J. J. Thomson\") found to consist of subatomic particles that would be called electrons.[\\[80\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-84)[\\[81\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-85)\n\n[![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/a\/a7\/Max_Planck_%281858-1947%29.jpg\/250px-Max_Planck_%281858-1947%29.jpg)](https:\/\/en.wikipedia.org\/wiki\/File:Max_Planck_\\(1858-1947\\).jpg)\n\n[Max Planck](https:\/\/en.wikipedia.org\/wiki\/Max_Planck \"Max Planck\") is considered the father of the quantum theory.\n\nThe [black-body radiation](https:\/\/en.wikipedia.org\/wiki\/Black-body_radiation \"Black-body radiation\") problem was discovered by [Gustav Kirchhoff](https:\/\/en.wikipedia.org\/wiki\/Gustav_Kirchhoff \"Gustav Kirchhoff\") in 1859. In 1900, Max Planck proposed the hypothesis that energy is radiated and absorbed in discrete \"quanta\" (or energy packets), yielding a calculation that precisely matched the observed patterns of black-body radiation.[\\[82\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-86) The word *quantum* derives from the [Latin](https:\/\/en.wikipedia.org\/wiki\/Latin \"Latin\"), meaning \"how great\" or \"how much\".[\\[83\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-87) According to Planck, quantities of energy could be thought of as divided into \"elements\" whose size (*E*) would be proportional to their [frequency](https:\/\/en.wikipedia.org\/wiki\/Frequency \"Frequency\") (*ν*): E \\= h ν {\\\\displaystyle E=h\\\\nu \\\\ } ![{\\\\displaystyle E=h\\\\nu \\\\ }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/24767663bb931daecf1eb628be4fea46f32e3622), where *h* is the [Planck constant](https:\/\/en.wikipedia.org\/wiki\/Planck_constant \"Planck constant\"). Planck cautiously insisted that this was only an aspect of the processes of absorption and emission of radiation and was not the *physical reality* of the radiation.[\\[84\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-88) In fact, he considered his quantum hypothesis a mathematical trick to get the right answer rather than a sizable discovery.[\\[85\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Kragh-89) However, in 1905 Albert Einstein interpreted Planck's quantum hypothesis [realistically](https:\/\/en.wikipedia.org\/wiki\/Local_realism \"Local realism\") and used it to explain the [photoelectric effect](https:\/\/en.wikipedia.org\/wiki\/Photoelectric_effect \"Photoelectric effect\"), in which shining light on certain materials can eject electrons from the material. Niels Bohr then developed Planck's ideas about radiation into a [model of the hydrogen atom](https:\/\/en.wikipedia.org\/wiki\/Bohr_model \"Bohr model\") that successfully predicted the [spectral lines](https:\/\/en.wikipedia.org\/wiki\/Spectral_line \"Spectral line\") of hydrogen.[\\[86\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-90) Einstein further developed this idea to show that an [electromagnetic wave](https:\/\/en.wikipedia.org\/wiki\/Electromagnetic_wave \"Electromagnetic wave\") such as light could also be described as a particle (later called the photon), with a discrete amount of energy that depends on its frequency.[\\[87\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-91) In his paper \"On the Quantum Theory of Radiation\", Einstein expanded on the interaction between energy and matter to explain the absorption and emission of energy by atoms. Although overshadowed at the time by his general theory of relativity, this paper articulated the mechanism underlying the stimulated emission of radiation,[\\[88\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-92) which became the basis of the laser.[\\[89\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-93)\n\n[![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/6\/6e\/Solvay_conference_1927.jpg\/500px-Solvay_conference_1927.jpg)](https:\/\/en.wikipedia.org\/wiki\/File:Solvay_conference_1927.jpg)\n\nThe 1927 [Solvay Conference](https:\/\/en.wikipedia.org\/wiki\/Solvay_Conference \"Solvay Conference\") in [Brussels](https:\/\/en.wikipedia.org\/wiki\/Brussels \"Brussels\") was the fifth world physics conference.\n\nThis phase is known as the [old quantum theory](https:\/\/en.wikipedia.org\/wiki\/Old_quantum_theory \"Old quantum theory\"). Never complete or self-consistent, the old quantum theory was rather a set of [heuristic](https:\/\/en.wikipedia.org\/wiki\/Heuristic \"Heuristic\") corrections to classical mechanics.[\\[90\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-terHaar-94)[\\[91\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-95) The theory is now understood as a [semi-classical approximation](https:\/\/en.wikipedia.org\/wiki\/WKB_approximation#Application_to_the_Schr%C3%B6dinger_equation \"WKB approximation\") to modern quantum mechanics.[\\[92\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-96)[\\[93\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-97) Notable results from this period include, in addition to the work of Planck, Einstein and Bohr mentioned above, Einstein and [Peter Debye](https:\/\/en.wikipedia.org\/wiki\/Peter_Debye \"Peter Debye\")'s work on the [specific heat](https:\/\/en.wikipedia.org\/wiki\/Specific_heat \"Specific heat\") of solids, Bohr and [Hendrika Johanna van Leeuwen](https:\/\/en.wikipedia.org\/wiki\/Hendrika_Johanna_van_Leeuwen \"Hendrika Johanna van Leeuwen\")'s [proof](https:\/\/en.wikipedia.org\/wiki\/Bohr%E2%80%93Van_Leeuwen_theorem \"Bohr–Van Leeuwen theorem\") that classical physics cannot account for [diamagnetism](https:\/\/en.wikipedia.org\/wiki\/Diamagnetism \"Diamagnetism\"), and [Arnold Sommerfeld](https:\/\/en.wikipedia.org\/wiki\/Arnold_Sommerfeld \"Arnold Sommerfeld\")'s extension of the Bohr model to include special-relativistic effects.[\\[90\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-terHaar-94)[\\[94\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Aharoni-98)\n\nIn the mid-1920s quantum mechanics was developed to become the standard formulation for atomic physics. In 1923, the French physicist [Louis de Broglie](https:\/\/en.wikipedia.org\/wiki\/Louis_de_Broglie \"Louis de Broglie\") put forward his theory of matter waves by stating that particles can exhibit wave characteristics and vice versa. Building on de Broglie's approach, modern quantum mechanics was born in 1925, when the German physicists Werner Heisenberg, Max Born, and [Pascual Jordan](https:\/\/en.wikipedia.org\/wiki\/Pascual_Jordan \"Pascual Jordan\")[\\[95\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Edwards79-99)[\\[96\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Edwards81-100) developed [matrix mechanics](https:\/\/en.wikipedia.org\/wiki\/Matrix_mechanics \"Matrix mechanics\") and the Austrian physicist Erwin Schrödinger invented [wave mechanics](https:\/\/en.wikipedia.org\/wiki\/Schr%C3%B6dinger_equation \"Schrödinger equation\"). Born introduced the probabilistic interpretation of Schrödinger's wave function in July 1926.[\\[97\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-101) Thus, the entire field of quantum physics emerged, leading to its wider acceptance at the Fifth [Solvay Conference](https:\/\/en.wikipedia.org\/wiki\/Solvay_Conference \"Solvay Conference\") in 1927.[\\[98\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-pais1997-102)\n\nBy 1930, quantum mechanics had been further unified and formalized by [David Hilbert](https:\/\/en.wikipedia.org\/wiki\/David_Hilbert \"David Hilbert\"), Paul Dirac and [John von Neumann](https:\/\/en.wikipedia.org\/wiki\/John_von_Neumann \"John von Neumann\")[\\[99\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-103) with greater emphasis on [measurement](https:\/\/en.wikipedia.org\/wiki\/Measurement_in_quantum_mechanics \"Measurement in quantum mechanics\"), the statistical nature of our knowledge of reality, and [philosophical speculation about the 'observer'](https:\/\/en.wikipedia.org\/wiki\/Interpretations_of_quantum_mechanics \"Interpretations of quantum mechanics\"). It has since permeated many disciplines, including quantum chemistry, [quantum electronics](https:\/\/en.wikipedia.org\/wiki\/Quantum_electronics \"Quantum electronics\"), [quantum optics](https:\/\/en.wikipedia.org\/wiki\/Quantum_optics \"Quantum optics\"), and [quantum information science](https:\/\/en.wikipedia.org\/wiki\/Quantum_information_science \"Quantum information science\"). It also provides a useful framework for many features of the modern [periodic table of elements](https:\/\/en.wikipedia.org\/wiki\/Periodic_table_of_elements \"Periodic table of elements\"), and describes the behaviors of [atoms](https:\/\/en.wikipedia.org\/wiki\/Atoms \"Atoms\") during [chemical bonding](https:\/\/en.wikipedia.org\/wiki\/Chemical_bond \"Chemical bond\") and the flow of electrons in computer [semiconductors](https:\/\/en.wikipedia.org\/wiki\/Semiconductor \"Semiconductor\"), and therefore plays a crucial role in many modern technologies. While quantum mechanics was constructed to describe the world of the very small, it is also needed to explain some [macroscopic](https:\/\/en.wikipedia.org\/wiki\/Macroscopic \"Macroscopic\") phenomena such as [superconductors](https:\/\/en.wikipedia.org\/wiki\/Superconductors \"Superconductors\")[\\[100\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-feynman2015-104) and [superfluids](https:\/\/en.wikipedia.org\/wiki\/Superfluid \"Superfluid\").[\\[101\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-105)\n\n## See also\n- [Bra–ket notation](https:\/\/en.wikipedia.org\/wiki\/Bra%E2%80%93ket_notation \"Bra–ket notation\")\n- [Einstein's thought experiments](https:\/\/en.wikipedia.org\/wiki\/Einstein%27s_thought_experiments \"Einstein's thought experiments\")\n- [List of textbooks on classical mechanics and quantum mechanics](https:\/\/en.wikipedia.org\/wiki\/List_of_textbooks_on_classical_mechanics_and_quantum_mechanics \"List of textbooks on classical mechanics and quantum mechanics\")\n- [Macroscopic quantum phenomena](https:\/\/en.wikipedia.org\/wiki\/Macroscopic_quantum_phenomena \"Macroscopic quantum phenomena\")\n- [Phase-space formulation](https:\/\/en.wikipedia.org\/wiki\/Phase-space_formulation \"Phase-space formulation\")\n- [Regularization (physics)](https:\/\/en.wikipedia.org\/wiki\/Regularization_\\(physics\\) \"Regularization (physics)\")\n- [Two-state quantum system](https:\/\/en.wikipedia.org\/wiki\/Two-state_quantum_system \"Two-state quantum system\")\n## Explanatory notes\n1. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-35)** A momentum eigenstate would be a perfectly monochromatic wave of infinite extent, which is not square-integrable. Likewise, a position eigenstate would be a [Dirac delta distribution](https:\/\/en.wikipedia.org\/wiki\/Dirac_delta_distribution \"Dirac delta distribution\"), not square-integrable and technically not a function at all. Consequently, neither can belong to the particle's Hilbert space. Physicists sometimes introduce fictitious \"bases\" for a Hilbert space comprising elements outside that space. These are invented for calculational convenience and do not represent physical states.[\\[26\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Cohen-Tannoudji-26): 100–105\n2. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-feynmanIII_40-0)** See, for example, [the Feynman Lectures on Physics](https:\/\/en.wikipedia.org\/wiki\/The_Feynman_Lectures_on_Physics \"The Feynman Lectures on Physics\") for some of the technological applications which use quantum mechanics, e.g., [transistors](https:\/\/en.wikipedia.org\/wiki\/Transistor \"Transistor\") (vol **III**, pp. 14–11 ff), [integrated circuits](https:\/\/en.wikipedia.org\/wiki\/Integrated_circuit \"Integrated circuit\"), which are follow-on technology in solid-state physics (vol **II**, pp. 8–6), and [lasers](https:\/\/en.wikipedia.org\/wiki\/Laser \"Laser\") (vol **III**, pp. 9–13).\n3. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-46)** See *[Macroscopic quantum phenomena](https:\/\/en.wikipedia.org\/wiki\/Macroscopic_quantum_phenomena \"Macroscopic quantum phenomena\")*, *[Bose–Einstein condensate](https:\/\/en.wikipedia.org\/wiki\/Bose%E2%80%93Einstein_condensate \"Bose–Einstein condensate\")*, and *[Quantum machine](https:\/\/en.wikipedia.org\/wiki\/Quantum_machine \"Quantum machine\")*\n4. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-68)** The published form of the EPR argument was due to Podolsky, and Einstein himself was not satisfied with it. In his own publications and correspondence, Einstein used a different argument to insist that quantum mechanics is an incomplete theory.[\\[61\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-spekkens-64)[\\[62\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-howard-65)[\\[63\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-66)[\\[64\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-67)\n## References\n1. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-Born1926_1-0)**\n   [Born, M.](https:\/\/en.wikipedia.org\/wiki\/Max_Born \"Max Born\") (1926). \"Zur Quantenmechanik der Stoßvorgänge\" \\[On the Quantum Mechanics of Collision Processes\\]. *Zeitschrift für Physik* (in German). **37** (12): 863–867\\. [Bibcode](https:\/\/en.wikipedia.org\/wiki\/Bibcode_\\(identifier\\) \"Bibcode (identifier)\"):[1926ZPhy...37..863B](https:\/\/ui.adsabs.harvard.edu\/abs\/1926ZPhy...37..863B). [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1007\/BF01397477](https:\/\/doi.org\/10.1007%2FBF01397477). [ISSN](https:\/\/en.wikipedia.org\/wiki\/ISSN_\\(identifier\\) \"ISSN (identifier)\") [1434-6001](https:\/\/search.worldcat.org\/issn\/1434-6001). [S2CID](https:\/\/en.wikipedia.org\/wiki\/S2CID_\\(identifier\\) \"S2CID (identifier)\") [119896026](https:\/\/api.semanticscholar.org\/CorpusID:119896026).\n2. ^ [***a***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-Feynman_2-0) [***b***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-Feynman_2-1) [***c***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-Feynman_2-2) [***d***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-Feynman_2-3)\n   Feynman, Richard; Leighton, Robert; Sands, Matthew (1964). [*The Feynman Lectures on Physics*](https:\/\/feynmanlectures.caltech.edu\/III_01.html). Vol. 3. California Institute of Technology. Retrieved 19 December 2020.\n   \n   Reprinted, Addison-Wesley, 1989,\n   \n   [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n   \n   [978-0-201-50064-6](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-201-50064-6 \"Special:BookSources\/978-0-201-50064-6\")\n3. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-3)**\n   Jaeger, Gregg (September 2014). \"What in the (quantum) world is macroscopic?\". *American Journal of Physics*. **82** (9): 896–905\\. [Bibcode](https:\/\/en.wikipedia.org\/wiki\/Bibcode_\\(identifier\\) \"Bibcode (identifier)\"):[2014AmJPh..82..896J](https:\/\/ui.adsabs.harvard.edu\/abs\/2014AmJPh..82..896J). [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1119\/1.4878358](https:\/\/doi.org\/10.1119%2F1.4878358).\n4. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-4)**\n   Fein, Yaakov Y.; Geyer, Philipp; Zwick, Patrick; Kiałka, Filip; Pedalino, Sebastian; Mayor, Marcel; Gerlich, Stefan; Arndt, Markus (September 2019). \"Quantum superposition of molecules beyond 25 kDa\". *Nature Physics*. **15** (12): 1242–1245\\. [Bibcode](https:\/\/en.wikipedia.org\/wiki\/Bibcode_\\(identifier\\) \"Bibcode (identifier)\"):[2019NatPh..15.1242F](https:\/\/ui.adsabs.harvard.edu\/abs\/2019NatPh..15.1242F). [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1038\/s41567-019-0663-9](https:\/\/doi.org\/10.1038%2Fs41567-019-0663-9). [S2CID](https:\/\/en.wikipedia.org\/wiki\/S2CID_\\(identifier\\) \"S2CID (identifier)\") [203638258](https:\/\/api.semanticscholar.org\/CorpusID:203638258).\n5. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-5)**\n   Bojowald, Martin (2015). \"Quantum cosmology: a review\". *Reports on Progress in Physics*. **78** (2) 023901. [arXiv](https:\/\/en.wikipedia.org\/wiki\/ArXiv_\\(identifier\\) \"ArXiv (identifier)\"):[1501\\.04899](https:\/\/arxiv.org\/abs\/1501.04899). [Bibcode](https:\/\/en.wikipedia.org\/wiki\/Bibcode_\\(identifier\\) \"Bibcode (identifier)\"):[2015RPPh...78b3901B](https:\/\/ui.adsabs.harvard.edu\/abs\/2015RPPh...78b3901B). [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1088\/0034-4885\/78\/2\/023901](https:\/\/doi.org\/10.1088%2F0034-4885%2F78%2F2%2F023901). [PMID](https:\/\/en.wikipedia.org\/wiki\/PMID_\\(identifier\\) \"PMID (identifier)\") [25582917](https:\/\/pubmed.ncbi.nlm.nih.gov\/25582917). [S2CID](https:\/\/en.wikipedia.org\/wiki\/S2CID_\\(identifier\\) \"S2CID (identifier)\") [18463042](https:\/\/api.semanticscholar.org\/CorpusID:18463042).\n6. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-6)**\n   Fan, X.; Myers, T. G.; Sukra, B. A. D.; Gabrielse, G. (2023-02-13). \"Measurement of the Electron Magnetic Moment\". *Physical Review Letters*. **130** (7) 071801. [arXiv](https:\/\/en.wikipedia.org\/wiki\/ArXiv_\\(identifier\\) \"ArXiv (identifier)\"):[2209\\.13084](https:\/\/arxiv.org\/abs\/2209.13084). [Bibcode](https:\/\/en.wikipedia.org\/wiki\/Bibcode_\\(identifier\\) \"Bibcode (identifier)\"):[2023PhRvL.130g1801F](https:\/\/ui.adsabs.harvard.edu\/abs\/2023PhRvL.130g1801F). [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1103\/PhysRevLett.130.071801](https:\/\/doi.org\/10.1103%2FPhysRevLett.130.071801). [PMID](https:\/\/en.wikipedia.org\/wiki\/PMID_\\(identifier\\) \"PMID (identifier)\") [36867820](https:\/\/pubmed.ncbi.nlm.nih.gov\/36867820).\n7. ^ [***a***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-Zwiebach2022_7-0) [***b***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-Zwiebach2022_7-1) [***c***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-Zwiebach2022_7-2) [***d***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-Zwiebach2022_7-3) [***e***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-Zwiebach2022_7-4) [***f***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-Zwiebach2022_7-5) [***g***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-Zwiebach2022_7-6) [***h***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-Zwiebach2022_7-7) [***i***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-Zwiebach2022_7-8) [***j***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-Zwiebach2022_7-9)\n   [Zwiebach, Barton](https:\/\/en.wikipedia.org\/wiki\/Barton_Zwiebach \"Barton Zwiebach\") (2022). *Mastering Quantum Mechanics: Essentials, Theory, and Applications*. MIT Press. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n   \n   [978-0-262-04613-8](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-262-04613-8 \"Special:BookSources\/978-0-262-04613-8\")\n   \n   .\n8. ^ [***a***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-Lederman_8-0) [***b***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-Lederman_8-1) [***c***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-Lederman_8-2)\n   Lederman, Leon M.; Hill, Christopher T. (2011). [*Quantum Physics for Poets*](https:\/\/books.google.com\/books?id=qY_yOwHg_WYC&pg=PA102). US: Prometheus Books. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n   \n   [978-1-61614-281-0](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-1-61614-281-0 \"Special:BookSources\/978-1-61614-281-0\")\n   \n   .\n9. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-M%C3%BCller-Kirsten_9-0)**\n   Müller-Kirsten, H. J. W. (2006). [*Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral*](https:\/\/books.google.com\/books?id=p1_Z81Le58MC&pg=PA14). US: World Scientific. p. 14. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n   \n   [978-981-256-691-1](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-981-256-691-1 \"Special:BookSources\/978-981-256-691-1\")\n   \n   .\n10. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-Plotnitsky_10-0)**\n    Plotnitsky, Arkady (2012). [*Niels Bohr and Complementarity: An Introduction*](https:\/\/books.google.com\/books?id=dmdUp97S4AYC&pg=PA75). US: Springer. pp. 75–76\\. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-1-4614-4517-3](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-1-4614-4517-3 \"Special:BookSources\/978-1-4614-4517-3\")\n    \n    .\n11. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-11)**\n    [Griffiths, David J.](https:\/\/en.wikipedia.org\/wiki\/David_J._Griffiths \"David J. Griffiths\") (1995). [*Introduction to Quantum Mechanics*](https:\/\/en.wikipedia.org\/wiki\/Introduction_to_Quantum_Mechanics_\\(book\\) \"Introduction to Quantum Mechanics (book)\"). Prentice Hall. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [0-13-124405-1](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/0-13-124405-1 \"Special:BookSources\/0-13-124405-1\")\n    \n    .\n12. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-Trixler2013_12-0)**\n    Trixler, F. (2013). [\"Quantum tunnelling to the origin and evolution of life\"](https:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC3768233). *Current Organic Chemistry*. **17** (16): 1758–1770\\. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.2174\/13852728113179990083](https:\/\/doi.org\/10.2174%2F13852728113179990083). [PMC](https:\/\/en.wikipedia.org\/wiki\/PMC_\\(identifier\\) \"PMC (identifier)\") [3768233](https:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC3768233). [PMID](https:\/\/en.wikipedia.org\/wiki\/PMID_\\(identifier\\) \"PMID (identifier)\") [24039543](https:\/\/pubmed.ncbi.nlm.nih.gov\/24039543).\n13. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-13)**\n    Phifer, Arnold (2012-03-27). [\"Developing more energy-efficient transistors through quantum tunneling\"](https:\/\/news.nd.edu\/news\/developing-more-energy-efficient-transistors-through-quantum-tunneling\/). *Notre Dame News*. Retrieved 2024-06-07.\n14. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-14)**\n    [Bub, Jeffrey](https:\/\/en.wikipedia.org\/wiki\/Jeffrey_Bub \"Jeffrey Bub\") (2019). [\"Quantum entanglement\"](https:\/\/plato.stanford.edu\/entries\/qt-entangle\/). In Zalta, Edward N. (ed.). *Stanford Encyclopedia of Philosophy*. Metaphysics Research Lab, Stanford University.\n15. ^ [***a***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-Caves_15-0) [***b***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-Caves_15-1)\n    [Caves, Carlton M.](https:\/\/en.wikipedia.org\/wiki\/Carlton_M._Caves \"Carlton M. Caves\") (2015). \"Quantum Information Science: Emerging No More\". In Kelley, Paul; Agrawal, Govind; Bass, Mike; Hecht, Jeff; Stroud, Carlos (eds.). *OSA Century of Optics*. [The Optical Society](https:\/\/en.wikipedia.org\/wiki\/The_Optical_Society \"The Optical Society\"). pp. 320–323\\. [arXiv](https:\/\/en.wikipedia.org\/wiki\/ArXiv_\\(identifier\\) \"ArXiv (identifier)\"):[1302\\.1864](https:\/\/arxiv.org\/abs\/1302.1864). [Bibcode](https:\/\/en.wikipedia.org\/wiki\/Bibcode_\\(identifier\\) \"Bibcode (identifier)\"):[2013arXiv1302.1864C](https:\/\/ui.adsabs.harvard.edu\/abs\/2013arXiv1302.1864C). [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-1-943580-04-0](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-1-943580-04-0 \"Special:BookSources\/978-1-943580-04-0\")\n    \n    .\n16. ^ [***a***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-wiseman15_16-0) [***b***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-wiseman15_16-1)\n    [Wiseman, Howard](https:\/\/en.wikipedia.org\/wiki\/Howard_M._Wiseman \"Howard M. Wiseman\") (October 2015). [\"Death by experiment for local realism\"](https:\/\/doi.org\/10.1038%2Fnature15631). *[Nature](https:\/\/en.wikipedia.org\/wiki\/Nature_\\(journal\\) \"Nature (journal)\")*. **526** (7575): 649–650\\. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1038\/nature15631](https:\/\/doi.org\/10.1038%2Fnature15631). [ISSN](https:\/\/en.wikipedia.org\/wiki\/ISSN_\\(identifier\\) \"ISSN (identifier)\") [0028-0836](https:\/\/search.worldcat.org\/issn\/0028-0836). [PMID](https:\/\/en.wikipedia.org\/wiki\/PMID_\\(identifier\\) \"PMID (identifier)\") [26503054](https:\/\/pubmed.ncbi.nlm.nih.gov\/26503054).\n17. ^ [***a***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-wolchover17_17-0) [***b***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-wolchover17_17-1)\n    [Wolchover, Natalie](https:\/\/en.wikipedia.org\/wiki\/Natalie_Wolchover \"Natalie Wolchover\") (7 February 2017). [\"Experiment Reaffirms Quantum Weirdness\"](https:\/\/www.quantamagazine.org\/20170207-bell-test-quantum-loophole\/). *[Quanta Magazine](https:\/\/en.wikipedia.org\/wiki\/Quanta_Magazine \"Quanta Magazine\")*. Retrieved 8 February 2020.\n18. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-18)**\n    [Baez, John C.](https:\/\/en.wikipedia.org\/wiki\/John_C._Baez \"John C. Baez\") (20 March 2020). [\"How to Learn Math and Physics\"](https:\/\/math.ucr.edu\/home\/baez\/books.html). *University of California, Riverside*. Retrieved 19 December 2020. \"there's no way to understand the interpretation of quantum mechanics without also being able to *solve quantum mechanics problems* – to understand the theory, you need to be able to use it (and vice versa)\"\n19. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-19)**\n    [Sagan, Carl](https:\/\/en.wikipedia.org\/wiki\/Carl_Sagan \"Carl Sagan\") (1996). [*The Demon-Haunted World: Science as a Candle in the Dark*](https:\/\/en.wikipedia.org\/wiki\/The_Demon-Haunted_World \"The Demon-Haunted World\"). Ballantine Books. p. 249. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [0-345-40946-9](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/0-345-40946-9 \"Special:BookSources\/0-345-40946-9\")\n    \n    . \"\"For most physics students, (the \"mathematical underpinning\" of quantum mechanics) might occupy them from, say, third grade to early graduate school – roughly 15 years. ... The job of the popularizer of science, trying to get across some idea of quantum mechanics to a general audience that has not gone through these initiation rites, is daunting. Indeed, there are no successful popularizations of quantum mechanics in my opinion – partly for this reason.\"\n20. ^ [***a***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-Holevo2001_20-0) [***b***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-Holevo2001_20-1) [***c***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-Holevo2001_20-2)\n    [Holevo, Alexander S.](https:\/\/en.wikipedia.org\/wiki\/Alexander_Holevo \"Alexander Holevo\") (2001). *Statistical Structure of Quantum Theory*. Lecture Notes in Physics. Springer. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [3-540-42082-7](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/3-540-42082-7 \"Special:BookSources\/3-540-42082-7\")\n    \n    . [OCLC](https:\/\/en.wikipedia.org\/wiki\/OCLC_\\(identifier\\) \"OCLC (identifier)\") [318268606](https:\/\/search.worldcat.org\/oclc\/318268606).\n21. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-21)**\n    Hall, Brian C. (2013). *Quantum Theory for Mathematicians*. [Graduate Texts in Mathematics](https:\/\/en.wikipedia.org\/wiki\/Graduate_Texts_in_Mathematics \"Graduate Texts in Mathematics\"). Vol. 267. Springer. p. 125. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-1-4614-7115-8](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-1-4614-7115-8 \"Special:BookSources\/978-1-4614-7115-8\")\n    \n    .\n22. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-22)**\n    Weinberg, Steven (2010). [*Dreams Of A Final Theory: The Search for The Fundamental Laws of Nature*](https:\/\/books.google.com\/books?id=OLrZkgPsZR0C). Random House. p. [82](https:\/\/books.google.com\/books?id=OLrZkgPsZR0C&pg=PT82). [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-1-4070-6396-6](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-1-4070-6396-6 \"Special:BookSources\/978-1-4070-6396-6\")\n    \n    .\n23. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-23)**\n    Zhang, Ruiqin; Deng, Conghao (1993). \"Exact solutions of the Schrödinger equation for some quantum-mechanical many-body systems\". *Physical Review A*. **47** (1): 71–77\\. [Bibcode](https:\/\/en.wikipedia.org\/wiki\/Bibcode_\\(identifier\\) \"Bibcode (identifier)\"):[1993PhRvA..47...71Z](https:\/\/ui.adsabs.harvard.edu\/abs\/1993PhRvA..47...71Z). [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1103\/PhysRevA.47.71](https:\/\/doi.org\/10.1103%2FPhysRevA.47.71). [ISSN](https:\/\/en.wikipedia.org\/wiki\/ISSN_\\(identifier\\) \"ISSN (identifier)\") [1050-2947](https:\/\/search.worldcat.org\/issn\/1050-2947). [PMID](https:\/\/en.wikipedia.org\/wiki\/PMID_\\(identifier\\) \"PMID (identifier)\") [9908895](https:\/\/pubmed.ncbi.nlm.nih.gov\/9908895).\n24. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-24)**\n    Li, Jing; Drummond, N. D.; Schuck, Peter; Olevano, Valerio (2019-04-01). [\"Comparing many-body approaches against the helium atom exact solution\"](https:\/\/doi.org\/10.21468%2FSciPostPhys.6.4.040). *SciPost Physics*. **6** (4): 40. [arXiv](https:\/\/en.wikipedia.org\/wiki\/ArXiv_\\(identifier\\) \"ArXiv (identifier)\"):[1801\\.09977](https:\/\/arxiv.org\/abs\/1801.09977). [Bibcode](https:\/\/en.wikipedia.org\/wiki\/Bibcode_\\(identifier\\) \"Bibcode (identifier)\"):[2019ScPP....6...40L](https:\/\/ui.adsabs.harvard.edu\/abs\/2019ScPP....6...40L). [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.21468\/SciPostPhys.6.4.040](https:\/\/doi.org\/10.21468%2FSciPostPhys.6.4.040). [ISSN](https:\/\/en.wikipedia.org\/wiki\/ISSN_\\(identifier\\) \"ISSN (identifier)\") [2542-4653](https:\/\/search.worldcat.org\/issn\/2542-4653).\n25. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-25)**\n    Drake, Gordon W. F. (2023). \"High Precision Calculations for Helium\". In Drake, Gordon W. F. (ed.). *Springer Handbook of Atomic, Molecular, and Optical Physics*. Springer Handbooks. Cham: Springer International Publishing. pp. 199–216\\. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1007\/978-3-030-73893-8\\_12](https:\/\/doi.org\/10.1007%2F978-3-030-73893-8_12). [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-3-030-73892-1](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-3-030-73892-1 \"Special:BookSources\/978-3-030-73892-1\")\n    \n    .\n26. ^ [***a***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-Cohen-Tannoudji_26-0) [***b***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-Cohen-Tannoudji_26-1) [***c***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-Cohen-Tannoudji_26-2) [***d***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-Cohen-Tannoudji_26-3)\n    [Cohen-Tannoudji, Claude](https:\/\/en.wikipedia.org\/wiki\/Claude_Cohen-Tannoudji \"Claude Cohen-Tannoudji\"); Diu, Bernard; Laloë, Franck (2005). *Quantum Mechanics*. Translated by Hemley, Susan Reid; Ostrowsky, Nicole; Ostrowsky, Dan. John Wiley & Sons. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [0-471-16433-X](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/0-471-16433-X \"Special:BookSources\/0-471-16433-X\")\n    \n    .\n27. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-L&L_27-0)**\n    [Landau, Lev D.](https:\/\/en.wikipedia.org\/wiki\/Lev_Landau \"Lev Landau\"); [Lifschitz, Evgeny M.](https:\/\/en.wikipedia.org\/wiki\/Evgeny_Lifshitz \"Evgeny Lifshitz\") (1977). [*Quantum Mechanics: Non-Relativistic Theory*](https:\/\/archive.org\/details\/QuantumMechanics_104). Vol. 3 (3rd ed.). [Pergamon Press](https:\/\/en.wikipedia.org\/wiki\/Pergamon_Press \"Pergamon Press\"). [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-0-08-020940-1](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-08-020940-1 \"Special:BookSources\/978-0-08-020940-1\")\n    \n    . [OCLC](https:\/\/en.wikipedia.org\/wiki\/OCLC_\\(identifier\\) \"OCLC (identifier)\") [2284121](https:\/\/search.worldcat.org\/oclc\/2284121).\n28. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-ballentine1970_28-0)**\n    Section 3.2 of\n    \n    Ballentine, Leslie E. (1970), \"The Statistical Interpretation of Quantum Mechanics\", *Reviews of Modern Physics*, **42** (4): 358–381, [Bibcode](https:\/\/en.wikipedia.org\/wiki\/Bibcode_\\(identifier\\) \"Bibcode (identifier)\"):[1970RvMP...42..358B](https:\/\/ui.adsabs.harvard.edu\/abs\/1970RvMP...42..358B), [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1103\/RevModPhys.42.358](https:\/\/doi.org\/10.1103%2FRevModPhys.42.358), [S2CID](https:\/\/en.wikipedia.org\/wiki\/S2CID_\\(identifier\\) \"S2CID (identifier)\") [120024263](https:\/\/api.semanticscholar.org\/CorpusID:120024263)\n    \n    . This fact is experimentally well-known for example in quantum optics; see e.g. chap. 2 and Fig. 2.1\n    \n    Leonhardt, Ulf (1997), [*Measuring the Quantum State of Light*](https:\/\/archive.org\/details\/measuringquantum0000leon), Cambridge: Cambridge University Press, [Bibcode](https:\/\/en.wikipedia.org\/wiki\/Bibcode_\\(identifier\\) \"Bibcode (identifier)\"):[1997mqsl.book.....L](https:\/\/ui.adsabs.harvard.edu\/abs\/1997mqsl.book.....L), [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [0-521-49730-2](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/0-521-49730-2 \"Special:BookSources\/0-521-49730-2\")\n    \n    .\n29. ^ [***a***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-:0_29-0) [***b***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-:0_29-1) [***c***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-:0_29-2)\n    [Nielsen, Michael A.](https:\/\/en.wikipedia.org\/wiki\/Michael_Nielsen \"Michael Nielsen\"); [Chuang, Isaac L.](https:\/\/en.wikipedia.org\/wiki\/Isaac_Chuang \"Isaac Chuang\") (2010). *Quantum Computation and Quantum Information* (2nd ed.). Cambridge University Press. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-1-107-00217-3](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-1-107-00217-3 \"Special:BookSources\/978-1-107-00217-3\")\n    \n    . [OCLC](https:\/\/en.wikipedia.org\/wiki\/OCLC_\\(identifier\\) \"OCLC (identifier)\") [844974180](https:\/\/search.worldcat.org\/oclc\/844974180).\n30. ^ [***a***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-:1_30-0) [***b***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-:1_30-1)\n    [Rieffel, Eleanor G.](https:\/\/en.wikipedia.org\/wiki\/Eleanor_Rieffel \"Eleanor Rieffel\"); Polak, Wolfgang H. (2011). [*Quantum Computing: A Gentle Introduction*](https:\/\/en.wikipedia.org\/wiki\/Quantum_Computing:_A_Gentle_Introduction \"Quantum Computing: A Gentle Introduction\"). MIT Press. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-0-262-01506-6](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-262-01506-6 \"Special:BookSources\/978-0-262-01506-6\")\n    \n    .\n31. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-wilde_31-0)**\n    Wilde, Mark M. (2017). *Quantum Information Theory* (2nd ed.). Cambridge University Press. [arXiv](https:\/\/en.wikipedia.org\/wiki\/ArXiv_\\(identifier\\) \"ArXiv (identifier)\"):[1106\\.1445](https:\/\/arxiv.org\/abs\/1106.1445). [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1017\/9781316809976.001](https:\/\/doi.org\/10.1017%2F9781316809976.001). [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-1-107-17616-4](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-1-107-17616-4 \"Special:BookSources\/978-1-107-17616-4\")\n    \n    . [OCLC](https:\/\/en.wikipedia.org\/wiki\/OCLC_\\(identifier\\) \"OCLC (identifier)\") [973404322](https:\/\/search.worldcat.org\/oclc\/973404322). [S2CID](https:\/\/en.wikipedia.org\/wiki\/S2CID_\\(identifier\\) \"S2CID (identifier)\") [2515538](https:\/\/api.semanticscholar.org\/CorpusID:2515538).\n32. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-32)**\n    Schlosshauer, Maximilian (October 2019). \"Quantum decoherence\". *Physics Reports*. **831**: 1–57\\. [arXiv](https:\/\/en.wikipedia.org\/wiki\/ArXiv_\\(identifier\\) \"ArXiv (identifier)\"):[1911\\.06282](https:\/\/arxiv.org\/abs\/1911.06282). [Bibcode](https:\/\/en.wikipedia.org\/wiki\/Bibcode_\\(identifier\\) \"Bibcode (identifier)\"):[2019PhR...831....1S](https:\/\/ui.adsabs.harvard.edu\/abs\/2019PhR...831....1S). [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1016\/j.physrep.2019.10.001](https:\/\/doi.org\/10.1016%2Fj.physrep.2019.10.001). [S2CID](https:\/\/en.wikipedia.org\/wiki\/S2CID_\\(identifier\\) \"S2CID (identifier)\") [208006050](https:\/\/api.semanticscholar.org\/CorpusID:208006050).\n33. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-33)**\n    [Rechenberg, Helmut](https:\/\/en.wikipedia.org\/wiki\/Helmut_Rechenberg \"Helmut Rechenberg\") (1987). [\"Erwin Schrödinger and the creation of wave mechanics\"](http:\/\/www.actaphys.uj.edu.pl\/fulltext?series=Reg&vol=19&page=683) (PDF). *[Acta Physica Polonica B](https:\/\/en.wikipedia.org\/wiki\/Acta_Physica_Polonica_B \"Acta Physica Polonica B\")*. **19** (8): 683–695. Retrieved 13 June 2016.\n34. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-34)**\n    Feynman, Richard P.; Hibbs, Albert R. (2005). Steyer, Daniel F. (ed.). *Quantum Mechanics and Path Integrals* (Emended ed.). McGraw-Hill. pp. v–vii. 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[doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1016\/0039-3681(85)90001-9](https:\/\/doi.org\/10.1016%2F0039-3681%2885%2990001-9).\n63. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-66)**\n    [Sauer, Tilman](https:\/\/en.wikipedia.org\/wiki\/Tilman_Sauer \"Tilman Sauer\") (1 December 2007). [\"An Einstein manuscript on the EPR paradox for spin observables\"](http:\/\/philsci-archive.pitt.edu\/3222\/). *Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics*. **38** (4): 879–887\\. [Bibcode](https:\/\/en.wikipedia.org\/wiki\/Bibcode_\\(identifier\\) \"Bibcode (identifier)\"):[2007SHPMP..38..879S](https:\/\/ui.adsabs.harvard.edu\/abs\/2007SHPMP..38..879S). [CiteSeerX](https:\/\/en.wikipedia.org\/wiki\/CiteSeerX_\\(identifier\\) \"CiteSeerX (identifier)\") [10\\.1.1.571.6089](https:\/\/citeseerx.ist.psu.edu\/viewdoc\/summary?doi=10.1.1.571.6089). [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1016\/j.shpsb.2007.03.002](https:\/\/doi.org\/10.1016%2Fj.shpsb.2007.03.002). [ISSN](https:\/\/en.wikipedia.org\/wiki\/ISSN_\\(identifier\\) \"ISSN (identifier)\") [1355-2198](https:\/\/search.worldcat.org\/issn\/1355-2198).\n64. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-67)**\n    Einstein, Albert (1949). \"Autobiographical Notes\". In Schilpp, Paul Arthur (ed.). *Albert Einstein: Philosopher-Scientist*. Open Court Publishing Company.\n65. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-69)**\n    [Bell, John Stewart](https:\/\/en.wikipedia.org\/wiki\/John_Stewart_Bell \"John Stewart Bell\") (1 November 1964). [\"On the Einstein Podolsky Rosen paradox\"](https:\/\/doi.org\/10.1103%2FPhysicsPhysiqueFizika.1.195). *[Physics Physique Fizika](https:\/\/en.wikipedia.org\/wiki\/Physics_Physique_Fizika \"Physics Physique Fizika\")*. **1** (3): 195–200\\. [Bibcode](https:\/\/en.wikipedia.org\/wiki\/Bibcode_\\(identifier\\) \"Bibcode (identifier)\"):[1964PhyNY...1..195B](https:\/\/ui.adsabs.harvard.edu\/abs\/1964PhyNY...1..195B). [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1103\/PhysicsPhysiqueFizika.1.195](https:\/\/doi.org\/10.1103%2FPhysicsPhysiqueFizika.1.195).\n66. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-70)**\n    Goldstein, Sheldon (2017). [\"Bohmian Mechanics\"](https:\/\/plato.stanford.edu\/entries\/qm-bohm\/). *Stanford Encyclopedia of Philosophy*. Metaphysics Research Lab, Stanford University.\n67. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-71)**\n    Barrett, Jeffrey (2018). [\"Everett's Relative-State Formulation of Quantum Mechanics\"](https:\/\/plato.stanford.edu\/entries\/qm-everett\/). In Zalta, Edward N. (ed.). *Stanford Encyclopedia of Philosophy*. Metaphysics Research Lab, Stanford University.\n68. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-dewitt73_72-0)**\n    [Everett, Hugh](https:\/\/en.wikipedia.org\/wiki\/Hugh_Everett_III \"Hugh Everett III\"); [Wheeler, J. A.](https:\/\/en.wikipedia.org\/wiki\/John_Archibald_Wheeler \"John Archibald Wheeler\"); [DeWitt, B. S.](https:\/\/en.wikipedia.org\/wiki\/Bryce_DeWitt \"Bryce DeWitt\"); [Cooper, L. N.](https:\/\/en.wikipedia.org\/wiki\/Leon_Cooper \"Leon Cooper\"); Van Vechten, D.; Graham, N. (1973). [DeWitt, Bryce](https:\/\/en.wikipedia.org\/wiki\/Bryce_DeWitt \"Bryce DeWitt\"); Graham, R. Neill (eds.). *The Many-Worlds Interpretation of Quantum Mechanics*. Princeton Series in Physics. Princeton, NJ: [Princeton University Press](https:\/\/en.wikipedia.org\/wiki\/Princeton_University_Press \"Princeton University Press\"). p. v. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [0-691-08131-X](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/0-691-08131-X \"Special:BookSources\/0-691-08131-X\")\n    \n    .\n69. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-wallace2003_73-0)**\n    Wallace, David (2003). \"Everettian Rationality: defending Deutsch's approach to probability in the Everett interpretation\". *Stud. Hist. Phil. Mod. Phys*. **34** (3): 415–438\\. [arXiv](https:\/\/en.wikipedia.org\/wiki\/ArXiv_\\(identifier\\) \"ArXiv (identifier)\"):[quant-ph\/0303050](https:\/\/arxiv.org\/abs\/quant-ph\/0303050). [Bibcode](https:\/\/en.wikipedia.org\/wiki\/Bibcode_\\(identifier\\) \"Bibcode (identifier)\"):[2003SHPMP..34..415W](https:\/\/ui.adsabs.harvard.edu\/abs\/2003SHPMP..34..415W). [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1016\/S1355-2198(03)00036-4](https:\/\/doi.org\/10.1016%2FS1355-2198%2803%2900036-4). [S2CID](https:\/\/en.wikipedia.org\/wiki\/S2CID_\\(identifier\\) \"S2CID (identifier)\") [1921913](https:\/\/api.semanticscholar.org\/CorpusID:1921913).\n70. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-ballentine1973_74-0)**\n    Ballentine, L. E. (1973). \"Can the statistical postulate of quantum theory be derived? – A critique of the many-universes interpretation\". *Foundations of Physics*. **3** (2): 229–240\\. [Bibcode](https:\/\/en.wikipedia.org\/wiki\/Bibcode_\\(identifier\\) \"Bibcode (identifier)\"):[1973FoPh....3..229B](https:\/\/ui.adsabs.harvard.edu\/abs\/1973FoPh....3..229B). [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1007\/BF00708440](https:\/\/doi.org\/10.1007%2FBF00708440). [S2CID](https:\/\/en.wikipedia.org\/wiki\/S2CID_\\(identifier\\) \"S2CID (identifier)\") [121747282](https:\/\/api.semanticscholar.org\/CorpusID:121747282).\n71. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-75)**\n    Landsman, N. P. (2008). [\"The Born rule and its interpretation\"](https:\/\/www.math.ru.nl\/~landsman\/Born.pdf) (PDF). In Weinert, F.; Hentschel, K.; Greenberger, D.; Falkenburg, B. (eds.). *Compendium of Quantum Physics*. Springer. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-3-540-70622-9](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-3-540-70622-9 \"Special:BookSources\/978-3-540-70622-9\")\n    \n    . \"The conclusion seems to be that no generally accepted derivation of the Born rule has been given to date, but this does not imply that such a derivation is impossible in principle.\"\n72. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-kent2009_76-0)**\n    [Kent, Adrian](https:\/\/en.wikipedia.org\/wiki\/Adrian_Kent \"Adrian Kent\") (2010). \"One world versus many: The inadequacy of Everettian accounts of evolution, probability, and scientific confirmation\". In S. Saunders; J. Barrett; A. Kent; D. Wallace (eds.). *Many Worlds? Everett, Quantum Theory and Reality*. Oxford University Press. [arXiv](https:\/\/en.wikipedia.org\/wiki\/ArXiv_\\(identifier\\) \"ArXiv (identifier)\"):[0905\\.0624](https:\/\/arxiv.org\/abs\/0905.0624). [Bibcode](https:\/\/en.wikipedia.org\/wiki\/Bibcode_\\(identifier\\) \"Bibcode (identifier)\"):[2009arXiv0905.0624K](https:\/\/ui.adsabs.harvard.edu\/abs\/2009arXiv0905.0624K).\n73. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-77)**\n    [Van Fraassen, Bas C.](https:\/\/en.wikipedia.org\/wiki\/Bas_van_Fraassen \"Bas van Fraassen\") (April 2010). [\"Rovelli's World\"](http:\/\/link.springer.com\/10.1007\/s10701-009-9326-5). *[Foundations of Physics](https:\/\/en.wikipedia.org\/wiki\/Foundations_of_Physics \"Foundations of Physics\")*. **40** (4): 390–417\\. [Bibcode](https:\/\/en.wikipedia.org\/wiki\/Bibcode_\\(identifier\\) \"Bibcode (identifier)\"):[2010FoPh...40..390V](https:\/\/ui.adsabs.harvard.edu\/abs\/2010FoPh...40..390V). [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1007\/s10701-009-9326-5](https:\/\/doi.org\/10.1007%2Fs10701-009-9326-5). [ISSN](https:\/\/en.wikipedia.org\/wiki\/ISSN_\\(identifier\\) \"ISSN (identifier)\") [0015-9018](https:\/\/search.worldcat.org\/issn\/0015-9018). [S2CID](https:\/\/en.wikipedia.org\/wiki\/S2CID_\\(identifier\\) \"S2CID (identifier)\") [17217776](https:\/\/api.semanticscholar.org\/CorpusID:17217776).\n74. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-78)**\n    [Calosi, Claudio](https:\/\/en.wikipedia.org\/wiki\/Claudio_Calosi_\\(philosopher\\) \"Claudio Calosi (philosopher)\"); Riedel, Timotheus (2024). [\"Relational Quantum Mechanics at the Crossroads\"](https:\/\/doi.org\/10.1007%2Fs10701-024-00810-5). *Foundations of Physics*. **54** (6): 74. [Bibcode](https:\/\/en.wikipedia.org\/wiki\/Bibcode_\\(identifier\\) \"Bibcode (identifier)\"):[2024FoPh...54...74C](https:\/\/ui.adsabs.harvard.edu\/abs\/2024FoPh...54...74C). [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1007\/s10701-024-00810-5](https:\/\/doi.org\/10.1007%2Fs10701-024-00810-5). [hdl](https:\/\/en.wikipedia.org\/wiki\/Hdl_\\(identifier\\) \"Hdl (identifier)\"):[10278\/5097808](https:\/\/hdl.handle.net\/10278%2F5097808).\n75. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-:23_79-0)**\n    Healey, Richard (2016). [\"Quantum-Bayesian and Pragmatist Views of Quantum Theory\"](https:\/\/plato.stanford.edu\/entries\/quantum-bayesian\/). In Zalta, Edward N. (ed.). *[Stanford Encyclopedia of Philosophy](https:\/\/en.wikipedia.org\/wiki\/Stanford_Encyclopedia_of_Philosophy \"Stanford Encyclopedia of Philosophy\")*. Metaphysics Research Lab, Stanford University.\n76. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-80)**\n    Stacey, Blake C. (2023). \"Toward a World Game-Flavored as a Hawk's Wing\". In Berghofer, Philipp; Wiltsche, Harald A. (eds.). *Phenomenology and QBism: New Approaches to Quantum Mechanics*. Routledge. pp. 49–77\\. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.4324\/9781003259008-3](https:\/\/doi.org\/10.4324%2F9781003259008-3). [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-1-032-19181-2](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-1-032-19181-2 \"Special:BookSources\/978-1-032-19181-2\")\n    \n    .\n77. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-Born_&_Wolf_81-0)**\n    [Born, Max](https:\/\/en.wikipedia.org\/wiki\/Max_Born \"Max Born\"); [Wolf, Emil](https:\/\/en.wikipedia.org\/wiki\/Emil_Wolf \"Emil Wolf\") (1999). [*Principles of Optics*](https:\/\/en.wikipedia.org\/wiki\/Principles_of_Optics \"Principles of Optics\"). Cambridge University Press. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [0-521-64222-1](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/0-521-64222-1 \"Special:BookSources\/0-521-64222-1\")\n    \n    . [OCLC](https:\/\/en.wikipedia.org\/wiki\/OCLC_\\(identifier\\) \"OCLC (identifier)\") [1151058062](https:\/\/search.worldcat.org\/oclc\/1151058062).\n78. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-82)**\n    Scheider, Walter (April 1986). [\"Bringing one of the great moments of science to the classroom\"](http:\/\/www.cavendishscience.org\/phys\/tyoung\/tyoung.htm). *[The Physics Teacher](https:\/\/en.wikipedia.org\/wiki\/The_Physics_Teacher \"The Physics Teacher\")*. **24** (4): 217–219\\. [Bibcode](https:\/\/en.wikipedia.org\/wiki\/Bibcode_\\(identifier\\) \"Bibcode (identifier)\"):[1986PhTea..24..217S](https:\/\/ui.adsabs.harvard.edu\/abs\/1986PhTea..24..217S). [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1119\/1.2341987](https:\/\/doi.org\/10.1119%2F1.2341987). [ISSN](https:\/\/en.wikipedia.org\/wiki\/ISSN_\\(identifier\\) \"ISSN (identifier)\") [0031-921X](https:\/\/search.worldcat.org\/issn\/0031-921X).\n79. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-Feynman-kinetic-theory_83-0)**\n    Feynman, Richard; Leighton, Robert; Sands, Matthew (1964). [*The Feynman Lectures on Physics*](https:\/\/feynmanlectures.caltech.edu\/I_40.html). Vol. 1. California Institute of Technology. Retrieved 30 September 2021.\n    \n    Reprinted, Addison-Wesley, 1989,\n    \n    [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-0-201-50064-6](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-201-50064-6 \"Special:BookSources\/978-0-201-50064-6\")\n80. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-84)**\n    Martin, Andre (1986), \"Cathode Ray Tubes for Industrial and Military Applications\", in Hawkes, Peter (ed.), *Advances in Electronics and Electron Physics, Volume 67*, Academic Press, p. 183, [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-0-08-057733-3](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-08-057733-3 \"Special:BookSources\/978-0-08-057733-3\")\n    \n    , \"Evidence for the existence of \"cathode-rays\" was first found by Plücker and Hittorf ...\"\n81. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-85)**\n    Dahl, Per F. (1997). [*Flash of the Cathode Rays: A History of J. J. Thomson's Electron*](https:\/\/books.google.com\/books?id=xUzaWGocMdMC). CRC Press. pp. 47–57\\. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-0-7503-0453-5](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-7503-0453-5 \"Special:BookSources\/978-0-7503-0453-5\")\n    \n    .\n82. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-86)**\n    [Mehra, J.](https:\/\/en.wikipedia.org\/wiki\/Jagdish_Mehra \"Jagdish Mehra\"); Rechenberg, H. (1982). *The Historical Development of Quantum Theory, Vol. 1: The Quantum Theory of Planck, Einstein, Bohr and Sommerfeld. Its Foundation and the Rise of Its Difficulties (1900–1925)*. New York: Springer-Verlag. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-0-387-90642-3](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-387-90642-3 \"Special:BookSources\/978-0-387-90642-3\")\n    \n    .\n83. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-87)**\n    [\"Quantum\"](http:\/\/www.merriam-webster.com\/dictionary\/quantum). *Merriam-Webster Dictionary*. [Archived](https:\/\/web.archive.org\/web\/20121026104456\/http:\/\/www.merriam-webster.com\/dictionary\/quantum) from the original on Oct 26, 2012. Retrieved 18 August 2012.\n84. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-88)**\n    [Kuhn, T. S.](https:\/\/en.wikipedia.org\/wiki\/Thomas_Samuel_Kuhn \"Thomas Samuel Kuhn\") (1978). *Black-body theory and the quantum discontinuity 1894–1912*. Oxford: Clarendon Press. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-0-19-502383-1](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-19-502383-1 \"Special:BookSources\/978-0-19-502383-1\")\n    \n    .\n85. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-Kragh_89-0)**\n    [Kragh, Helge](https:\/\/en.wikipedia.org\/wiki\/Helge_Kragh \"Helge Kragh\") (1 December 2000). [\"Max Planck: the reluctant revolutionary\"](https:\/\/physicsworld.com\/a\/max-planck-the-reluctant-revolutionary\/). *[Physics World](https:\/\/en.wikipedia.org\/wiki\/Physics_World \"Physics World\")*. Retrieved 12 December 2020.\n86. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-90)**\n    [Stachel, John](https:\/\/en.wikipedia.org\/wiki\/John_Stachel \"John Stachel\") (2009). \"Bohr and the Photon\". *Quantum Reality, Relativistic Causality, and Closing the Epistemic Circle*. The Western Ontario Series in Philosophy of Science. Vol. 73. Dordrecht: Springer. pp. 69–83\\. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1007\/978-1-4020-9107-0\\_5](https:\/\/doi.org\/10.1007%2F978-1-4020-9107-0_5). [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-1-4020-9106-3](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-1-4020-9106-3 \"Special:BookSources\/978-1-4020-9106-3\")\n    \n    .\n87. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-91)**\n    Einstein, Albert (1905). [\"Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt\"](https:\/\/doi.org\/10.1002%2Fandp.19053220607) \\[On a heuristic point of view concerning the production and transformation of light\\]. *[Annalen der Physik](https:\/\/en.wikipedia.org\/wiki\/Annalen_der_Physik \"Annalen der Physik\")* (in German). **17** (6): 132–148\\. [Bibcode](https:\/\/en.wikipedia.org\/wiki\/Bibcode_\\(identifier\\) \"Bibcode (identifier)\"):[1905AnP...322..132E](https:\/\/ui.adsabs.harvard.edu\/abs\/1905AnP...322..132E). [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1002\/andp.19053220607](https:\/\/doi.org\/10.1002%2Fandp.19053220607).\n    \n    Reprinted in\n    \n    [Stachel, John](https:\/\/en.wikipedia.org\/wiki\/John_Stachel \"John Stachel\"), ed. (1989). *The Collected Papers of Albert Einstein* (in German). Vol. 2. Princeton University Press. pp. 149–166\\.\n    \n    See also \"Einstein's early work on the quantum hypothesis\", ibid. pp. 134–148.\n88. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-92)**\n    [Einstein, Albert](https:\/\/en.wikipedia.org\/wiki\/Albert_Einstein \"Albert Einstein\") (1917). \"Zur Quantentheorie der Strahlung\" \\[On the Quantum Theory of Radiation\\]. *[Physikalische Zeitschrift](https:\/\/en.wikipedia.org\/wiki\/Physikalische_Zeitschrift \"Physikalische Zeitschrift\")* (in German). **18**: 121–128\\. [Bibcode](https:\/\/en.wikipedia.org\/wiki\/Bibcode_\\(identifier\\) \"Bibcode (identifier)\"):[1917PhyZ...18..121E](https:\/\/ui.adsabs.harvard.edu\/abs\/1917PhyZ...18..121E).\n    \n    Translated in\n    \n    Einstein, A. (1967). \"On the Quantum Theory of Radiation\". *The Old Quantum Theory*. Elsevier. pp. 167–183\\. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1016\/b978-0-08-012102-4.50018-8](https:\/\/doi.org\/10.1016%2Fb978-0-08-012102-4.50018-8). [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-0-08-012102-4](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-08-012102-4 \"Special:BookSources\/978-0-08-012102-4\")\n    \n    .\n89. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-93)**\n    [Ball, Philip](https:\/\/en.wikipedia.org\/wiki\/Philip_Ball \"Philip Ball\") (2017-08-31). [\"A century ago Einstein sparked the notion of the laser\"](https:\/\/physicsworld.com\/a\/a-century-ago-einstein-sparked-the-notion-of-the-laser\/). *[Physics World](https:\/\/en.wikipedia.org\/wiki\/Physics_World \"Physics World\")*. Retrieved 2024-03-23.\n90. ^ [***a***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-terHaar_94-0) [***b***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-terHaar_94-1)\n    ter Haar, D. (1967). [*The Old Quantum Theory*](https:\/\/archive.org\/details\/oldquantumtheory0000haar). Pergamon Press. pp. 3–75\\. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-0-08-012101-7](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-08-012101-7 \"Special:BookSources\/978-0-08-012101-7\")\n    \n    . [LCCN](https:\/\/en.wikipedia.org\/wiki\/LCCN_\\(identifier\\) \"LCCN (identifier)\") [66-29628](https:\/\/lccn.loc.gov\/66-29628).\n91. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-95)**\n    Bokulich, Alisa; Bokulich, Peter (2020-08-13). [\"Bohr's Correspondence Principle\"](https:\/\/plato.stanford.edu\/entries\/bohr-correspondence\/). In [Zalta, Edward N.](https:\/\/en.wikipedia.org\/wiki\/Edward_N._Zalta \"Edward N. Zalta\") (ed.). *[Stanford Encyclopedia of Philosophy](https:\/\/en.wikipedia.org\/wiki\/Stanford_Encyclopedia_of_Philosophy \"Stanford Encyclopedia of Philosophy\")*. [ISSN](https:\/\/en.wikipedia.org\/wiki\/ISSN_\\(identifier\\) \"ISSN (identifier)\") [1095-5054](https:\/\/search.worldcat.org\/issn\/1095-5054). [OCLC](https:\/\/en.wikipedia.org\/wiki\/OCLC_\\(identifier\\) \"OCLC (identifier)\") [429049174](https:\/\/search.worldcat.org\/oclc\/429049174).\n92. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-96)**\n    [\"Semi-classical approximation\"](https:\/\/web.archive.org\/web\/20221007190530\/https:\/\/encyclopediaofmath.org\/index.php?title=Semi-classical_approximation). *Encyclopedia of Mathematics*. Archived from [the original](https:\/\/www.encyclopediaofmath.org\/index.php?title=Semi-classical_approximation) on 7 October 2022. Retrieved 1 February 2020.\n93. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-97)**\n    [Sakurai, J. J.](https:\/\/en.wikipedia.org\/wiki\/J._J._Sakurai \"J. J. Sakurai\"); Napolitano, J. (2014). \"Quantum Dynamics\". [*Modern Quantum Mechanics*](https:\/\/en.wikipedia.org\/wiki\/Modern_Quantum_Mechanics \"Modern Quantum Mechanics\"). Pearson. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-1-292-02410-3](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-1-292-02410-3 \"Special:BookSources\/978-1-292-02410-3\")\n    \n    . [OCLC](https:\/\/en.wikipedia.org\/wiki\/OCLC_\\(identifier\\) \"OCLC (identifier)\") [929609283](https:\/\/search.worldcat.org\/oclc\/929609283).\n94. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-Aharoni_98-0)**\n    [Aharoni, Amikam](https:\/\/en.wikipedia.org\/wiki\/Amikam_Aharoni \"Amikam Aharoni\") (1996). [*Introduction to the Theory of Ferromagnetism*](https:\/\/archive.org\/details\/introductiontoth00ahar\/page\/6). [Clarendon Press](https:\/\/en.wikipedia.org\/wiki\/Clarendon_Press \"Clarendon Press\"). pp. [6–7](https:\/\/archive.org\/details\/introductiontoth00ahar\/page\/6). [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [0-19-851791-2](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/0-19-851791-2 \"Special:BookSources\/0-19-851791-2\")\n    \n    .\n95. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-Edwards79_99-0)** David Edwards, \"The Mathematical Foundations of Quantum Mechanics\", *Synthese*, Volume 42, Number 1\/September, 1979, pp. 1–70.\n96. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-Edwards81_100-0)** David Edwards, \"The Mathematical Foundations of Quantum Field Theory: Fermions, Gauge Fields, and Super-symmetry, Part I: Lattice Field Theories\", *International Journal of Theoretical Physics*, Vol. 20, No. 7 (1981).\n97. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-101)**\n    [Bernstein, Jeremy](https:\/\/en.wikipedia.org\/wiki\/Jeremy_Bernstein \"Jeremy Bernstein\") (November 2005). [\"Max Born and the quantum theory\"](https:\/\/doi.org\/10.1119%2F1.2060717). *[American Journal of Physics](https:\/\/en.wikipedia.org\/wiki\/American_Journal_of_Physics \"American Journal of Physics\")*. **73** (11): 999–1008\\. [Bibcode](https:\/\/en.wikipedia.org\/wiki\/Bibcode_\\(identifier\\) \"Bibcode (identifier)\"):[2005AmJPh..73..999B](https:\/\/ui.adsabs.harvard.edu\/abs\/2005AmJPh..73..999B). [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1119\/1.2060717](https:\/\/doi.org\/10.1119%2F1.2060717). [ISSN](https:\/\/en.wikipedia.org\/wiki\/ISSN_\\(identifier\\) \"ISSN (identifier)\") [0002-9505](https:\/\/search.worldcat.org\/issn\/0002-9505).\n98. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-pais1997_102-0)**\n    [Pais, Abraham](https:\/\/en.wikipedia.org\/wiki\/Abraham_Pais \"Abraham Pais\") (1997). [*A Tale of Two Continents: A Physicist's Life in a Turbulent World*](https:\/\/archive.org\/details\/taleoftwocontine00pais). Princeton, New Jersey: Princeton University Press. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [0-691-01243-1](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/0-691-01243-1 \"Special:BookSources\/0-691-01243-1\")\n    \n    .\n99. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-103)**\n    Van Hove, Leon (1958). [\"Von Neumann's contributions to quantum mechanics\"](https:\/\/www.ams.org\/journals\/bull\/1958-64-03\/S0002-9904-1958-10206-2\/S0002-9904-1958-10206-2.pdf) (PDF). *[Bulletin of the American Mathematical Society](https:\/\/en.wikipedia.org\/wiki\/Bulletin_of_the_American_Mathematical_Society \"Bulletin of the American Mathematical Society\")*. **64** (3): Part 2:95–99. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1090\/s0002-9904-1958-10206-2](https:\/\/doi.org\/10.1090%2Fs0002-9904-1958-10206-2). [Archived](https:\/\/web.archive.org\/web\/20240120073106\/https:\/\/www.ams.org\/journals\/bull\/1958-64-03\/S0002-9904-1958-10206-2\/S0002-9904-1958-10206-2.pdf) (PDF) from the original on Jan 20, 2024.\n100. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-feynman2015_104-0)**\n     [Feynman, Richard](https:\/\/en.wikipedia.org\/wiki\/Richard_Feynman \"Richard Feynman\"). [\"The Feynman Lectures on Physics Vol. III Ch. 21: The Schrödinger Equation in a Classical Context: A Seminar on Superconductivity, 21-4\"](https:\/\/feynmanlectures.caltech.edu\/III_21.html#Ch21-S5). [California Institute of Technology](https:\/\/en.wikipedia.org\/wiki\/California_Institute_of_Technology \"California Institute of Technology\"). Retrieved 24 November 2015. \"...it was long believed that the wave function of the Schrödinger equation would never have a macroscopic representation analogous to the macroscopic representation of the amplitude for photons. On the other hand, it is now realized that the phenomena of superconductivity presents us with just this situation.\"\n     \n     `{{cite web}}`: CS1 maint: url-status ([link](https:\/\/en.wikipedia.org\/wiki\/Category:CS1_maint:_url-status \"Category:CS1 maint: url-status\"))\n101. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-105)**\n     Packard, Richard (2006). [\"Berkeley Experiments on Superfluid Macroscopic Quantum Effects\"](https:\/\/web.archive.org\/web\/20151125112132\/http:\/\/research.physics.berkeley.edu\/packard\/publications\/Articles\/LT24_Berk_expts_on_macro_sup_effects.pdf) (PDF). Physics Department, University of California, Berkeley. Archived from [the original](http:\/\/physics.berkeley.edu\/sites\/default\/files\/_\/lt24_berk_expts_on_macro_sup_effects.pdf) (PDF) on 25 November 2015. Retrieved 24 November 2015.\n## Further reading\nThe following titles, all by working physicists, attempt to communicate quantum theory to lay people, using a minimum of technical apparatus:\n\n- [Chester, Marvin](https:\/\/en.wikipedia.org\/wiki\/Marvin_Chester \"Marvin Chester\") (1987). *Primer of Quantum Mechanics*. John Wiley.\n  [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  [0-486-42878-8](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/0-486-42878-8 \"Special:BookSources\/0-486-42878-8\")\n- [Cox, Brian](https:\/\/en.wikipedia.org\/wiki\/Brian_Cox_\\(physicist\\) \"Brian Cox (physicist)\"); [Forshaw, Jeff](https:\/\/en.wikipedia.org\/wiki\/Jeff_Forshaw \"Jeff Forshaw\") (2011). [*The Quantum Universe: Everything That Can Happen Does Happen*](https:\/\/en.wikipedia.org\/wiki\/The_Quantum_Universe \"The Quantum Universe\"). Allen Lane. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  \n  [978-1-84614-432-5](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-1-84614-432-5 \"Special:BookSources\/978-1-84614-432-5\")\n  \n  .\n- [Richard Feynman](https:\/\/en.wikipedia.org\/wiki\/Richard_Feynman \"Richard Feynman\"), 1985. *[QED: The Strange Theory of Light and Matter](https:\/\/en.wikipedia.org\/wiki\/QED:_The_Strange_Theory_of_Light_and_Matter \"QED: The Strange Theory of Light and Matter\")*, Princeton University Press.\n  [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  [0-691-08388-6](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/0-691-08388-6 \"Special:BookSources\/0-691-08388-6\")\n  . Four elementary lectures on quantum electrodynamics and [quantum field theory](https:\/\/en.wikipedia.org\/wiki\/Quantum_field_theory \"Quantum field theory\"), yet containing many insights for the expert.\n- [Ghirardi, GianCarlo](https:\/\/en.wikipedia.org\/wiki\/Giancarlo_Ghirardi \"Giancarlo Ghirardi\"), 2004. *Sneaking a Look at God's Cards*, Gerald Malsbary, trans. Princeton Univ. Press. The most technical of the works cited here. Passages using [algebra](https:\/\/en.wikipedia.org\/wiki\/Algebra \"Algebra\"), [trigonometry](https:\/\/en.wikipedia.org\/wiki\/Trigonometry \"Trigonometry\"), and [bra–ket notation](https:\/\/en.wikipedia.org\/wiki\/Bra%E2%80%93ket_notation \"Bra–ket notation\") can be passed over on a first reading.\n- [N. David Mermin](https:\/\/en.wikipedia.org\/wiki\/N._David_Mermin \"N. David Mermin\"), 1990, \"Spooky actions at a distance: mysteries of the QT\" in his *Boojums All the Way Through*. Cambridge University Press: 110–76.\n- [Victor Stenger](https:\/\/en.wikipedia.org\/wiki\/Victor_Stenger \"Victor Stenger\"), 2000. *Timeless Reality: Symmetry, Simplicity, and Multiple Universes*. Buffalo, New York: Prometheus Books. Chpts. 5–8. Includes cosmological and philosophical considerations.\n\nMore technical:\n\n- [Bernstein, Jeremy](https:\/\/en.wikipedia.org\/wiki\/Jeremy_Bernstein \"Jeremy Bernstein\") (2009). [*Quantum Leaps*](https:\/\/books.google.com\/books?id=j0Me3brYOL0C). Cambridge, Massachusetts: Belknap Press of Harvard University Press. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  \n  [978-0-674-03541-6](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-674-03541-6 \"Special:BookSources\/978-0-674-03541-6\")\n  \n  .\n- [Bohm, David](https:\/\/en.wikipedia.org\/wiki\/David_Bohm \"David Bohm\") (1989). [*Quantum Theory*](https:\/\/archive.org\/details\/quantumtheory0000bohm). Dover Publications. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  \n  [978-0-486-65969-5](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-486-65969-5 \"Special:BookSources\/978-0-486-65969-5\")\n  \n  .\n- [Binney, James](https:\/\/en.wikipedia.org\/wiki\/James_Binney \"James Binney\"); Skinner, David (2008). *The Physics of Quantum Mechanics*. Oxford University Press. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  \n  [978-0-19-968857-9](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-19-968857-9 \"Special:BookSources\/978-0-19-968857-9\")\n  \n  .\n- Eisberg, Robert; [Resnick, Robert](https:\/\/en.wikipedia.org\/wiki\/Robert_Resnick \"Robert Resnick\") (1985). [*Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles*](https:\/\/archive.org\/details\/quantumphysicsof00eisb) (2nd ed.). Wiley. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  \n  [978-0-471-87373-0](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-471-87373-0 \"Special:BookSources\/978-0-471-87373-0\")\n  \n  .\n- [Bryce DeWitt](https:\/\/en.wikipedia.org\/wiki\/Bryce_DeWitt \"Bryce DeWitt\"), R. Neill Graham, eds., 1973. *The Many-Worlds Interpretation of Quantum Mechanics*, Princeton Series in Physics, Princeton University Press.\n  [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  [0-691-08131-X](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/0-691-08131-X \"Special:BookSources\/0-691-08131-X\")\n- [Everett, Hugh](https:\/\/en.wikipedia.org\/wiki\/Hugh_Everett \"Hugh Everett\") (1957). \"Relative State Formulation of Quantum Mechanics\". *Reviews of Modern Physics*. **29** (3): 454–462\\. [Bibcode](https:\/\/en.wikipedia.org\/wiki\/Bibcode_\\(identifier\\) \"Bibcode (identifier)\"):[1957RvMP...29..454E](https:\/\/ui.adsabs.harvard.edu\/abs\/1957RvMP...29..454E). [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1103\/RevModPhys.29.454](https:\/\/doi.org\/10.1103%2FRevModPhys.29.454). [S2CID](https:\/\/en.wikipedia.org\/wiki\/S2CID_\\(identifier\\) \"S2CID (identifier)\") [17178479](https:\/\/api.semanticscholar.org\/CorpusID:17178479).\n- [Feynman, Richard P.](https:\/\/en.wikipedia.org\/wiki\/Richard_Feynman \"Richard Feynman\"); [Leighton, Robert B.](https:\/\/en.wikipedia.org\/wiki\/Robert_B._Leighton \"Robert B. Leighton\"); Sands, Matthew (1965). [*The Feynman Lectures on Physics*](https:\/\/en.wikipedia.org\/wiki\/The_Feynman_Lectures_on_Physics \"The Feynman Lectures on Physics\"). Vol. 1–3\\. Addison-Wesley. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  \n  [978-0-7382-0008-8](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-7382-0008-8 \"Special:BookSources\/978-0-7382-0008-8\")\n  \n  .\n- [D. Greenberger](https:\/\/en.wikipedia.org\/wiki\/Daniel_Greenberger \"Daniel Greenberger\"), [K. Hentschel](https:\/\/en.wikipedia.org\/wiki\/Klaus_Hentschel \"Klaus Hentschel\"), F. Weinert, eds., 2009. *Compendium of quantum physics, Concepts, experiments, history and philosophy*, Springer-Verlag, Berlin, Heidelberg. Short articles on many QM topics.\n- [Griffiths, David J.](https:\/\/en.wikipedia.org\/wiki\/David_J._Griffiths \"David J. Griffiths\") (2004). *[Introduction to Quantum Mechanics](https:\/\/en.wikipedia.org\/wiki\/Introduction_to_Quantum_Mechanics_\\(book\\) \"Introduction to Quantum Mechanics (book)\")* (2nd ed.). Prentice Hall. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  \n  [978-0-13-111892-8](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-13-111892-8 \"Special:BookSources\/978-0-13-111892-8\")\n  \n  . [OCLC](https:\/\/en.wikipedia.org\/wiki\/OCLC_\\(identifier\\) \"OCLC (identifier)\") [40251748](https:\/\/search.worldcat.org\/oclc\/40251748).\n  A standard undergraduate text.\n- [Max Jammer](https:\/\/en.wikipedia.org\/wiki\/Max_Jammer \"Max Jammer\"), 1966. *The Conceptual Development of Quantum Mechanics*. McGraw Hill.\n- [Hagen Kleinert](https:\/\/en.wikipedia.org\/wiki\/Hagen_Kleinert \"Hagen Kleinert\"), 2004. *Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets*, 3rd ed. Singapore: World Scientific. [Draft of 4th edition.](http:\/\/www.physik.fu-berlin.de\/~kleinert\/b5) [Archived](https:\/\/web.archive.org\/web\/20080615134934\/http:\/\/www.physik.fu-berlin.de\/~kleinert\/b5) 2008-06-15 at the [Wayback Machine](https:\/\/en.wikipedia.org\/wiki\/Wayback_Machine \"Wayback Machine\")\n- Landau, L. D.; Lifshitz, E. M. (1977). *Quantum Mechanics: Non-Relativistic Theory*. Vol. 3 (3rd ed.). [Pergamon Press](https:\/\/en.wikipedia.org\/wiki\/Pergamon_Press \"Pergamon Press\"). [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  \n  [978-0-08-020940-1](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-08-020940-1 \"Special:BookSources\/978-0-08-020940-1\")\n  \n  .\n  [Online copy](https:\/\/archive.org\/details\/QuantumMechanics_104)\n- [Liboff, Richard L.](https:\/\/en.wikipedia.org\/wiki\/Liboff,_Richard \"Liboff, Richard\") (2002). *Introductory Quantum Mechanics*. Addison-Wesley. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  \n  [978-0-8053-8714-8](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-8053-8714-8 \"Special:BookSources\/978-0-8053-8714-8\")\n  \n  .\n- Gunther Ludwig, 1968. *Wave Mechanics*. London: Pergamon Press.\n  [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  [0-08-203204-1](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/0-08-203204-1 \"Special:BookSources\/0-08-203204-1\")\n- [George Mackey](https:\/\/en.wikipedia.org\/wiki\/George_Mackey \"George Mackey\") (2004). *The mathematical foundations of quantum mechanics*. Dover Publications.\n  [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  [0-486-43517-2](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/0-486-43517-2 \"Special:BookSources\/0-486-43517-2\")\n  .\n- [Merzbacher, Eugen](https:\/\/en.wikipedia.org\/wiki\/Eugen_Merzbacher \"Eugen Merzbacher\") (1998). *Quantum Mechanics*. Wiley, John & Sons, Inc. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  \n  [978-0-471-88702-7](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-471-88702-7 \"Special:BookSources\/978-0-471-88702-7\")\n  \n  .\n- [Albert Messiah](https:\/\/en.wikipedia.org\/wiki\/Albert_Messiah \"Albert Messiah\"), 1966. *Quantum Mechanics* (Vol. I), English translation from French by G. M. Temmer. North Holland, John Wiley & Sons. Cf. chpt. IV, section III.\n- [Omnès, Roland](https:\/\/en.wikipedia.org\/wiki\/Roland_Omn%C3%A8s \"Roland Omnès\") (1999). [*Understanding Quantum Mechanics*](https:\/\/archive.org\/details\/understandingqua00omne). Princeton University Press. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  \n  [978-0-691-00435-8](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-691-00435-8 \"Special:BookSources\/978-0-691-00435-8\")\n  \n  . [OCLC](https:\/\/en.wikipedia.org\/wiki\/OCLC_\\(identifier\\) \"OCLC (identifier)\") [39849482](https:\/\/search.worldcat.org\/oclc\/39849482).\n- [Scerri, Eric. R.](https:\/\/en.wikipedia.org\/wiki\/Eric_R._Scerri \"Eric R. Scerri\") (2006). *The Periodic Table: Its Story and Its Significance*. Oxford University Press. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  \n  [0-19-530573-6](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/0-19-530573-6 \"Special:BookSources\/0-19-530573-6\")\n  \n  .\n  Considers the extent to which chemistry and the periodic system have been reduced to quantum mechanics.\n- [Schiff, Leonard I.](https:\/\/en.wikipedia.org\/wiki\/Leonard_I._Schiff \"Leonard I. Schiff\") (1955). *Quantum Mechanics*. McGraw Hill.\n- [Shankar, R.](https:\/\/en.wikipedia.org\/wiki\/Ramamurti_Shankar \"Ramamurti Shankar\") (1994). *Principles of Quantum Mechanics*. Springer. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  \n  [978-0-306-44790-7](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-306-44790-7 \"Special:BookSources\/978-0-306-44790-7\")\n  \n  .\n- [Stone, A. Douglas](https:\/\/en.wikipedia.org\/wiki\/A._Douglas_Stone \"A. Douglas Stone\") (2013). [*Einstein and the Quantum*](https:\/\/archive.org\/details\/einsteinquantumq0000ston). Princeton University Press. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  \n  [978-0-691-13968-5](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-691-13968-5 \"Special:BookSources\/978-0-691-13968-5\")\n  \n  .\n- *What is Quantum Mechanics? A Physics Adventure*. Boston: [Transnational College](https:\/\/en.wikipedia.org\/w\/index.php?title=Transnational_College&action=edit&redlink=1 \"Transnational College (page does not exist)\"), Language Research Foundation. 1996. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  \n  [978-0-9643504-1-0](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-9643504-1-0 \"Special:BookSources\/978-0-9643504-1-0\")\n  \n  . [OCLC](https:\/\/en.wikipedia.org\/wiki\/OCLC_\\(identifier\\) \"OCLC (identifier)\") [34661512](https:\/\/search.worldcat.org\/oclc\/34661512).\n- [Veltman, Martinus J. G.](https:\/\/en.wikipedia.org\/wiki\/Martinus_J._G._Veltman \"Martinus J. G. Veltman\") (2003), *Facts and Mysteries in Elementary Particle Physics*.\n## External links\n**Quantum mechanics** at Wikipedia's [sister projects](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:Wikimedia_sister_projects \"Wikipedia:Wikimedia sister projects\")\n\n- [![Wiktionary logo](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/9\/99\/Wiktionary-logo-en-v2.svg\/40px-Wiktionary-logo-en-v2.svg.png)](https:\/\/en.wikipedia.org\/wiki\/File:Wiktionary-logo-en-v2.svg)[Definitions](https:\/\/en.wiktionary.org\/wiki\/Special:Search\/Quantum_mechanics \"wikt:Special:Search\/Quantum mechanics\") from Wiktionary\n- [![Wikimedia Commons logo](https:\/\/upload.wikimedia.org\/wikipedia\/en\/thumb\/4\/4a\/Commons-logo.svg\/20px-Commons-logo.svg.png)](https:\/\/en.wikipedia.org\/wiki\/File:Commons-logo.svg)[Media](https:\/\/commons.wikimedia.org\/wiki\/Category:Quantum_mechanics \"c:Category:Quantum mechanics\") from Commons\n- [![Wikinews logo](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/2\/24\/Wikinews-logo.svg\/40px-Wikinews-logo.svg.png)](https:\/\/en.wikipedia.org\/wiki\/File:Wikinews-logo.svg)[News](https:\/\/en.wikinews.org\/wiki\/Special:Search\/Quantum_mechanics \"n:Special:Search\/Quantum mechanics\") from Wikinews\n- ![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/f\/fa\/Wikiquote-logo.svg\/40px-Wikiquote-logo.svg.png)[Quotations](https:\/\/en.wikiquote.org\/wiki\/Quantum_mechanics \"q:Quantum mechanics\") from Wikiquote\n- [![Wikisource logo](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/4\/4c\/Wikisource-logo.svg\/40px-Wikisource-logo.svg.png)](https:\/\/en.wikipedia.org\/wiki\/File:Wikisource-logo.svg)[Texts](https:\/\/en.wikisource.org\/wiki\/Quantum_mechanics \"s:Quantum mechanics\") from Wikisource\n- [![Wikibooks logo](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/f\/fa\/Wikibooks-logo.svg\/40px-Wikibooks-logo.svg.png)](https:\/\/en.wikipedia.org\/wiki\/File:Wikibooks-logo.svg)[Textbooks](https:\/\/en.wikibooks.org\/wiki\/Quantum_Mechanics \"b:Quantum Mechanics\") from Wikibooks\n- [![Wikiversity logo](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/0\/0b\/Wikiversity_logo_2017.svg\/40px-Wikiversity_logo_2017.svg.png)](https:\/\/en.wikipedia.org\/wiki\/File:Wikiversity_logo_2017.svg)[Resources](https:\/\/en.wikiversity.org\/wiki\/Quantum_mechanics \"v:Quantum mechanics\") from Wikiversity\n\n- [Introduction to quantum mechanics by Timon Idema](https:\/\/interactivetextbooks.tudelft.nl\/introduction-to-quantum-mechanics\/)\n- [Quantum Physics Made Relatively Simple](https:\/\/bethe.cornell.edu\/): three video lectures by [Hans Bethe](https:\/\/en.wikipedia.org\/wiki\/Hans_Bethe \"Hans Bethe\").\n\n**Course material**\n\n- [Quantum Cook Book](http:\/\/oyc.yale.edu\/sites\/default\/files\/notes_quantum_cookbook.pdf) and [PHYS 201: Fundamentals of Physics II](http:\/\/oyc.yale.edu\/physics\/phys-201#sessions) by [Ramamurti Shankar](https:\/\/en.wikipedia.org\/wiki\/Ramamurti_Shankar \"Ramamurti Shankar\"), Yale OpenCourseware.\n- *[Modern Physics: With waves, thermodynamics, and optics](https:\/\/lightandmatter.com\/mod\/)* – an online textbook.\n- [MIT OpenCourseWare](https:\/\/en.wikipedia.org\/wiki\/MIT_OpenCourseWare \"MIT OpenCourseWare\"): [Chemistry](https:\/\/ocw.mit.edu\/courses\/chemistry\/) and [Physics](https:\/\/ocw.mit.edu\/courses\/physics\/). See [8\\.04](https:\/\/ocw.mit.edu\/courses\/physics\/8-04-quantum-physics-i-spring-2016\/), [8\\.05](https:\/\/ocw.mit.edu\/courses\/physics\/8-05-quantum-physics-ii-fall-2013\/index.htm) and [8\\.06](https:\/\/ocw.mit.edu\/courses\/physics\/8-06-quantum-physics-iii-spring-2018\/index.htm).\n- [⁠5\\+1\/2⁠ Examples in Quantum Mechanics](http:\/\/physics.csbsju.edu\/QM\/).\n\n**Philosophy**\n\n- Ismael, Jenann. [\"Quantum Mechanics\"](https:\/\/plato.stanford.edu\/entries\/qm\/). In [Zalta, Edward N.](https:\/\/en.wikipedia.org\/wiki\/Edward_N._Zalta \"Edward N. Zalta\") (ed.). *[Stanford Encyclopedia of Philosophy](https:\/\/en.wikipedia.org\/wiki\/Stanford_Encyclopedia_of_Philosophy \"Stanford Encyclopedia of Philosophy\")*. [ISSN](https:\/\/en.wikipedia.org\/wiki\/ISSN_\\(identifier\\) \"ISSN (identifier)\") [1095-5054](https:\/\/search.worldcat.org\/issn\/1095-5054). [OCLC](https:\/\/en.wikipedia.org\/wiki\/OCLC_\\(identifier\\) \"OCLC (identifier)\") [429049174](https:\/\/search.worldcat.org\/oclc\/429049174).\n- [Zalta, Edward N.](https:\/\/en.wikipedia.org\/wiki\/Edward_N._Zalta \"Edward N. Zalta\") (ed.). [\"Philosophical Issues in Quantum Theory\"](https:\/\/plato.stanford.edu\/entries\/qt-issues\/). *[Stanford Encyclopedia of Philosophy](https:\/\/en.wikipedia.org\/wiki\/Stanford_Encyclopedia_of_Philosophy \"Stanford Encyclopedia of Philosophy\")*. [ISSN](https:\/\/en.wikipedia.org\/wiki\/ISSN_\\(identifier\\) \"ISSN (identifier)\") [1095-5054](https:\/\/search.worldcat.org\/issn\/1095-5054). [OCLC](https:\/\/en.wikipedia.org\/wiki\/OCLC_\\(identifier\\) \"OCLC (identifier)\") [429049174](https:\/\/search.worldcat.org\/oclc\/429049174).\n\n| [v](https:\/\/en.wikipedia.org\/wiki\/Template:Quantum_mechanics_topics \"Template:Quantum mechanics topics\") [t](https:\/\/en.wikipedia.org\/wiki\/Template_talk:Quantum_mechanics_topics \"Template talk:Quantum mechanics topics\") [e](https:\/\/en.wikipedia.org\/wiki\/Special:EditPage\/Template:Quantum_mechanics_topics \"Special:EditPage\/Template:Quantum mechanics topics\")[Quantum mechanics]() |   |\n|---|---|\n| Background | [Introduction](https:\/\/en.wikipedia.org\/wiki\/Introduction_to_quantum_mechanics \"Introduction to quantum mechanics\") [History](https:\/\/en.wikipedia.org\/wiki\/History_of_quantum_mechanics \"History of quantum mechanics\") [Timeline](https:\/\/en.wikipedia.org\/wiki\/Timeline_of_quantum_mechanics \"Timeline of quantum mechanics\") [Classical mechanics](https:\/\/en.wikipedia.org\/wiki\/Classical_mechanics \"Classical mechanics\") [Old quantum theory](https:\/\/en.wikipedia.org\/wiki\/Old_quantum_theory \"Old quantum theory\") [Glossary](https:\/\/en.wikipedia.org\/wiki\/Glossary_of_elementary_quantum_mechanics \"Glossary of elementary quantum mechanics\") |\n| Fundamentals | [Born rule](https:\/\/en.wikipedia.org\/wiki\/Born_rule \"Born rule\") [Bra–ket notation](https:\/\/en.wikipedia.org\/wiki\/Bra%E2%80%93ket_notation \"Bra–ket notation\") [Complementarity](https:\/\/en.wikipedia.org\/wiki\/Complementarity_\\(physics\\) \"Complementarity (physics)\") [Density matrix](https:\/\/en.wikipedia.org\/wiki\/Density_matrix \"Density matrix\") [Energy level](https:\/\/en.wikipedia.org\/wiki\/Energy_level \"Energy level\") [Ground state](https:\/\/en.wikipedia.org\/wiki\/Ground_state \"Ground state\") [Excited state](https:\/\/en.wikipedia.org\/wiki\/Excited_state \"Excited state\") [Degenerate levels](https:\/\/en.wikipedia.org\/wiki\/Degenerate_energy_levels \"Degenerate energy levels\") [Zero-point energy](https:\/\/en.wikipedia.org\/wiki\/Zero-point_energy \"Zero-point energy\") [Entanglement](https:\/\/en.wikipedia.org\/wiki\/Quantum_entanglement \"Quantum entanglement\") [Hamiltonian](https:\/\/en.wikipedia.org\/wiki\/Hamiltonian_\\(quantum_mechanics\\) \"Hamiltonian (quantum mechanics)\") [Interference](https:\/\/en.wikipedia.org\/wiki\/Wave_interference \"Wave interference\") [Decoherence](https:\/\/en.wikipedia.org\/wiki\/Quantum_decoherence \"Quantum decoherence\") [Measurement](https:\/\/en.wikipedia.org\/wiki\/Measurement_in_quantum_mechanics \"Measurement in quantum mechanics\") [Nonlocality](https:\/\/en.wikipedia.org\/wiki\/Quantum_nonlocality \"Quantum nonlocality\") [Quantum state](https:\/\/en.wikipedia.org\/wiki\/Quantum_state \"Quantum state\") [quantum jump](https:\/\/en.wikipedia.org\/wiki\/Quantum_jump \"Quantum jump\") [Superposition](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition \"Quantum superposition\") [Tunnelling](https:\/\/en.wikipedia.org\/wiki\/Quantum_tunnelling \"Quantum tunnelling\") [Scattering theory](https:\/\/en.wikipedia.org\/wiki\/Scattering#Theory \"Scattering\") [Symmetry in quantum mechanics](https:\/\/en.wikipedia.org\/wiki\/Symmetry_in_quantum_mechanics \"Symmetry in quantum mechanics\") [Uncertainty](https:\/\/en.wikipedia.org\/wiki\/Uncertainty_principle \"Uncertainty principle\") [Wave function](https:\/\/en.wikipedia.org\/wiki\/Wave_function \"Wave function\") [Collapse](https:\/\/en.wikipedia.org\/wiki\/Wave_function_collapse \"Wave function collapse\") [Wave–particle duality](https:\/\/en.wikipedia.org\/wiki\/Wave%E2%80%93particle_duality \"Wave–particle duality\") |\n| Formulations | [Formulations](https:\/\/en.wikipedia.org\/wiki\/Mathematical_formulation_of_quantum_mechanics \"Mathematical formulation of quantum mechanics\") [Heisenberg](https:\/\/en.wikipedia.org\/wiki\/Heisenberg_picture \"Heisenberg picture\") [Interaction](https:\/\/en.wikipedia.org\/wiki\/Interaction_picture \"Interaction picture\") [Matrix mechanics](https:\/\/en.wikipedia.org\/wiki\/Matrix_mechanics \"Matrix mechanics\") [Schrödinger](https:\/\/en.wikipedia.org\/wiki\/Schr%C3%B6dinger_picture \"Schrödinger picture\") [Path integral formulation](https:\/\/en.wikipedia.org\/wiki\/Path_integral_formulation \"Path integral formulation\") [Phase space](https:\/\/en.wikipedia.org\/wiki\/Phase-space_formulation \"Phase-space formulation\") |\n| Equations | [Klein–Gordon](https:\/\/en.wikipedia.org\/wiki\/Klein%E2%80%93Gordon_equation \"Klein–Gordon equation\") [Dirac](https:\/\/en.wikipedia.org\/wiki\/Dirac_equation \"Dirac equation\") [Weyl](https:\/\/en.wikipedia.org\/wiki\/Weyl_equation \"Weyl equation\") [Majorana](https:\/\/en.wikipedia.org\/wiki\/Majorana_equation \"Majorana equation\") [Rarita–Schwinger](https:\/\/en.wikipedia.org\/wiki\/Rarita%E2%80%93Schwinger_equation \"Rarita–Schwinger equation\") [Pauli](https:\/\/en.wikipedia.org\/wiki\/Pauli_equation \"Pauli equation\") [Rydberg](https:\/\/en.wikipedia.org\/wiki\/Rydberg_formula \"Rydberg formula\") [Schrödinger](https:\/\/en.wikipedia.org\/wiki\/Schr%C3%B6dinger_equation \"Schrödinger equation\") |\n| [Interpretations](https:\/\/en.wikipedia.org\/wiki\/Interpretations_of_quantum_mechanics \"Interpretations of quantum mechanics\") | [Bayesian](https:\/\/en.wikipedia.org\/wiki\/Quantum_Bayesianism \"Quantum Bayesianism\") [Consciousness causes collapse](https:\/\/en.wikipedia.org\/wiki\/Consciousness_causes_collapse \"Consciousness causes collapse\") [Consistent histories](https:\/\/en.wikipedia.org\/wiki\/Consistent_histories \"Consistent histories\") [Copenhagen](https:\/\/en.wikipedia.org\/wiki\/Copenhagen_interpretation \"Copenhagen interpretation\") [de Broglie–Bohm](https:\/\/en.wikipedia.org\/wiki\/De_Broglie%E2%80%93Bohm_theory \"De Broglie–Bohm theory\") [Ensemble](https:\/\/en.wikipedia.org\/wiki\/Ensemble_interpretation \"Ensemble interpretation\") [Hidden-variable](https:\/\/en.wikipedia.org\/wiki\/Hidden-variable_theory \"Hidden-variable theory\") [Local](https:\/\/en.wikipedia.org\/wiki\/Local_hidden-variable_theory \"Local hidden-variable theory\") [Superdeterminism](https:\/\/en.wikipedia.org\/wiki\/Superdeterminism \"Superdeterminism\") [Many-worlds](https:\/\/en.wikipedia.org\/wiki\/Many-worlds_interpretation \"Many-worlds interpretation\") [Objective collapse](https:\/\/en.wikipedia.org\/wiki\/Objective-collapse_theory \"Objective-collapse theory\") [Quantum logic](https:\/\/en.wikipedia.org\/wiki\/Quantum_logic \"Quantum logic\") [Relational](https:\/\/en.wikipedia.org\/wiki\/Relational_quantum_mechanics \"Relational quantum mechanics\") [Transactional](https:\/\/en.wikipedia.org\/wiki\/Transactional_interpretation \"Transactional interpretation\") |\n| Experiments | [Bell test](https:\/\/en.wikipedia.org\/wiki\/Bell_test \"Bell test\") [Davisson–Germer](https:\/\/en.wikipedia.org\/wiki\/Davisson%E2%80%93Germer_experiment \"Davisson–Germer experiment\") [Delayed-choice quantum eraser](https:\/\/en.wikipedia.org\/wiki\/Delayed-choice_quantum_eraser \"Delayed-choice quantum eraser\") [Double-slit](https:\/\/en.wikipedia.org\/wiki\/Double-slit_experiment \"Double-slit experiment\") [Franck–Hertz](https:\/\/en.wikipedia.org\/wiki\/Franck%E2%80%93Hertz_experiment \"Franck–Hertz experiment\") [Mach–Zehnder interferometer](https:\/\/en.wikipedia.org\/wiki\/Mach%E2%80%93Zehnder_interferometer \"Mach–Zehnder interferometer\") [Elitzur–Vaidman](https:\/\/en.wikipedia.org\/wiki\/Elitzur%E2%80%93Vaidman_bomb_tester \"Elitzur–Vaidman bomb tester\") [Popper](https:\/\/en.wikipedia.org\/wiki\/Popper%27s_experiment \"Popper's experiment\") [Quantum eraser](https:\/\/en.wikipedia.org\/wiki\/Quantum_eraser_experiment \"Quantum eraser experiment\") [Stern–Gerlach](https:\/\/en.wikipedia.org\/wiki\/Stern%E2%80%93Gerlach_experiment \"Stern–Gerlach experiment\") [Wheeler's delayed choice](https:\/\/en.wikipedia.org\/wiki\/Wheeler%27s_delayed-choice_experiment \"Wheeler's delayed-choice experiment\") |\n| [Science](https:\/\/en.wikipedia.org\/wiki\/Nanotechnology \"Nanotechnology\") | [Quantum biology](https:\/\/en.wikipedia.org\/wiki\/Quantum_biology \"Quantum biology\") [Quantum chemistry](https:\/\/en.wikipedia.org\/wiki\/Quantum_chemistry \"Quantum chemistry\") [Quantum chaos](https:\/\/en.wikipedia.org\/wiki\/Quantum_chaos \"Quantum chaos\") [Quantum cosmology](https:\/\/en.wikipedia.org\/wiki\/Quantum_cosmology \"Quantum cosmology\") [Quantum differential calculus](https:\/\/en.wikipedia.org\/wiki\/Quantum_differential_calculus \"Quantum differential calculus\") [Quantum dynamics](https:\/\/en.wikipedia.org\/wiki\/Quantum_dynamics \"Quantum dynamics\") [Quantum geometry](https:\/\/en.wikipedia.org\/wiki\/Quantum_geometry \"Quantum geometry\") [Quantum measurement problem](https:\/\/en.wikipedia.org\/wiki\/Measurement_problem \"Measurement problem\") [Quantum mind](https:\/\/en.wikipedia.org\/wiki\/Quantum_mind \"Quantum mind\") [Quantum stochastic calculus](https:\/\/en.wikipedia.org\/wiki\/Quantum_stochastic_calculus \"Quantum stochastic calculus\") [Quantum spacetime](https:\/\/en.wikipedia.org\/wiki\/Quantum_spacetime \"Quantum spacetime\") |\n| [Technology](https:\/\/en.wikipedia.org\/wiki\/Quantum_engineering \"Quantum engineering\") | [Quantum algorithms](https:\/\/en.wikipedia.org\/wiki\/Quantum_algorithm \"Quantum algorithm\") [Quantum amplifier](https:\/\/en.wikipedia.org\/wiki\/Quantum_amplifier \"Quantum amplifier\") [Quantum bus](https:\/\/en.wikipedia.org\/wiki\/Quantum_bus \"Quantum bus\") [Quantum cellular automata](https:\/\/en.wikipedia.org\/wiki\/Quantum_cellular_automaton \"Quantum cellular automaton\") [Quantum finite automata](https:\/\/en.wikipedia.org\/wiki\/Quantum_finite_automaton \"Quantum finite automaton\") [Quantum channel](https:\/\/en.wikipedia.org\/wiki\/Quantum_channel \"Quantum channel\") [Quantum circuit](https:\/\/en.wikipedia.org\/wiki\/Quantum_circuit \"Quantum circuit\") [Quantum complexity theory](https:\/\/en.wikipedia.org\/wiki\/Quantum_complexity_theory \"Quantum complexity theory\") [Quantum computing](https:\/\/en.wikipedia.org\/wiki\/Quantum_computing \"Quantum computing\") [Timeline](https:\/\/en.wikipedia.org\/wiki\/Timeline_of_quantum_computing_and_communication \"Timeline of quantum computing and communication\") [Quantum cryptography](https:\/\/en.wikipedia.org\/wiki\/Quantum_cryptography \"Quantum cryptography\") [Quantum electronics](https:\/\/en.wikipedia.org\/wiki\/Quantum_optics#Quantum_electronics \"Quantum optics\") [Quantum error correction](https:\/\/en.wikipedia.org\/wiki\/Quantum_error_correction \"Quantum error correction\") [Quantum imaging](https:\/\/en.wikipedia.org\/wiki\/Quantum_imaging \"Quantum imaging\") [Quantum image processing](https:\/\/en.wikipedia.org\/wiki\/Quantum_image_processing \"Quantum image processing\") [Quantum information](https:\/\/en.wikipedia.org\/wiki\/Quantum_information \"Quantum information\") [Quantum key distribution](https:\/\/en.wikipedia.org\/wiki\/Quantum_key_distribution \"Quantum key distribution\") [Quantum logic](https:\/\/en.wikipedia.org\/wiki\/Quantum_logic \"Quantum logic\") [Quantum logic gates](https:\/\/en.wikipedia.org\/wiki\/Quantum_logic_gate \"Quantum logic gate\") [Quantum machine](https:\/\/en.wikipedia.org\/wiki\/Quantum_machine \"Quantum machine\") [Quantum machine learning](https:\/\/en.wikipedia.org\/wiki\/Quantum_machine_learning \"Quantum machine learning\") [Quantum metamaterial](https:\/\/en.wikipedia.org\/wiki\/Quantum_metamaterial \"Quantum metamaterial\") [Quantum metrology](https:\/\/en.wikipedia.org\/wiki\/Quantum_metrology \"Quantum metrology\") [Quantum network](https:\/\/en.wikipedia.org\/wiki\/Quantum_network \"Quantum network\") [Quantum neural network](https:\/\/en.wikipedia.org\/wiki\/Quantum_neural_network \"Quantum neural network\") [Quantum optics](https:\/\/en.wikipedia.org\/wiki\/Quantum_optics \"Quantum optics\") [Quantum programming](https:\/\/en.wikipedia.org\/wiki\/Quantum_programming \"Quantum programming\") [Quantum sensing](https:\/\/en.wikipedia.org\/wiki\/Quantum_sensor \"Quantum sensor\") [Quantum simulator](https:\/\/en.wikipedia.org\/wiki\/Quantum_simulator \"Quantum simulator\") [Quantum teleportation](https:\/\/en.wikipedia.org\/wiki\/Quantum_teleportation \"Quantum teleportation\") |\n| Extensions | [Quantum fluctuation](https:\/\/en.wikipedia.org\/wiki\/Quantum_fluctuation \"Quantum fluctuation\") [Casimir effect](https:\/\/en.wikipedia.org\/wiki\/Casimir_effect \"Casimir effect\") [Quantum statistical mechanics](https:\/\/en.wikipedia.org\/wiki\/Quantum_statistical_mechanics \"Quantum statistical mechanics\") [Quantum field theory](https:\/\/en.wikipedia.org\/wiki\/Quantum_field_theory \"Quantum field theory\") [History](https:\/\/en.wikipedia.org\/wiki\/History_of_quantum_field_theory \"History of quantum field theory\") [Quantum gravity](https:\/\/en.wikipedia.org\/wiki\/Quantum_gravity \"Quantum gravity\") [Relativistic quantum mechanics](https:\/\/en.wikipedia.org\/wiki\/Relativistic_quantum_mechanics \"Relativistic quantum mechanics\") |\n| Related | [Schrödinger's cat](https:\/\/en.wikipedia.org\/wiki\/Schr%C3%B6dinger%27s_cat \"Schrödinger's cat\") [in popular culture](https:\/\/en.wikipedia.org\/wiki\/Schr%C3%B6dinger%27s_cat_in_popular_culture \"Schrödinger's cat in popular culture\") [Wigner's friend](https:\/\/en.wikipedia.org\/wiki\/Wigner%27s_friend \"Wigner's friend\") [EPR paradox](https:\/\/en.wikipedia.org\/wiki\/Einstein%E2%80%93Podolsky%E2%80%93Rosen_paradox \"Einstein–Podolsky–Rosen paradox\") [Quantum mysticism](https:\/\/en.wikipedia.org\/wiki\/Quantum_mysticism \"Quantum mysticism\") |\n| ![](https:\/\/upload.wikimedia.org\/wikipedia\/en\/thumb\/9\/96\/Symbol_category_class.svg\/20px-Symbol_category_class.svg.png) [Category](https:\/\/en.wikipedia.org\/wiki\/Category:Quantum_mechanics \"Category:Quantum mechanics\") |   |\n\n| [v](https:\/\/en.wikipedia.org\/wiki\/Template:Branches_of_physics \"Template:Branches of physics\") [t](https:\/\/en.wikipedia.org\/wiki\/Template_talk:Branches_of_physics \"Template talk:Branches of physics\") [e](https:\/\/en.wikipedia.org\/wiki\/Special:EditPage\/Template:Branches_of_physics \"Special:EditPage\/Template:Branches of physics\")Major [branches of physics](https:\/\/en.wikipedia.org\/wiki\/Branches_of_physics \"Branches of physics\") |   |\n|---|---|\n| Divisions | [Pure](https:\/\/en.wikipedia.org\/wiki\/Basic_research \"Basic research\") [Applied](https:\/\/en.wikipedia.org\/wiki\/Applied_physics \"Applied physics\") [Engineering](https:\/\/en.wikipedia.org\/wiki\/Engineering_physics \"Engineering physics\") |\n| Approaches | [Experimental](https:\/\/en.wikipedia.org\/wiki\/Experimental_physics \"Experimental physics\") [Theoretical](https:\/\/en.wikipedia.org\/wiki\/Theoretical_physics \"Theoretical physics\") [Computational](https:\/\/en.wikipedia.org\/wiki\/Computational_physics \"Computational physics\") |\n| [Classical](https:\/\/en.wikipedia.org\/wiki\/Classical_physics \"Classical physics\") | [Classical mechanics](https:\/\/en.wikipedia.org\/wiki\/Classical_mechanics \"Classical mechanics\") [Newtonian](https:\/\/en.wikipedia.org\/wiki\/Newton%27s_laws_of_motion \"Newton's laws of motion\") [Analytical](https:\/\/en.wikipedia.org\/wiki\/Analytical_mechanics \"Analytical mechanics\") [Celestial](https:\/\/en.wikipedia.org\/wiki\/Celestial_mechanics \"Celestial mechanics\") [Continuum](https:\/\/en.wikipedia.org\/wiki\/Continuum_mechanics \"Continuum mechanics\") [Acoustics](https:\/\/en.wikipedia.org\/wiki\/Acoustics \"Acoustics\") [Classical electromagnetism](https:\/\/en.wikipedia.org\/wiki\/Classical_electromagnetism \"Classical electromagnetism\") [Classical optics](https:\/\/en.wikipedia.org\/wiki\/Classical_optics \"Classical optics\") [Ray](https:\/\/en.wikipedia.org\/wiki\/Geometrical_optics \"Geometrical optics\") [Wave](https:\/\/en.wikipedia.org\/wiki\/Physical_optics \"Physical optics\") [Thermodynamics](https:\/\/en.wikipedia.org\/wiki\/Thermodynamics \"Thermodynamics\") [Statistical](https:\/\/en.wikipedia.org\/wiki\/Statistical_mechanics \"Statistical mechanics\") [Non-equilibrium](https:\/\/en.wikipedia.org\/wiki\/Non-equilibrium_thermodynamics \"Non-equilibrium thermodynamics\") |\n| [Modern](https:\/\/en.wikipedia.org\/wiki\/Modern_physics \"Modern physics\") | [Relativistic mechanics](https:\/\/en.wikipedia.org\/wiki\/Relativistic_mechanics \"Relativistic mechanics\") [Special](https:\/\/en.wikipedia.org\/wiki\/Special_relativity \"Special relativity\") [General](https:\/\/en.wikipedia.org\/wiki\/General_relativity \"General relativity\") [Nuclear physics](https:\/\/en.wikipedia.org\/wiki\/Nuclear_physics \"Nuclear physics\") [Particle physics](https:\/\/en.wikipedia.org\/wiki\/Particle_physics \"Particle physics\") [Quantum mechanics]() [Atomic, molecular, and optical physics](https:\/\/en.wikipedia.org\/wiki\/Atomic,_molecular,_and_optical_physics \"Atomic, molecular, and optical physics\") [Atomic](https:\/\/en.wikipedia.org\/wiki\/Atomic_physics \"Atomic physics\") [Molecular](https:\/\/en.wikipedia.org\/wiki\/Molecular_physics \"Molecular physics\") [Modern optics](https:\/\/en.wikipedia.org\/wiki\/Optics#Modern_optics \"Optics\") [Condensed matter physics](https:\/\/en.wikipedia.org\/wiki\/Condensed_matter_physics \"Condensed matter physics\") [Solid-state physics](https:\/\/en.wikipedia.org\/wiki\/Solid-state_physics \"Solid-state physics\") 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For the company, see [Quantum-Systems](https:\/\/en.wikipedia.org\/wiki\/Quantum-Systems \"Quantum-Systems\").\n\n[![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/e\/e7\/Hydrogen_Density_Plots.png\/500px-Hydrogen_Density_Plots.png)](https:\/\/en.wikipedia.org\/wiki\/File:Hydrogen_Density_Plots.png)\n\n[Wave functions](https:\/\/en.wikipedia.org\/wiki\/Wave_function \"Wave function\") of the [electron](https:\/\/en.wikipedia.org\/wiki\/Electron \"Electron\") in a hydrogen atom at different energy levels. Quantum mechanics cannot predict the exact location of a particle in space, only the probability of finding it at different locations.[\\[1\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Born1926-1) The brighter areas represent a higher probability of finding the electron.\n\n**Quantum mechanics** is the fundamental physical [theory](https:\/\/en.wikipedia.org\/wiki\/Scientific_theory \"Scientific theory\") that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of [atoms](https:\/\/en.wikipedia.org\/wiki\/Atom \"Atom\").[\\[2\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Feynman-2): 1.1  It is the foundation of all **quantum physics**, which includes [quantum chemistry](https:\/\/en.wikipedia.org\/wiki\/Quantum_chemistry \"Quantum chemistry\"), [quantum biology](https:\/\/en.wikipedia.org\/wiki\/Quantum_biology \"Quantum biology\"), [quantum field theory](https:\/\/en.wikipedia.org\/wiki\/Quantum_field_theory \"Quantum field theory\"), [quantum technology](https:\/\/en.wikipedia.org\/wiki\/Quantum_technology \"Quantum technology\"), and [quantum information science](https:\/\/en.wikipedia.org\/wiki\/Quantum_information_science \"Quantum information science\").\n\nQuantum mechanics can describe many systems that [classical physics](https:\/\/en.wikipedia.org\/wiki\/Classical_physics \"Classical physics\") cannot. Classical physics can describe many aspects of nature at an ordinary ([macroscopic](https:\/\/en.wikipedia.org\/wiki\/Macroscopic \"Macroscopic\") and [(optical) microscopic](https:\/\/en.wikipedia.org\/wiki\/Microscopic_scale \"Microscopic scale\")) scale, however is insufficient for describing them at very small [submicroscopic](https:\/\/en.wikipedia.org\/wiki\/Submicroscopic \"Submicroscopic\") (atomic and [subatomic](https:\/\/en.wikipedia.org\/wiki\/Subatomic \"Subatomic\")) scales. Classical mechanics can be derived from quantum mechanics as an approximation that is valid at ordinary scales.[\\[3\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-3)\n\n**Quantum systems** have [bound](https:\/\/en.wikipedia.org\/wiki\/Bound_state \"Bound state\") states that are [quantized](https:\/\/en.wikipedia.org\/wiki\/Quantization_\\(physics\\) \"Quantization (physics)\") to [discrete values](https:\/\/en.wikipedia.org\/wiki\/Discrete_mathematics \"Discrete mathematics\") of [energy](https:\/\/en.wikipedia.org\/wiki\/Energy \"Energy\"), [momentum](https:\/\/en.wikipedia.org\/wiki\/Momentum \"Momentum\"), [angular momentum](https:\/\/en.wikipedia.org\/wiki\/Angular_momentum \"Angular momentum\"), and other quantities, in contrast to classical systems where these quantities can be measured continuously. Measurements of quantum systems show characteristics of both [particles](https:\/\/en.wikipedia.org\/wiki\/Particle \"Particle\") and [waves](https:\/\/en.wikipedia.org\/wiki\/Wave \"Wave\") ([wave–particle duality](https:\/\/en.wikipedia.org\/wiki\/Wave%E2%80%93particle_duality \"Wave–particle duality\")), and there are limits to how accurately the value of a physical quantity can be predicted prior to its measurement, given a complete set of initial conditions (the [uncertainty principle](https:\/\/en.wikipedia.org\/wiki\/Uncertainty_principle \"Uncertainty principle\")).\n\nQuantum mechanics arose gradually from theories to explain observations that could not be reconciled with [classical physics](https:\/\/en.wikipedia.org\/wiki\/Classical_physics \"Classical physics\"), such as [Max Planck](https:\/\/en.wikipedia.org\/wiki\/Max_Planck \"Max Planck\")'s solution in 1900 to the [black-body radiation](https:\/\/en.wikipedia.org\/wiki\/Black-body_radiation \"Black-body radiation\") problem, and the correspondence between energy and frequency in [Albert Einstein](https:\/\/en.wikipedia.org\/wiki\/Albert_Einstein \"Albert Einstein\")'s [1905 paper](https:\/\/en.wikipedia.org\/wiki\/Annus_Mirabilis_papers#Photoelectric_effect \"Annus Mirabilis papers\"), which explained the [photoelectric effect](https:\/\/en.wikipedia.org\/wiki\/Photoelectric_effect \"Photoelectric effect\"). These early attempts to understand microscopic phenomena, now known as the \"[old quantum theory](https:\/\/en.wikipedia.org\/wiki\/Old_quantum_theory \"Old quantum theory\")\", led to the full development of quantum mechanics in the mid-1920s by [Niels Bohr](https:\/\/en.wikipedia.org\/wiki\/Niels_Bohr \"Niels Bohr\"), [Erwin Schrödinger](https:\/\/en.wikipedia.org\/wiki\/Erwin_Schr%C3%B6dinger \"Erwin Schrödinger\"), [Werner Heisenberg](https:\/\/en.wikipedia.org\/wiki\/Werner_Heisenberg \"Werner Heisenberg\"), [Max Born](https:\/\/en.wikipedia.org\/wiki\/Max_Born \"Max Born\"), [Paul Dirac](https:\/\/en.wikipedia.org\/wiki\/Paul_Dirac \"Paul Dirac\") and others. The modern theory is formulated in various [specially developed mathematical formalisms](https:\/\/en.wikipedia.org\/wiki\/Mathematical_formulations_of_quantum_mechanics \"Mathematical formulations of quantum mechanics\"). In one of them, a mathematical entity called the [wave function](https:\/\/en.wikipedia.org\/wiki\/Wave_function \"Wave function\") provides information, in the form of [probability amplitudes](https:\/\/en.wikipedia.org\/wiki\/Probability_amplitude \"Probability amplitude\"), about what measurements of a particle's energy, momentum, and other physical properties may yield.\n\nOverview and fundamental concepts\n\nQuantum mechanics allows the calculation of properties and behaviour of [physical systems](https:\/\/en.wikipedia.org\/wiki\/Physical_systems \"Physical systems\"). It is typically applied to microscopic systems: [molecules](https:\/\/en.wikipedia.org\/wiki\/Molecules \"Molecules\"), [atoms](https:\/\/en.wikipedia.org\/wiki\/Atoms \"Atoms\") and [subatomic particles](https:\/\/en.wikipedia.org\/wiki\/Subatomic_particle \"Subatomic particle\"). It has been demonstrated to hold for complex molecules with thousands of atoms,[\\[4\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-4) but its application to human beings raises philosophical problems, such as [Wigner's friend](https:\/\/en.wikipedia.org\/wiki\/Wigner%27s_friend \"Wigner's friend\"), and its application to the universe as a whole remains speculative.[\\[5\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-5) Predictions of quantum mechanics have been verified experimentally to an extremely high degree of [accuracy](https:\/\/en.wikipedia.org\/wiki\/Accuracy \"Accuracy\"). For example, the refinement of quantum mechanics for the interaction of light and matter, known as [quantum electrodynamics](https:\/\/en.wikipedia.org\/wiki\/Quantum_electrodynamics \"Quantum electrodynamics\") (QED), has been [shown to agree with experiment](https:\/\/en.wikipedia.org\/wiki\/Precision_tests_of_QED \"Precision tests of QED\") to within 1 part in 1012 when predicting the magnetic properties of an electron.[\\[6\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-6)\n\nA fundamental feature of the theory is that it usually cannot predict with certainty what will happen, but only gives probabilities. Mathematically, a probability is found by taking the square of the absolute value of a [complex number](https:\/\/en.wikipedia.org\/wiki\/Complex_number \"Complex number\"), known as a probability amplitude. This is known as the [Born rule](https:\/\/en.wikipedia.org\/wiki\/Born_rule \"Born rule\"), named after physicist [Max Born](https:\/\/en.wikipedia.org\/wiki\/Max_Born \"Max Born\"). For example, a quantum particle like an [electron](https:\/\/en.wikipedia.org\/wiki\/Electron \"Electron\") can be described by a wave function, which associates to each point in space a probability amplitude. Applying the Born rule to these amplitudes gives a [probability density function](https:\/\/en.wikipedia.org\/wiki\/Probability_density_function \"Probability density function\") for the position that the electron will be found to have when an experiment is performed to measure it. This is the best the theory can do; it cannot say for certain where the electron will be found. The [Schrödinger equation](https:\/\/en.wikipedia.org\/wiki\/Schr%C3%B6dinger_equation \"Schrödinger equation\") relates the collection of probability amplitudes that pertain to one moment of time to the collection of probability amplitudes that pertain to another.[\\[7\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Zwiebach2022-7): 67–87\n\nOne consequence of the mathematical rules of quantum mechanics is a tradeoff in predictability between measurable quantities. The most famous form of this [uncertainty principle](https:\/\/en.wikipedia.org\/wiki\/Uncertainty_principle \"Uncertainty principle\") says that no matter how a quantum particle is prepared or how carefully experiments upon it are arranged, it is impossible to have a precise prediction for a measurement of its position and also at the same time for a measurement of its [momentum](https:\/\/en.wikipedia.org\/wiki\/Momentum \"Momentum\").[\\[7\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Zwiebach2022-7): 427–435\n\n[![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/c\/cd\/Double-slit.svg\/330px-Double-slit.svg.png)](https:\/\/en.wikipedia.org\/wiki\/File:Double-slit.svg)\n\nAn illustration of the [double-slit experiment](https:\/\/en.wikipedia.org\/wiki\/Double-slit_experiment \"Double-slit experiment\")\n\nAnother consequence of the mathematical rules of quantum mechanics is the phenomenon of [quantum interference](https:\/\/en.wikipedia.org\/wiki\/Quantum_interference \"Quantum interference\"), which is often illustrated with the [double-slit experiment](https:\/\/en.wikipedia.org\/wiki\/Double-slit_experiment \"Double-slit experiment\"). In the basic version of this experiment, a [coherent light source](https:\/\/en.wikipedia.org\/wiki\/Coherence_\\(physics\\) \"Coherence (physics)\"), such as a [laser](https:\/\/en.wikipedia.org\/wiki\/Laser \"Laser\") beam, illuminates a plate pierced by two parallel slits, and the light passing through the slits is observed on a screen behind the plate.[\\[8\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Lederman-8): 102–111 [\\[2\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Feynman-2): 1.1–1.8  The wave nature of light causes the light waves passing through the two slits to [interfere](https:\/\/en.wikipedia.org\/wiki\/Interference_\\(wave_propagation\\) \"Interference (wave propagation)\"), producing bright and dark bands on the screen – a result that would not be expected if light consisted of classical particles.[\\[8\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Lederman-8) However, the light is always found to be absorbed at the screen at discrete points, as individual particles rather than waves; the interference pattern appears via the varying density of these particle hits on the screen. Furthermore, versions of the experiment that include detectors at the slits find that each detected [photon](https:\/\/en.wikipedia.org\/wiki\/Photon \"Photon\") passes through one slit (as would a classical particle), and not through both slits (as would a wave).[\\[8\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Lederman-8): 109 [\\[9\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-M%C3%BCller-Kirsten-9)[\\[10\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Plotnitsky-10) However, [such experiments](https:\/\/en.wikipedia.org\/wiki\/Double-slit_experiment#Which_way \"Double-slit experiment\") demonstrate that particles do not form the interference pattern if one detects which slit they pass through. This behavior is known as [wave–particle duality](https:\/\/en.wikipedia.org\/wiki\/Wave%E2%80%93particle_duality \"Wave–particle duality\"). In addition to light, [electrons](https:\/\/en.wikipedia.org\/wiki\/Electrons \"Electrons\"), [atoms](https:\/\/en.wikipedia.org\/wiki\/Atoms \"Atoms\"), and [molecules](https:\/\/en.wikipedia.org\/wiki\/Molecules \"Molecules\") are all found to exhibit the same dual behavior when fired towards a double slit.[\\[2\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Feynman-2)\n\n[![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/5\/5f\/QuantumTunnel.jpg\/330px-QuantumTunnel.jpg)](https:\/\/en.wikipedia.org\/wiki\/File:QuantumTunnel.jpg)\n\nA simplified diagram of [quantum tunneling](https:\/\/en.wikipedia.org\/wiki\/Quantum_tunneling \"Quantum tunneling\"), a phenomenon by which a particle may move through a barrier which would be impossible under classical mechanics\n\nAnother non-classical phenomenon predicted by quantum mechanics is [quantum tunnelling](https:\/\/en.wikipedia.org\/wiki\/Quantum_tunnelling \"Quantum tunnelling\"): a particle that goes up against a [potential barrier](https:\/\/en.wikipedia.org\/wiki\/Potential_barrier \"Potential barrier\") can cross it, even if its kinetic energy is smaller than the maximum of the potential.[\\[11\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-11) In classical mechanics this particle would be trapped. Quantum tunnelling has several important consequences, enabling [radioactive decay](https:\/\/en.wikipedia.org\/wiki\/Radioactive_decay \"Radioactive decay\"), [nuclear fusion](https:\/\/en.wikipedia.org\/wiki\/Nuclear_fusion \"Nuclear fusion\") in stars, and applications such as [scanning tunnelling microscopy](https:\/\/en.wikipedia.org\/wiki\/Scanning_tunnelling_microscopy \"Scanning tunnelling microscopy\"), [tunnel diode](https:\/\/en.wikipedia.org\/wiki\/Tunnel_diode \"Tunnel diode\") and [tunnel field-effect transistor](https:\/\/en.wikipedia.org\/wiki\/Tunnel_field-effect_transistor \"Tunnel field-effect transistor\").[\\[12\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Trixler2013-12)[\\[13\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-13)\n\nWhen quantum systems interact, the result can be the creation of [quantum entanglement](https:\/\/en.wikipedia.org\/wiki\/Quantum_entanglement \"Quantum entanglement\"): their properties become so intertwined that a description of the whole solely in terms of the individual parts is no longer possible. Erwin Schrödinger called entanglement \"...*the* characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought\".[\\[14\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-14) Quantum entanglement enables [quantum computing](https:\/\/en.wikipedia.org\/wiki\/Quantum_computing \"Quantum computing\") and is part of quantum communication protocols, such as [quantum key distribution](https:\/\/en.wikipedia.org\/wiki\/Quantum_key_distribution \"Quantum key distribution\") and [superdense coding](https:\/\/en.wikipedia.org\/wiki\/Superdense_coding \"Superdense coding\").[\\[15\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Caves-15) Contrary to popular misconception, entanglement does not allow sending signals [faster than light](https:\/\/en.wikipedia.org\/wiki\/Faster_than_light \"Faster than light\"), as demonstrated by the [no-communication theorem](https:\/\/en.wikipedia.org\/wiki\/No-communication_theorem \"No-communication theorem\").[\\[15\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Caves-15)\n\nAnother possibility opened by entanglement is testing for \"[hidden variables](https:\/\/en.wikipedia.org\/wiki\/Hidden_variable_theory \"Hidden variable theory\")\", hypothetical properties more fundamental than the quantities addressed in quantum theory itself, knowledge of which would allow more exact predictions than quantum theory provides. A collection of results, most significantly [Bell's theorem](https:\/\/en.wikipedia.org\/wiki\/Bell%27s_theorem \"Bell's theorem\"), have demonstrated that broad classes of such hidden-variable theories are in fact incompatible with quantum physics. According to Bell's theorem, if nature actually operates in accord with any theory of *local* hidden variables, then the results of a [Bell test](https:\/\/en.wikipedia.org\/wiki\/Bell_test \"Bell test\") will be constrained in a particular, quantifiable way. Many Bell tests have been performed and they have shown results incompatible with the constraints imposed by local hidden variables.[\\[16\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-wiseman15-16)[\\[17\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-wolchover17-17)\n\nIt is not possible to present these concepts in more than a superficial way without introducing the mathematics involved; understanding quantum mechanics requires not only manipulating complex numbers, but also [linear algebra](https:\/\/en.wikipedia.org\/wiki\/Linear_algebra \"Linear algebra\"), [differential equations](https:\/\/en.wikipedia.org\/wiki\/Differential_equation \"Differential equation\"), [group theory](https:\/\/en.wikipedia.org\/wiki\/Group_theory \"Group theory\"), and other more advanced subjects.[\\[18\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-18)[\\[19\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-19) Accordingly, this article will present a mathematical formulation of quantum mechanics and survey its application to some useful and oft-studied examples.\n\nMathematical formulation\n\nIn the mathematically rigorous formulation of quantum mechanics, the state of a quantum mechanical system is a vector ![{\\\\displaystyle \\\\psi }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a) belonging to a ([separable](https:\/\/en.wikipedia.org\/wiki\/Separable_space \"Separable space\")) complex [Hilbert space](https:\/\/en.wikipedia.org\/wiki\/Hilbert_space \"Hilbert space\") ![{\\\\displaystyle {\\\\mathcal {H}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/19ef4c7b923a5125ac91aa491838a95ee15b804f). This vector is postulated to be normalized under the Hilbert space inner product, that is, it obeys ![{\\\\displaystyle \\\\langle \\\\psi ,\\\\psi \\\\rangle =1}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/7d5758e7a60b4e54bc46e01b0618919c65b787a0), and it is well-defined up to a complex number of modulus 1 (the global phase), that is, ![{\\\\displaystyle \\\\psi }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a) and ![{\\\\displaystyle e^{i\\\\alpha }\\\\psi }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/7193e9b0479da798b9cf2dcc492f49166d0d7103) represent the same physical system. In other words, the possible states are points in the [projective space](https:\/\/en.wikipedia.org\/wiki\/Projective_space \"Projective space\") of a Hilbert space, usually called the [complex projective space](https:\/\/en.wikipedia.org\/wiki\/Complex_projective_space \"Complex projective space\"). The exact nature of this Hilbert space is dependent on the system – for example, for describing position and momentum the Hilbert space is the space of complex [square-integrable](https:\/\/en.wikipedia.org\/wiki\/Square-integrable \"Square-integrable\") functions ![{\\\\displaystyle L^{2}(\\\\mathbb {C} )}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/3a3649087af50f42a95da2163656076529ca3139),[\\[20\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Holevo2001-20): 13  while the Hilbert space for the [spin](https:\/\/en.wikipedia.org\/wiki\/Spin_\\(physics\\) \"Spin (physics)\") of a single proton is simply the space of two-dimensional complex vectors ![{\\\\displaystyle \\\\mathbb {C} ^{2}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/6f43d6ec8a1e1fe5a85aec0dd9bdcd45ae09b06b) with the usual inner product.[\\[20\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Holevo2001-20): 20\n\nPhysical quantities of interest – position, momentum, energy, spin – are represented by observables, which are [Hermitian](https:\/\/en.wikipedia.org\/wiki\/Hermitian_adjoint#Hermitian_operators \"Hermitian adjoint\") (more precisely, [self-adjoint](https:\/\/en.wikipedia.org\/wiki\/Self-adjoint_operator \"Self-adjoint operator\")) linear [operators](https:\/\/en.wikipedia.org\/wiki\/Operator_\\(physics\\) \"Operator (physics)\") acting on the Hilbert space.[\\[20\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Holevo2001-20): 17  A quantum state can be an [eigenvector](https:\/\/en.wikipedia.org\/wiki\/Eigenvector \"Eigenvector\") of an observable, in which case it is called an [eigenstate](https:\/\/en.wikipedia.org\/wiki\/Eigenstate \"Eigenstate\"), and the associated [eigenvalue](https:\/\/en.wikipedia.org\/wiki\/Eigenvalue \"Eigenvalue\") corresponds to the value of the observable in that eigenstate. More generally, a quantum state will be a linear combination of the eigenstates, known as a [quantum superposition](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition \"Quantum superposition\"). When an observable is measured, the result will be one of its eigenvalues with probability given by the [Born rule](https:\/\/en.wikipedia.org\/wiki\/Born_rule \"Born rule\"): in the simplest case the eigenvalue ![{\\\\displaystyle \\\\lambda }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/b43d0ea3c9c025af1be9128e62a18fa74bedda2a) is non-degenerate and the probability is given by ![{\\\\displaystyle \\|\\\\langle {\\\\vec {\\\\lambda }},\\\\psi \\\\rangle \\|^{2}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/de4b465224a805b558697f44dcf3824fb2fbfe20), where ![{\\\\displaystyle {\\\\vec {\\\\lambda }}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/d8c69573b9c55ec8ea726662102b8e693bf32f01) is its associated unit-length eigenvector. More generally, the eigenvalue is degenerate and the probability is given by ![{\\\\displaystyle \\\\langle \\\\psi ,P\\_{\\\\lambda }\\\\psi \\\\rangle }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/6ba3648378b52e2e710937b36948cd908b2d6531), where ![{\\\\displaystyle P\\_{\\\\lambda }}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/330591f9b6fffc93ca78514576fd0d8cfac6f0c7) is the projector onto its associated eigenspace.[\\[21\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-21) In the continuous case, these formulas give instead the [probability density](https:\/\/en.wikipedia.org\/wiki\/Probability_density \"Probability density\").\n\nAfter the [measurement](https:\/\/en.wikipedia.org\/wiki\/Measurement_in_quantum_mechanics \"Measurement in quantum mechanics\"), if result ![{\\\\displaystyle \\\\lambda }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/b43d0ea3c9c025af1be9128e62a18fa74bedda2a) was obtained, the quantum state is postulated to [collapse](https:\/\/en.wikipedia.org\/wiki\/Collapse_of_the_wavefunction \"Collapse of the wavefunction\") to ![{\\\\displaystyle {\\\\vec {\\\\lambda }}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/d8c69573b9c55ec8ea726662102b8e693bf32f01), in the non-degenerate case, or to ![{\\\\textstyle P\\_{\\\\lambda }\\\\psi {\\\\big \/}\\\\!{\\\\sqrt {\\\\langle \\\\psi ,P\\_{\\\\lambda }\\\\psi \\\\rangle }}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/e38d8eb9122b686c4d8f937fe59e2ec511be43ed), in the general case. The [probabilistic](https:\/\/en.wikipedia.org\/wiki\/Probabilistic \"Probabilistic\") nature of quantum mechanics thus stems from the act of measurement. This is one of the most debated aspects of quantum theory, with different [interpretations of quantum mechanics](https:\/\/en.wikipedia.org\/wiki\/Interpretations_of_quantum_mechanics \"Interpretations of quantum mechanics\") giving radically different answers to questions regarding quantum-state collapse, as discussed [below](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#Philosophical_implications).\n\nTime evolution of a quantum state\n\nThe time evolution of a quantum state is described by the Schrödinger equation: ![{\\\\displaystyle i\\\\hbar {\\\\frac {\\\\partial }{\\\\partial t}}\\\\psi (t)=H\\\\psi (t).}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/5c41b5a62afa3aa83e6ea98e69d692ac0f77c2c6) Here ![{\\\\displaystyle H}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/75a9edddcca2f782014371f75dca39d7e13a9c1b) denotes the [Hamiltonian](https:\/\/en.wikipedia.org\/wiki\/Hamiltonian_\\(quantum_mechanics\\) \"Hamiltonian (quantum mechanics)\"), the observable corresponding to the [total energy](https:\/\/en.wikipedia.org\/wiki\/Total_energy \"Total energy\") of the system, and ![{\\\\displaystyle \\\\hbar }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/de68de3a92517953436c93b5a76461d49160cc41) is the reduced [Planck constant](https:\/\/en.wikipedia.org\/wiki\/Planck_constant \"Planck constant\"). The constant ![{\\\\displaystyle i\\\\hbar }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/3d724d600903f701570a9f97c2a8378a6f3a80e5) is introduced so that the Hamiltonian is reduced to the [classical Hamiltonian](https:\/\/en.wikipedia.org\/wiki\/Hamiltonian_mechanics \"Hamiltonian mechanics\") in cases where the quantum system can be approximated by a classical system; the ability to make such an approximation in certain limits is called the [correspondence principle](https:\/\/en.wikipedia.org\/wiki\/Correspondence_principle \"Correspondence principle\").\n\nThe solution of this differential equation is given by ![{\\\\displaystyle \\\\psi (t)=e^{-iHt\/\\\\hbar }\\\\psi (0).}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/3b0a822b842f8d952c4e5827f312597e72a45f4d) The operator ![{\\\\displaystyle U(t)=e^{-iHt\/\\\\hbar }}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/17a17e78953fa7d61788cc8bc41a25ef254fc8e0) is known as the time-evolution operator, and has the crucial property that it is [unitary](https:\/\/en.wikipedia.org\/wiki\/Unitarity_\\(physics\\) \"Unitarity (physics)\"). This time evolution is [deterministic](https:\/\/en.wikipedia.org\/wiki\/Deterministic \"Deterministic\") in the sense that – given an initial quantum state ![{\\\\displaystyle \\\\psi (0)}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/fcfdf3fefa6feee8b2192815ac15a2650d8a8db2) – it makes a definite prediction of what the quantum state ![{\\\\displaystyle \\\\psi (t)}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/6cf4a36b5f945be90a527b3dbe3d55d3f0439cdb) will be at any later time.[\\[22\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-22)\n\n[![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/3\/35\/Atomic-orbital-clouds_spd_m0.png\/330px-Atomic-orbital-clouds_spd_m0.png)](https:\/\/en.wikipedia.org\/wiki\/File:Atomic-orbital-clouds_spd_m0.png)\n\nFig. 1: [Probability densities](https:\/\/en.wikipedia.org\/wiki\/Probability_density_function \"Probability density function\") corresponding to the wave functions of an electron in a hydrogen atom possessing definite energy levels (increasing from the top of the image to the bottom: *n* = 1, 2, 3, ...) and angular momenta (increasing across from left to right: *s*, *p*, *d*, ...). Denser areas correspond to higher probability density in a position measurement.Such wave functions are directly comparable to [Chladni's figures](https:\/\/en.wikipedia.org\/wiki\/Chladni%27s_figures \"Chladni's figures\") of [acoustic](https:\/\/en.wikipedia.org\/wiki\/Acoustics \"Acoustics\") modes of vibration in classical physics and are modes of oscillation as well, possessing a sharp energy and thus, a definite frequency. The [angular momentum](https:\/\/en.wikipedia.org\/wiki\/Angular_momentum \"Angular momentum\") and energy are [quantized](https:\/\/en.wikipedia.org\/wiki\/Quantization_\\(physics\\) \"Quantization (physics)\") and take *only* discrete values like those shown – as is the case for [resonant frequencies](https:\/\/en.wikipedia.org\/wiki\/Resonant_frequencies \"Resonant frequencies\") in acoustics.\n\nSome wave functions produce probability distributions that are independent of time, such as [eigenstates](https:\/\/en.wikipedia.org\/wiki\/Eigenstate \"Eigenstate\") of the Hamiltonian.[\\[7\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Zwiebach2022-7): 133–137  Many systems that are treated dynamically in classical mechanics are described by such \"static\" wave functions. For example, a single electron in an unexcited atom is pictured classically as a particle moving in a circular trajectory around the [atomic nucleus](https:\/\/en.wikipedia.org\/wiki\/Atomic_nucleus \"Atomic nucleus\"), whereas in quantum mechanics, it is described by a static wave function surrounding the nucleus. For example, the electron wave function for an unexcited hydrogen atom is a spherically symmetric function known as an [*s* orbital](https:\/\/en.wikipedia.org\/wiki\/Atomic_orbital \"Atomic orbital\") ([Fig. 1](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#fig1)).\n\nAnalytic solutions of the Schrödinger equation are known for [very few relatively simple model Hamiltonians](https:\/\/en.wikipedia.org\/wiki\/List_of_quantum-mechanical_systems_with_analytical_solutions \"List of quantum-mechanical systems with analytical solutions\") including the [quantum harmonic oscillator](https:\/\/en.wikipedia.org\/wiki\/Quantum_harmonic_oscillator \"Quantum harmonic oscillator\"), the [particle in a box](https:\/\/en.wikipedia.org\/wiki\/Particle_in_a_box \"Particle in a box\"), the [dihydrogen cation](https:\/\/en.wikipedia.org\/wiki\/Dihydrogen_cation \"Dihydrogen cation\"), and the [hydrogen atom](https:\/\/en.wikipedia.org\/wiki\/Hydrogen_atom \"Hydrogen atom\"). Even the [helium](https:\/\/en.wikipedia.org\/wiki\/Helium \"Helium\") atom – which contains just two electrons – has defied all attempts at a fully analytic treatment, admitting no solution in [closed form](https:\/\/en.wikipedia.org\/wiki\/Closed-form_expression \"Closed-form expression\").[\\[23\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-23)[\\[24\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-24)[\\[25\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-25)\n\nHowever, there are techniques for finding approximate solutions. One method, called [perturbation theory](https:\/\/en.wikipedia.org\/wiki\/Perturbation_theory_\\(quantum_mechanics\\) \"Perturbation theory (quantum mechanics)\"), uses the analytic result for a simple quantum mechanical model to create a result for a related but more complicated model by (for example) the addition of a weak [potential energy](https:\/\/en.wikipedia.org\/wiki\/Potential_energy \"Potential energy\").[\\[7\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Zwiebach2022-7): 793  Another approximation method applies to systems for which quantum mechanics produces only small deviations from classical behavior. These deviations can then be computed based on the classical motion.[\\[7\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Zwiebach2022-7): 849\n\nUncertainty principle\n\nOne consequence of the basic quantum formalism is the uncertainty principle. In its most familiar form, this states that no preparation of a quantum particle can imply simultaneously precise predictions both for a measurement of its position and for a measurement of its momentum.[\\[26\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Cohen-Tannoudji-26)[\\[27\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-L&L-27) Both position and momentum are observables, meaning that they are represented by [Hermitian operators](https:\/\/en.wikipedia.org\/wiki\/Hermitian_operators \"Hermitian operators\"). The position operator ![{\\\\displaystyle {\\\\hat {X}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/acc59ad6d9a06d55b96b65beb0fdfc89acc1e40e) and momentum operator ![{\\\\displaystyle {\\\\hat {P}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/7a46a8cf7bc789e8c4f8de7ca0d9946a46fb8842) do not commute, but rather satisfy the [canonical commutation relation](https:\/\/en.wikipedia.org\/wiki\/Canonical_commutation_relation \"Canonical commutation relation\"): ![{\\\\displaystyle \\[{\\\\hat {X}},{\\\\hat {P}}\\]=i\\\\hbar .}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/803fe39b0eeaff8d1570df480e738cf5a968cc71) Given a quantum state, the Born rule lets us compute expectation values for both ![{\\\\displaystyle X}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/68baa052181f707c662844a465bfeeb135e82bab) and ![{\\\\displaystyle P}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/b4dc73bf40314945ff376bd363916a738548d40a), and moreover for powers of them. Defining the uncertainty for an observable by a [standard deviation](https:\/\/en.wikipedia.org\/wiki\/Standard_deviation \"Standard deviation\"), we have ![{\\\\displaystyle \\\\sigma \\_{X}={\\\\textstyle {\\\\sqrt {\\\\left\\\\langle X^{2}\\\\right\\\\rangle -\\\\left\\\\langle X\\\\right\\\\rangle ^{2}}}},}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/457ec20972d63dfb1ecc9087e18d1f949f908c8f) and likewise for the momentum: ![{\\\\displaystyle \\\\sigma \\_{P}={\\\\sqrt {\\\\left\\\\langle P^{2}\\\\right\\\\rangle -\\\\left\\\\langle P\\\\right\\\\rangle ^{2}}}.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/63ec8f5f7b9e5957ea6d06c56068b06244acc184) The uncertainty principle states that ![{\\\\displaystyle \\\\sigma \\_{X}\\\\sigma \\_{P}\\\\geq {\\\\frac {\\\\hbar }{2}}.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/538cada7fa57155ece387e5d53d90ca366e323fe) Either standard deviation can in principle be made arbitrarily small, but not both simultaneously.[\\[28\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-ballentine1970-28) This inequality generalizes to arbitrary pairs of self-adjoint operators ![{\\\\displaystyle A}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/7daff47fa58cdfd29dc333def748ff5fa4c923e3) and ![{\\\\displaystyle B}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/47136aad860d145f75f3eed3022df827cee94d7a). The [commutator](https:\/\/en.wikipedia.org\/wiki\/Commutator \"Commutator\") of these two operators is ![{\\\\displaystyle \\[A,B\\]=AB-BA,}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/b2a47259b42e63c048c65f67d304404867841951) and this provides the lower bound on the product of standard deviations: ![{\\\\displaystyle \\\\sigma \\_{A}\\\\sigma \\_{B}\\\\geq {\\\\tfrac {1}{2}}\\\\left\\|{\\\\bigl \\\\langle }\\[A,B\\]{\\\\bigr \\\\rangle }\\\\right\\|.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/d0fd768b447334e150b8b98181f74b475e41ee52)\n\nAnother consequence of the canonical commutation relation is that the position and momentum operators are [Fourier transforms](https:\/\/en.wikipedia.org\/wiki\/Fourier_transform#Uncertainty_principle \"Fourier transform\") of each other, so that a description of an object according to its momentum is the Fourier transform of its description according to its position. The fact that dependence in momentum is the Fourier transform of the dependence in position means that the momentum operator is equivalent (up to an ![{\\\\displaystyle i\/\\\\hbar }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/44a04e1f19b5e7bea2bfa8002a841bf8d1b4e66a) factor) to taking the derivative according to the position, since in Fourier analysis [differentiation corresponds to multiplication in the dual space](https:\/\/en.wikipedia.org\/wiki\/Fourier_transform#Differentiation \"Fourier transform\"). This is why in quantum equations in position space, the momentum ![{\\\\displaystyle p\\_{i}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/5bab39399bf5424f25d957cdc57c84a0622626d2) is replaced by ![{\\\\displaystyle -i\\\\hbar {\\\\frac {\\\\partial }{\\\\partial x}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/ab7fffcee704fc55eb36b137e0cc25132b5dc1bf), and in particular in the [non-relativistic Schrödinger equation in position space](https:\/\/en.wikipedia.org\/wiki\/Schr%C3%B6dinger_equation#Equation \"Schrödinger equation\") the momentum-squared term is replaced with a Laplacian times ![{\\\\displaystyle -\\\\hbar ^{2}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/185934c9e69984a0905432abe63d3b41741342b4).[\\[26\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Cohen-Tannoudji-26)\n\nComposite systems and entanglement\n\nWhen two different quantum systems are considered together, the Hilbert space of the combined system is the [tensor product](https:\/\/en.wikipedia.org\/wiki\/Tensor_product \"Tensor product\") of the Hilbert spaces of the two components. For example, let A and B be two quantum systems, with Hilbert spaces ![{\\\\displaystyle {\\\\mathcal {H}}\\_{A}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/4611f8542d6c72789b7e82bfcf29014d44c13aa3) and ![{\\\\displaystyle {\\\\mathcal {H}}\\_{B}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/bfea1102a35aff8c6435147dc9dffc21b4cfe4ef), respectively. The Hilbert space of the composite system is then ![{\\\\displaystyle {\\\\mathcal {H}}\\_{AB}={\\\\mathcal {H}}\\_{A}\\\\otimes {\\\\mathcal {H}}\\_{B}.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/54e4f6aff2be5ec9f3eb4657a133d1be17f6e2b3) If the state for the first system is the vector ![{\\\\displaystyle \\\\psi \\_{A}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/a04e49109107e7765357dd92e975895840fa7700) and the state for the second system is ![{\\\\displaystyle \\\\psi \\_{B}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/7ca3b90bba42c451129c4969322e175f5a1d3368), then the state of the composite system is ![{\\\\displaystyle \\\\psi \\_{A}\\\\otimes \\\\psi \\_{B}.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/852c1e09787e91af3af1a10d1eee4eadc232d48c) Not all states in the joint Hilbert space ![{\\\\displaystyle {\\\\mathcal {H}}\\_{AB}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/951b54ea87d5bd62464abaf1be530e6a37a86f1b) can be written in this form, however, because the superposition principle implies that linear combinations of these \"separable\" or \"product states\" are also valid. For example, if ![{\\\\displaystyle \\\\psi \\_{A}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/a04e49109107e7765357dd92e975895840fa7700) and ![{\\\\displaystyle \\\\phi \\_{A}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/5e38d84dcc21ea769a51ab502691f566c830b430) are both possible states for system ![{\\\\displaystyle A}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/7daff47fa58cdfd29dc333def748ff5fa4c923e3), and likewise ![{\\\\displaystyle \\\\psi \\_{B}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/7ca3b90bba42c451129c4969322e175f5a1d3368) and ![{\\\\displaystyle \\\\phi \\_{B}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/a829a7a85bb29904dec5e93605d0b976baf45755) are both possible states for system ![{\\\\displaystyle B}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/47136aad860d145f75f3eed3022df827cee94d7a), then ![{\\\\displaystyle {\\\\tfrac {1}{\\\\sqrt {2}}}\\\\left(\\\\psi \\_{A}\\\\otimes \\\\psi \\_{B}+\\\\phi \\_{A}\\\\otimes \\\\phi \\_{B}\\\\right)}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/53d226ba585a99942ec855197c23ed885b635258) is a valid joint state that is not separable. States that are not separable are called [entangled](https:\/\/en.wikipedia.org\/wiki\/Quantum_entanglement \"Quantum entanglement\").[\\[29\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-:0-29)[\\[30\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-:1-30)\n\nIf the state for a composite system is entangled, it is impossible to describe either component system A or system B by a state vector. One can instead define [reduced density matrices](https:\/\/en.wikipedia.org\/wiki\/Reduced_density_matrix \"Reduced density matrix\") that describe the statistics that can be obtained by making measurements on either component system alone. This necessarily causes a loss of information, though: knowing the reduced density matrices of the individual systems is not enough to reconstruct the state of the composite system.[\\[29\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-:0-29)[\\[30\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-:1-30) Just as density matrices specify the state of a subsystem of a larger system, analogously, [positive operator-valued measures](https:\/\/en.wikipedia.org\/wiki\/POVM \"POVM\") (POVMs) describe the effect on a subsystem of a measurement performed on a larger system. POVMs are extensively used in quantum information theory.[\\[29\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-:0-29)[\\[31\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-wilde-31)\n\nAs described above, entanglement is a key feature of models of measurement processes in which an apparatus becomes entangled with the system being measured. Systems interacting with the environment in which they reside generally become entangled with that environment, a phenomenon known as [quantum decoherence](https:\/\/en.wikipedia.org\/wiki\/Quantum_decoherence \"Quantum decoherence\"). This can explain why, in practice, quantum effects are difficult to observe in systems larger than microscopic.[\\[32\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-32)\n\nEquivalence between formulations\n\nThere are many mathematically equivalent formulations of quantum mechanics. One of the oldest and most common is the \"[transformation theory](https:\/\/en.wikipedia.org\/wiki\/Transformation_theory_\\(quantum_mechanics\\) \"Transformation theory (quantum mechanics)\")\" proposed by [Paul Dirac](https:\/\/en.wikipedia.org\/wiki\/Paul_Dirac \"Paul Dirac\"), which unifies and generalizes the two earliest formulations of quantum mechanics – [matrix mechanics](https:\/\/en.wikipedia.org\/wiki\/Matrix_mechanics \"Matrix mechanics\") (invented by [Werner Heisenberg](https:\/\/en.wikipedia.org\/wiki\/Werner_Heisenberg \"Werner Heisenberg\")) and wave mechanics (invented by [Erwin Schrödinger](https:\/\/en.wikipedia.org\/wiki\/Erwin_Schr%C3%B6dinger \"Erwin Schrödinger\")).[\\[33\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-33) An alternative formulation of quantum mechanics is [Feynman](https:\/\/en.wikipedia.org\/wiki\/Feynman \"Feynman\")'s [path integral formulation](https:\/\/en.wikipedia.org\/wiki\/Path_integral_formulation \"Path integral formulation\"), in which a quantum-mechanical amplitude is considered as a sum over all possible classical and non-classical paths between the initial and final states. This is the quantum-mechanical counterpart of the [action principle](https:\/\/en.wikipedia.org\/wiki\/Action_principle \"Action principle\") in classical mechanics.[\\[34\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-34)\n\nSymmetries and conservation laws\n\nThe Hamiltonian ![{\\\\displaystyle H}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/75a9edddcca2f782014371f75dca39d7e13a9c1b) is known as the *generator* of time evolution, since it defines a unitary time-evolution operator ![{\\\\displaystyle U(t)=e^{-iHt\/\\\\hbar }}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/17a17e78953fa7d61788cc8bc41a25ef254fc8e0) for each value of ![{\\\\displaystyle t}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/65658b7b223af9e1acc877d848888ecdb4466560). From this relation between ![{\\\\displaystyle U(t)}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/666c639df532e88616357c4991cabce9a57b5611) and ![{\\\\displaystyle H}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/75a9edddcca2f782014371f75dca39d7e13a9c1b), it follows that any observable ![{\\\\displaystyle A}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/7daff47fa58cdfd29dc333def748ff5fa4c923e3) that commutes with ![{\\\\displaystyle H}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/75a9edddcca2f782014371f75dca39d7e13a9c1b) will be *conserved*: its expectation value will not change over time.[\\[7\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Zwiebach2022-7): 471  This statement generalizes, as mathematically, any Hermitian operator ![{\\\\displaystyle A}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/7daff47fa58cdfd29dc333def748ff5fa4c923e3) can generate a family of unitary operators parameterized by a variable ![{\\\\displaystyle t}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/65658b7b223af9e1acc877d848888ecdb4466560). Under the evolution generated by ![{\\\\displaystyle A}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/7daff47fa58cdfd29dc333def748ff5fa4c923e3), any observable ![{\\\\displaystyle B}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/47136aad860d145f75f3eed3022df827cee94d7a) that commutes with ![{\\\\displaystyle A}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/7daff47fa58cdfd29dc333def748ff5fa4c923e3) will be conserved. Moreover, if ![{\\\\displaystyle B}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/47136aad860d145f75f3eed3022df827cee94d7a) is conserved by evolution under ![{\\\\displaystyle A}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/7daff47fa58cdfd29dc333def748ff5fa4c923e3), then ![{\\\\displaystyle A}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/7daff47fa58cdfd29dc333def748ff5fa4c923e3) is conserved under the evolution generated by ![{\\\\displaystyle B}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/47136aad860d145f75f3eed3022df827cee94d7a). This implies a quantum version of the result proven by [Emmy Noether](https:\/\/en.wikipedia.org\/wiki\/Emmy_Noether \"Emmy Noether\") in classical ([Lagrangian](https:\/\/en.wikipedia.org\/wiki\/Lagrangian_mechanics \"Lagrangian mechanics\")) mechanics: for every [differentiable](https:\/\/en.wikipedia.org\/wiki\/Differentiable \"Differentiable\") [symmetry](https:\/\/en.wikipedia.org\/wiki\/Symmetry_\\(physics\\) \"Symmetry (physics)\") of a Hamiltonian, there exists a corresponding [conservation law](https:\/\/en.wikipedia.org\/wiki\/Conservation_law \"Conservation law\").\n\nExamples\n\nFree particle\n\n[![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/5\/56\/Guassian_Dispersion.gif)](https:\/\/en.wikipedia.org\/wiki\/File:Guassian_Dispersion.gif)\n\nPosition space probability density of a Gaussian [wave packet](https:\/\/en.wikipedia.org\/wiki\/Wave_packet \"Wave packet\") moving in one dimension in free space\n\nThe simplest example of a quantum system with a position degree of freedom is a free particle in a single spatial dimension. A free particle is one which is not subject to external influences, so that its Hamiltonian consists only of its kinetic energy: ![{\\\\displaystyle H={\\\\frac {1}{2m}}P^{2}=-{\\\\frac {\\\\hbar ^{2}}{2m}}{\\\\frac {d^{2}}{dx^{2}}}.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/084f585ee5c6cd8a34e323a6de7943227128afc3) The general solution of the Schrödinger equation is given by ![{\\\\displaystyle \\\\psi (x,t)={\\\\frac {1}{\\\\sqrt {2\\\\pi }}}\\\\int \\_{-\\\\infty }^{\\\\infty }{\\\\hat {\\\\psi }}(k,0)e^{i(kx-{\\\\frac {\\\\hbar k^{2}}{2m}}t)}\\\\mathrm {d} k,}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/ef4f021ba945856e3676808b11724109a8a74dad) which is a superposition of all possible [plane waves](https:\/\/en.wikipedia.org\/wiki\/Plane_wave \"Plane wave\") ![{\\\\displaystyle e^{i(kx-{\\\\frac {\\\\hbar k^{2}}{2m}}t)}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/fb4cd9a9984c84a493ce547babcf58e31b04f7e7), which are eigenstates of the momentum operator with momentum ![{\\\\displaystyle p=\\\\hbar k}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/24fee69175538303b28ac54e907baf53d0a58dbf). The coefficients of the superposition are ![{\\\\displaystyle {\\\\hat {\\\\psi }}(k,0)}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/8b8323c08418da8bc376c6d78b578d4729b927ea), which is the Fourier transform of the initial quantum state ![{\\\\displaystyle \\\\psi (x,0)}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/55ad442e07ca2d7986ef0787f9129fc325cde137).\n\nIt is not possible for the solution to be a single momentum eigenstate, or a single position eigenstate, as these are not normalizable quantum states.[\\[note 1\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-35) Instead, we can consider a Gaussian [wave packet](https:\/\/en.wikipedia.org\/wiki\/Wave_packet \"Wave packet\"): ![{\\\\displaystyle \\\\psi (x,0)={\\\\frac {1}{\\\\sqrt\\[{4}\\]{\\\\pi a}}}e^{-{\\\\frac {x^{2}}{2a}}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/a4c2dae82312897d5fd4c58986c426a6009e6840) which has Fourier transform, and therefore momentum distribution ![{\\\\displaystyle {\\\\hat {\\\\psi }}(k,0)={\\\\sqrt\\[{4}\\]{\\\\frac {a}{\\\\pi }}}e^{-{\\\\frac {ak^{2}}{2}}}.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/c4991535bba434314af8c27c16fff74f49ce367e) We see that as we make ![{\\\\displaystyle a}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/ffd2487510aa438433a2579450ab2b3d557e5edc) smaller the spread in position gets smaller, but the spread in momentum gets larger. Conversely, by making ![{\\\\displaystyle a}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/ffd2487510aa438433a2579450ab2b3d557e5edc) larger we make the spread in momentum smaller, but the spread in position gets larger. This illustrates the uncertainty principle.\n\nAs we let the Gaussian wave packet evolve in time, we see that its center moves through space at a constant velocity (like a classical particle with no forces acting on it). However, the wave packet will also spread out as time progresses, which means that the position becomes more and more uncertain. The uncertainty in momentum, however, stays constant.[\\[35\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-36)\n\nParticle in a box\n\n[![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/2\/27\/Infinite_potential_well.svg\/250px-Infinite_potential_well.svg.png)](https:\/\/en.wikipedia.org\/wiki\/File:Infinite_potential_well.svg)\n\n1-dimensional potential energy box (or infinite potential well)\n\nThe particle in a one-dimensional potential energy box is the most mathematically simple example where restraints lead to the quantization of energy levels. The box is defined as having zero potential energy everywhere *inside* a certain region, and therefore infinite potential energy everywhere *outside* that region.[\\[26\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Cohen-Tannoudji-26): 77–78  For the one-dimensional case in the ![{\\\\displaystyle x}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/87f9e315fd7e2ba406057a97300593c4802b53e4) direction, the time-independent Schrödinger equation may be written ![{\\\\displaystyle -{\\\\frac {\\\\hbar ^{2}}{2m}}{\\\\frac {d^{2}\\\\psi }{dx^{2}}}=E\\\\psi .}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/cc73e9216faf1390c3ed550b72be21fc068ec747)\n\nWith the differential operator defined by ![{\\\\displaystyle {\\\\hat {p}}\\_{x}=-i\\\\hbar {\\\\frac {d}{dx}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/2511885975007002c7582a9c8175689076df210a)the previous equation is evocative of the [classic kinetic energy analogue](https:\/\/en.wikipedia.org\/wiki\/Kinetic_energy#Kinetic_energy_of_rigid_bodies \"Kinetic energy\"), ![{\\\\displaystyle {\\\\frac {1}{2m}}{\\\\hat {p}}\\_{x}^{2}=E,}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/422f57813ede91b8e348e786616c6297742b4fd7) with state ![{\\\\displaystyle \\\\psi }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a) in this case having energy ![{\\\\displaystyle E}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b) coincident with the kinetic energy of the particle.\n\nThe general solutions of the Schrödinger equation for the particle in a box are ![{\\\\displaystyle \\\\psi (x)=Ae^{ikx}+Be^{-ikx}\\\\qquad \\\\qquad E={\\\\frac {\\\\hbar ^{2}k^{2}}{2m}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/b5fb1b2f1d5afb42edb4eb98bf89791d283c1e53) or, from [Euler's formula](https:\/\/en.wikipedia.org\/wiki\/Euler%27s_formula \"Euler's formula\"), ![{\\\\displaystyle \\\\psi (x)=C\\\\sin(kx)+D\\\\cos(kx).\\\\!}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/229ebbbe587e0c678318cb85089a74ed289c971a)\n\nThe infinite potential walls of the box determine the values of ![{\\\\displaystyle C,D,}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/6e153d2c75e3492a32d4fafefec88846862c3b9b) and ![{\\\\displaystyle k}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/c3c9a2c7b599b37105512c5d570edc034056dd40) at ![{\\\\displaystyle x=0}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/953917eaf52f2e1baad54c8c9e3d6f9bb3710cdc) and ![{\\\\displaystyle x=L}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/e5fe40c588800aaab69041986b49a59664cd767a) where ![{\\\\displaystyle \\\\psi }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a) must be zero. Thus, at ![{\\\\displaystyle x=0}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/953917eaf52f2e1baad54c8c9e3d6f9bb3710cdc), ![{\\\\displaystyle \\\\psi (0)=0=C\\\\sin(0)+D\\\\cos(0)=D}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/34188e64cbf80486b3ad311b53b512fa36de59c6) and ![{\\\\displaystyle D=0}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/d375dfda80ee8df1d1d7aa8b962114044e464305). At ![{\\\\displaystyle x=L}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/e5fe40c588800aaab69041986b49a59664cd767a), ![{\\\\displaystyle \\\\psi (L)=0=C\\\\sin(kL),}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/c7a5bc9ef330efb5b6eba5d89c2ee026bbeea6a8) in which ![{\\\\displaystyle C}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/4fc55753007cd3c18576f7933f6f089196732029) cannot be zero as this would conflict with the postulate that ![{\\\\displaystyle \\\\psi }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a) has norm 1. Therefore, since ![{\\\\displaystyle \\\\sin(kL)=0}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/e5177ccdb2057c5c1be728af20b8ef3d61f79999), ![{\\\\displaystyle kL}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/d8865c1184b2c1dff6226dae50d3be91f4f01cfe) must be an integer multiple of ![{\\\\displaystyle \\\\pi }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/9be4ba0bb8df3af72e90a0535fabcc17431e540a), ![{\\\\displaystyle k={\\\\frac {n\\\\pi }{L}}\\\\qquad \\\\qquad n=1,2,3,\\\\ldots .}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/3fd91af3024e1f59cf57e04884494fd2c55664f8)\n\nThis constraint on ![{\\\\displaystyle k}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/c3c9a2c7b599b37105512c5d570edc034056dd40) implies a constraint on the energy levels, yielding ![{\\\\displaystyle E\\_{n}={\\\\frac {\\\\hbar ^{2}\\\\pi ^{2}n^{2}}{2mL^{2}}}={\\\\frac {n^{2}h^{2}}{8mL^{2}}}.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/fe507fd1f92ebfe133aeb4f1da46ea27f569f38b)\n\nA [finite potential well](https:\/\/en.wikipedia.org\/wiki\/Finite_potential_well \"Finite potential well\") is the generalization of the infinite potential well problem to potential wells having finite depth. The finite potential well problem is mathematically more complicated than the infinite particle-in-a-box problem as the wave function is not pinned to zero at the walls of the well. Instead, the wave function must satisfy more complicated mathematical boundary conditions as it is nonzero in regions outside the well. Another related problem is that of the [rectangular potential barrier](https:\/\/en.wikipedia.org\/wiki\/Rectangular_potential_barrier \"Rectangular potential barrier\"), which furnishes a model for the [quantum tunneling](https:\/\/en.wikipedia.org\/wiki\/Quantum_tunneling \"Quantum tunneling\") effect that plays an important role in the performance of modern technologies such as [flash memory](https:\/\/en.wikipedia.org\/wiki\/Flash_memory \"Flash memory\") and [scanning tunneling microscopy](https:\/\/en.wikipedia.org\/wiki\/Scanning_tunneling_microscopy \"Scanning tunneling microscopy\").\n\nHarmonic oscillator\n\n[![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/9\/90\/QuantumHarmonicOscillatorAnimation.gif)](https:\/\/en.wikipedia.org\/wiki\/File:QuantumHarmonicOscillatorAnimation.gif)\n\nSome trajectories of a [harmonic oscillator](https:\/\/en.wikipedia.org\/wiki\/Harmonic_oscillator \"Harmonic oscillator\") (i.e. a ball attached to a [spring](https:\/\/en.wikipedia.org\/wiki\/Hooke%27s_law \"Hooke's law\")) in [classical mechanics](https:\/\/en.wikipedia.org\/wiki\/Classical_mechanics \"Classical mechanics\") (A-B) and quantum mechanics (C-H). In quantum mechanics, the position of the ball is represented by a [wave](https:\/\/en.wikipedia.org\/wiki\/Wave \"Wave\") (called the wave function), with the [real part](https:\/\/en.wikipedia.org\/wiki\/Real_part \"Real part\") shown in blue and the [imaginary part](https:\/\/en.wikipedia.org\/wiki\/Imaginary_part \"Imaginary part\") shown in red. Some of the trajectories (such as C, D, E, and F) are [standing waves](https:\/\/en.wikipedia.org\/wiki\/Standing_wave \"Standing wave\") (or \"[stationary states](https:\/\/en.wikipedia.org\/wiki\/Stationary_state \"Stationary state\")\"). Each standing-wave frequency is proportional to a possible [energy level](https:\/\/en.wikipedia.org\/wiki\/Energy_level \"Energy level\") of the oscillator. This \"energy quantization\" does not occur in classical physics, where the oscillator can have *any* energy.\n\nAs in the classical case, the potential for the quantum harmonic oscillator is given by[\\[7\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Zwiebach2022-7): 234  ![{\\\\displaystyle V(x)={\\\\frac {1}{2}}m\\\\omega ^{2}x^{2}.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/ac9d8735e10198c090b576c765dc2778fe458b9c)\n\nThis problem can either be treated by directly solving the Schrödinger equation, which is not trivial, or by using the more elegant \"ladder method\" first proposed by Paul Dirac. The [eigenstates](https:\/\/en.wikipedia.org\/wiki\/Eigenstate \"Eigenstate\") are given by ![{\\\\displaystyle \\\\psi \\_{n}(x)={\\\\sqrt {\\\\frac {1}{2^{n}\\\\,n!}}}\\\\cdot \\\\left({\\\\frac {m\\\\omega }{\\\\pi \\\\hbar }}\\\\right)^{1\/4}\\\\cdot e^{-{\\\\frac {m\\\\omega x^{2}}{2\\\\hbar }}}\\\\cdot H\\_{n}\\\\left({\\\\sqrt {\\\\frac {m\\\\omega }{\\\\hbar }}}x\\\\right),\\\\qquad }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/8a0031b1b0c9fa4a9e4e6957718d509846522c4d) ![{\\\\displaystyle n=0,1,2,\\\\ldots .}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/d652f9756f144ea772e4e632cb1398652d36372c) where *Hn* are the [Hermite polynomials](https:\/\/en.wikipedia.org\/wiki\/Hermite_polynomials \"Hermite polynomials\") ![{\\\\displaystyle H\\_{n}(x)=(-1)^{n}e^{x^{2}}{\\\\frac {d^{n}}{dx^{n}}}\\\\left(e^{-x^{2}}\\\\right),}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/0da8b4a24291ce8b32bcc0665f490f3c76afc7d3) and the corresponding energy levels are ![{\\\\displaystyle E\\_{n}=\\\\hbar \\\\omega \\\\left(n+{1 \\\\over 2}\\\\right).}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/635fb242f4c34cd5ba947f4c8132ddd47f2872c0)\n\nThis is another example illustrating the discretization of energy for [bound states](https:\/\/en.wikipedia.org\/wiki\/Bound_state \"Bound state\").\n\nMach–Zehnder interferometer\n\n[![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/c\/c9\/Mach-Zehnder_interferometer.svg\/330px-Mach-Zehnder_interferometer.svg.png)](https:\/\/en.wikipedia.org\/wiki\/File:Mach-Zehnder_interferometer.svg)\n\nSchematic of a Mach–Zehnder interferometer\n\nThe [Mach–Zehnder interferometer](https:\/\/en.wikipedia.org\/wiki\/Mach%E2%80%93Zehnder_interferometer \"Mach–Zehnder interferometer\") (MZI) illustrates the concepts of superposition and interference with linear algebra in dimension 2, rather than differential equations. It can be seen as a simplified version of the double-slit experiment, but it is of interest in its own right, for example in the [delayed choice quantum eraser](https:\/\/en.wikipedia.org\/wiki\/Delayed_choice_quantum_eraser \"Delayed choice quantum eraser\"), the [Elitzur–Vaidman bomb tester](https:\/\/en.wikipedia.org\/wiki\/Elitzur%E2%80%93Vaidman_bomb_tester \"Elitzur–Vaidman bomb tester\"), and in studies of quantum entanglement.[\\[36\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Paris1999-37)[\\[37\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Haack2010-38)\n\nWe can model a photon going through the interferometer by considering that at each point it can be in a superposition of only two paths: the \"lower\" path which starts from the left, goes straight through both beam splitters, and ends at the top, and the \"upper\" path which starts from the bottom, goes straight through both beam splitters, and ends at the right. The quantum state of the photon is therefore a vector ![{\\\\displaystyle \\\\psi \\\\in \\\\mathbb {C} ^{2}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/a7a51a8280039fc22cd88d90915a952f8e020f47) that is a superposition of the \"lower\" path ![{\\\\displaystyle \\\\psi \\_{l}={\\\\begin{pmatrix}1\\\\\\\\0\\\\end{pmatrix}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/feca39f1f03b4dc63be6f7a7c2060430b1217e2f) and the \"upper\" path ![{\\\\displaystyle \\\\psi \\_{u}={\\\\begin{pmatrix}0\\\\\\\\1\\\\end{pmatrix}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/400e9751fcd7816718398d3892bec2ad26bb4713), that is, ![{\\\\displaystyle \\\\psi =\\\\alpha \\\\psi \\_{l}+\\\\beta \\\\psi \\_{u}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/fb88364b79f3c611ca2ca1edb5356139bf4e1085) for complex ![{\\\\displaystyle \\\\alpha ,\\\\beta }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/e4b46b57cfa0011b643037751809904d915c1b48). In order to respect the postulate that ![{\\\\displaystyle \\\\langle \\\\psi ,\\\\psi \\\\rangle =1}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/7d5758e7a60b4e54bc46e01b0618919c65b787a0) we require that ![{\\\\displaystyle \\|\\\\alpha \\|^{2}+\\|\\\\beta \\|^{2}=1}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/18cd7473cdb894839d10852890517b1fb687c73b).\n\nBoth [beam splitters](https:\/\/en.wikipedia.org\/wiki\/Beam_splitter \"Beam splitter\") are modelled as the unitary matrix ![{\\\\displaystyle B={\\\\frac {1}{\\\\sqrt {2}}}{\\\\begin{pmatrix}1\\&i\\\\\\\\i&1\\\\end{pmatrix}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/ddf502efcb65d0cbac5bb8ef1a6f163ac9cf2145), which means that when a photon meets the beam splitter it will either stay on the same path with a probability amplitude of ![{\\\\displaystyle 1\/{\\\\sqrt {2}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/75a0bbdb60fcb73ac67d9970a5eb0808b87fd37d), or be reflected to the other path with a probability amplitude of ![{\\\\displaystyle i\/{\\\\sqrt {2}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/e846b6a5731f3799960a4968399d85bc0b7fb9fd). The phase shifter on the upper arm is modelled as the unitary matrix ![{\\\\displaystyle P={\\\\begin{pmatrix}1&0\\\\\\\\0\\&e^{i\\\\Delta \\\\Phi }\\\\end{pmatrix}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/3df9457946dd8035c51a39e0926be9f07c7f0a3e), which means that if the photon is on the \"upper\" path it will gain a relative phase of ![{\\\\displaystyle \\\\Delta \\\\Phi }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/20cedb08e6edea3cad9b2829ef67311bbe518dd2), and it will stay unchanged if it is in the lower path.\n\nA photon that enters the interferometer from the left will then be acted upon with a beam splitter ![{\\\\displaystyle B}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/47136aad860d145f75f3eed3022df827cee94d7a), a phase shifter ![{\\\\displaystyle P}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/b4dc73bf40314945ff376bd363916a738548d40a), and another beam splitter ![{\\\\displaystyle B}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/47136aad860d145f75f3eed3022df827cee94d7a), and so end up in the state ![{\\\\displaystyle BPB\\\\psi \\_{l}=ie^{i\\\\Delta \\\\Phi \/2}{\\\\begin{pmatrix}-\\\\sin(\\\\Delta \\\\Phi \/2)\\\\\\\\\\\\cos(\\\\Delta \\\\Phi \/2)\\\\end{pmatrix}},}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/7927a94da54f5d57b8accffdb9ad456a1e3b5033) and the probabilities that it will be detected at the right or at the top are given respectively by ![{\\\\displaystyle p(u)=\\|\\\\langle \\\\psi \\_{u},BPB\\\\psi \\_{l}\\\\rangle \\|^{2}=\\\\cos ^{2}{\\\\frac {\\\\Delta \\\\Phi }{2}},}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/110acb8ee7dc4e309de846470778f4767fe97f8f) ![{\\\\displaystyle p(l)=\\|\\\\langle \\\\psi \\_{l},BPB\\\\psi \\_{l}\\\\rangle \\|^{2}=\\\\sin ^{2}{\\\\frac {\\\\Delta \\\\Phi }{2}}.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/86880ce53051688a4f591ebfb5183fbae013deee) One can therefore use the Mach–Zehnder interferometer to estimate the [phase shift](https:\/\/en.wikipedia.org\/wiki\/Phase_\\(waves\\) \"Phase (waves)\") by estimating these probabilities.\n\nIt is interesting to consider what would happen if the photon were definitely in either the \"lower\" or \"upper\" paths between the beam splitters. This can be accomplished by blocking one of the paths, or equivalently by removing the first beam splitter (and feeding the photon from the left or the bottom, as desired). In both cases, there will be no interference between the paths anymore, and the probabilities are given by ![{\\\\displaystyle p(u)=p(l)=1\/2}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/d2a9b3426984de1a64c07261c3e14b485320fee4), independently of the phase ![{\\\\displaystyle \\\\Delta \\\\Phi }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/20cedb08e6edea3cad9b2829ef67311bbe518dd2). From this we can conclude that the photon does not take one path or another after the first beam splitter, but rather that it is in a genuine quantum superposition of the two paths.[\\[38\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-vedral-39)\n\nApplications\n\nQuantum mechanics has had enormous success in explaining many of the features of our universe, with regard to small-scale and discrete quantities and interactions which cannot be explained by [classical methods](https:\/\/en.wikipedia.org\/wiki\/Classical_physics \"Classical physics\").[\\[note 2\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-feynmanIII-40) Quantum mechanics is often the only theory that can reveal the individual behaviors of the subatomic particles that make up all forms of matter (electrons, [protons](https:\/\/en.wikipedia.org\/wiki\/Proton \"Proton\"), [neutrons](https:\/\/en.wikipedia.org\/wiki\/Neutron \"Neutron\"), [photons](https:\/\/en.wikipedia.org\/wiki\/Photon \"Photon\"), and others). [Solid-state physics](https:\/\/en.wikipedia.org\/wiki\/Solid-state_physics \"Solid-state physics\") and [materials science](https:\/\/en.wikipedia.org\/wiki\/Materials_science \"Materials science\") are dependent upon quantum mechanics.[\\[39\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-marvincohen2008-41)\n\nIn many aspects, modern technology operates at a scale where quantum effects are significant. Important applications of quantum theory include [quantum chemistry](https:\/\/en.wikipedia.org\/wiki\/Quantum_chemistry \"Quantum chemistry\"), [quantum optics](https:\/\/en.wikipedia.org\/wiki\/Quantum_optics \"Quantum optics\"), [quantum computing](https:\/\/en.wikipedia.org\/wiki\/Quantum_computing \"Quantum computing\"), [superconducting magnets](https:\/\/en.wikipedia.org\/wiki\/Superconducting_magnet \"Superconducting magnet\"), [light-emitting diodes](https:\/\/en.wikipedia.org\/wiki\/Light-emitting_diode \"Light-emitting diode\"), the [optical amplifier](https:\/\/en.wikipedia.org\/wiki\/Optical_amplifier \"Optical amplifier\") and the laser, the [transistor](https:\/\/en.wikipedia.org\/wiki\/Transistor \"Transistor\") and [semiconductors](https:\/\/en.wikipedia.org\/wiki\/Semiconductor \"Semiconductor\") such as the [microprocessor](https:\/\/en.wikipedia.org\/wiki\/Microprocessor \"Microprocessor\"), [medical and research imaging](https:\/\/en.wikipedia.org\/wiki\/Medical_imaging \"Medical imaging\") such as [magnetic resonance imaging](https:\/\/en.wikipedia.org\/wiki\/Magnetic_resonance_imaging \"Magnetic resonance imaging\") and [electron microscopy](https:\/\/en.wikipedia.org\/wiki\/Electron_microscopy \"Electron microscopy\").[\\[40\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-42) Explanations for many biological and physical phenomena are rooted in the nature of the chemical bond, most notably the macro-molecule [DNA](https:\/\/en.wikipedia.org\/wiki\/DNA \"DNA\").\n\nRelation to other scientific theories\n\nClassical mechanics\n\nThe rules of quantum mechanics assert that the state space of a system is a Hilbert space and that observables of the system are Hermitian operators acting on vectors in that space – although they do not tell us which Hilbert space or which operators. These can be chosen appropriately in order to obtain a quantitative description of a quantum system, a necessary step in making physical predictions. An important guide for making these choices is the [correspondence principle](https:\/\/en.wikipedia.org\/wiki\/Correspondence_principle \"Correspondence principle\"), a heuristic which states that the predictions of quantum mechanics reduce to those of [classical mechanics](https:\/\/en.wikipedia.org\/wiki\/Classical_mechanics \"Classical mechanics\") in the regime of large [quantum numbers](https:\/\/en.wikipedia.org\/wiki\/Quantum_number \"Quantum number\").[\\[41\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Tipler-43) One can also start from an established classical model of a particular system, and then try to guess the underlying quantum model that would give rise to the classical model in the correspondence limit. This approach is known as [quantization](https:\/\/en.wikipedia.org\/wiki\/Canonical_quantization \"Canonical quantization\").[\\[42\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Peres1993-44): 299 [\\[43\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-45)\n\nWhen quantum mechanics was originally formulated, it was applied to models whose correspondence limit was [non-relativistic](https:\/\/en.wikipedia.org\/wiki\/Theory_of_relativity \"Theory of relativity\") classical mechanics. For instance, the well-known model of the [quantum harmonic oscillator](https:\/\/en.wikipedia.org\/wiki\/Quantum_harmonic_oscillator \"Quantum harmonic oscillator\") uses an explicitly non-relativistic expression for the [kinetic energy](https:\/\/en.wikipedia.org\/wiki\/Kinetic_energy \"Kinetic energy\") of the oscillator, and is thus a quantum version of the [classical harmonic oscillator](https:\/\/en.wikipedia.org\/wiki\/Harmonic_oscillator \"Harmonic oscillator\").[\\[7\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Zwiebach2022-7): 234\n\nComplications arise with [chaotic systems](https:\/\/en.wikipedia.org\/wiki\/Chaotic_systems \"Chaotic systems\"), which do not have good quantum numbers, and [quantum chaos](https:\/\/en.wikipedia.org\/wiki\/Quantum_chaos \"Quantum chaos\") studies the relationship between classical and quantum descriptions in these systems.[\\[42\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Peres1993-44): 353\n\n[Quantum decoherence](https:\/\/en.wikipedia.org\/wiki\/Quantum_decoherence \"Quantum decoherence\") is a mechanism through which quantum systems lose [coherence](https:\/\/en.wikipedia.org\/wiki\/Quantum_coherence \"Quantum coherence\"), and thus become incapable of displaying many typically quantum effects: [quantum superpositions](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition \"Quantum superposition\") become simply probabilistic mixtures, and quantum entanglement becomes simply classical correlations.[\\[7\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Zwiebach2022-7): 687–730  Quantum coherence is not typically evident at macroscopic scales, though at temperatures approaching [absolute zero](https:\/\/en.wikipedia.org\/wiki\/Absolute_zero \"Absolute zero\") quantum behavior may manifest macroscopically.[\\[note 3\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-46)\n\nMany macroscopic properties of a classical system are a direct consequence of the quantum behavior of its parts. For example, the stability of bulk matter (consisting of atoms and [molecules](https:\/\/en.wikipedia.org\/wiki\/Molecule \"Molecule\") which would quickly collapse under electric forces alone), the rigidity of solids, and the mechanical, thermal, chemical, optical and magnetic properties of matter are all results of the interaction of [electric charges](https:\/\/en.wikipedia.org\/wiki\/Electric_charge \"Electric charge\") under the rules of quantum mechanics.[\\[44\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-47)\n\nSpecial relativity and electrodynamics\n\nEarly attempts to merge quantum mechanics with [special relativity](https:\/\/en.wikipedia.org\/wiki\/Special_relativity \"Special relativity\") involved the replacement of the Schrödinger equation with a covariant equation such as the [Klein–Gordon equation](https:\/\/en.wikipedia.org\/wiki\/Klein%E2%80%93Gordon_equation \"Klein–Gordon equation\") or the [Dirac equation](https:\/\/en.wikipedia.org\/wiki\/Dirac_equation \"Dirac equation\"). While these theories were successful in explaining many experimental results, they had certain unsatisfactory qualities stemming from their neglect of the relativistic creation and annihilation of particles. A fully relativistic quantum theory required the development of quantum field theory, which applies quantization to a field (rather than a fixed set of particles). The first complete quantum field theory, [quantum electrodynamics](https:\/\/en.wikipedia.org\/wiki\/Quantum_electrodynamics \"Quantum electrodynamics\"), provides a fully quantum description of the [electromagnetic interaction](https:\/\/en.wikipedia.org\/wiki\/Electromagnetic_interaction \"Electromagnetic interaction\"). Quantum electrodynamics is, along with [general relativity](https:\/\/en.wikipedia.org\/wiki\/General_relativity \"General relativity\"), one of the most accurate physical theories ever devised.[\\[45\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-48)[\\[46\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-49)\n\nThe full apparatus of quantum field theory is often unnecessary for describing electrodynamic systems. A simpler approach, one that has been used since the inception of quantum mechanics, is to treat [charged](https:\/\/en.wikipedia.org\/wiki\/Electric_charge \"Electric charge\") particles as quantum mechanical objects being acted on by a classical [electromagnetic field](https:\/\/en.wikipedia.org\/wiki\/Electromagnetic_field \"Electromagnetic field\"). For example, the elementary quantum model of the [hydrogen atom](https:\/\/en.wikipedia.org\/wiki\/Hydrogen_atom \"Hydrogen atom\") describes the [electric field](https:\/\/en.wikipedia.org\/wiki\/Electric_field \"Electric field\") of the hydrogen atom using a classical ![{\\\\displaystyle \\\\textstyle -e^{2}\/(4\\\\pi \\\\epsilon \\_{\\_{0}}r)}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/7c4876353362da0a234af137590c6379d8f8c273) [Coulomb potential](https:\/\/en.wikipedia.org\/wiki\/Electric_potential \"Electric potential\").[\\[7\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Zwiebach2022-7): 285  Likewise, in a [Stern–Gerlach experiment](https:\/\/en.wikipedia.org\/wiki\/Stern%E2%80%93Gerlach_experiment \"Stern–Gerlach experiment\"), a charged particle is modeled as a quantum system, while the background magnetic field is described classically.[\\[42\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Peres1993-44): 26  This \"semi-classical\" approach fails if quantum fluctuations in the electromagnetic field play an important role, such as in the emission of photons by [charged particles](https:\/\/en.wikipedia.org\/wiki\/Charged_particle \"Charged particle\").\n\n[Quantum field](https:\/\/en.wikipedia.org\/wiki\/Field_\\(physics\\) \"Field (physics)\") theories for the [strong nuclear force](https:\/\/en.wikipedia.org\/wiki\/Strong_nuclear_force \"Strong nuclear force\") and the [weak nuclear force](https:\/\/en.wikipedia.org\/wiki\/Weak_nuclear_force \"Weak nuclear force\") have also been developed. The quantum field theory of the strong nuclear force is called [quantum chromodynamics](https:\/\/en.wikipedia.org\/wiki\/Quantum_chromodynamics \"Quantum chromodynamics\"), and describes the interactions of subnuclear particles such as [quarks](https:\/\/en.wikipedia.org\/wiki\/Quark \"Quark\") and [gluons](https:\/\/en.wikipedia.org\/wiki\/Gluon \"Gluon\"). The weak nuclear force and the electromagnetic force were unified, in their quantized forms, into a single quantum field theory (known as [electroweak theory](https:\/\/en.wikipedia.org\/wiki\/Electroweak_theory \"Electroweak theory\")), by the physicists [Abdus Salam](https:\/\/en.wikipedia.org\/wiki\/Abdus_Salam \"Abdus Salam\"), [Sheldon Glashow](https:\/\/en.wikipedia.org\/wiki\/Sheldon_Glashow \"Sheldon Glashow\") and [Steven Weinberg](https:\/\/en.wikipedia.org\/wiki\/Steven_Weinberg \"Steven Weinberg\").[\\[47\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-50)\n\nRelation to general relativity\n\nEven though the predictions of both quantum theory and general relativity have been supported by rigorous and repeated [empirical evidence](https:\/\/en.wikipedia.org\/wiki\/Empirical_evidence \"Empirical evidence\"), their abstract formalisms contradict each other and they have proven extremely difficult to incorporate into one consistent, cohesive model. Gravity is negligible in many areas of particle physics, so that unification between general relativity and quantum mechanics is not an urgent issue in those particular applications. However, the lack of a correct theory of [quantum gravity](https:\/\/en.wikipedia.org\/wiki\/Quantum_gravity \"Quantum gravity\") is an important issue in [physical cosmology](https:\/\/en.wikipedia.org\/wiki\/Physical_cosmology \"Physical cosmology\") and the search by physicists for an elegant \"[Theory of Everything](https:\/\/en.wikipedia.org\/wiki\/Theory_of_Everything \"Theory of Everything\")\" (TOE). Consequently, resolving the inconsistencies between both theories has been a major goal of 20th- and 21st-century physics. This TOE would combine not only the models of subatomic physics but also derive the four fundamental forces of nature from a single force or phenomenon.[\\[48\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-NYT-20221010-51)\n\nOne proposal for doing so is [string theory](https:\/\/en.wikipedia.org\/wiki\/String_theory \"String theory\"), which posits that the [point-like particles](https:\/\/en.wikipedia.org\/wiki\/Point_particle \"Point particle\") of [particle physics](https:\/\/en.wikipedia.org\/wiki\/Particle_physics \"Particle physics\") are replaced by [one-dimensional](https:\/\/en.wikipedia.org\/wiki\/Dimension_\\(mathematics_and_physics\\) \"Dimension (mathematics and physics)\") objects called [strings](https:\/\/en.wikipedia.org\/wiki\/String_\\(physics\\) \"String (physics)\"). String theory describes how these strings propagate through space and interact with each other. On distance scales larger than the string scale, a string looks just like an ordinary particle, with its [mass](https:\/\/en.wikipedia.org\/wiki\/Mass \"Mass\"), [charge](https:\/\/en.wikipedia.org\/wiki\/Charge_\\(physics\\) \"Charge (physics)\"), and other properties determined by the [vibrational](https:\/\/en.wikipedia.org\/wiki\/Vibration \"Vibration\") state of the string. In string theory, one of the many vibrational states of the string corresponds to the [graviton](https:\/\/en.wikipedia.org\/wiki\/Graviton \"Graviton\"), a quantum mechanical particle that carries gravitational force.[\\[49\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-52)[\\[50\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-53)\n\nAnother popular theory is [loop quantum gravity](https:\/\/en.wikipedia.org\/wiki\/Loop_quantum_gravity \"Loop quantum gravity\") (LQG), which describes quantum properties of gravity and is thus a theory of [quantum spacetime](https:\/\/en.wikipedia.org\/wiki\/Quantum_spacetime \"Quantum spacetime\"). LQG is an attempt to merge and adapt standard quantum mechanics and standard general relativity. This theory describes space as an extremely fine fabric \"woven\" of finite loops called [spin networks](https:\/\/en.wikipedia.org\/wiki\/Spin_network \"Spin network\"). The evolution of a spin network over time is called a [spin foam](https:\/\/en.wikipedia.org\/wiki\/Spin_foam \"Spin foam\"). The characteristic length scale of a spin foam is the [Planck length](https:\/\/en.wikipedia.org\/wiki\/Planck_length \"Planck length\"), approximately 1.616×10−35 m, and so lengths shorter than the Planck length are not physically meaningful in LQG.[\\[51\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-54)\n\nPhilosophical implications\n\nUnsolved problem in physics\n\nIs there a preferred interpretation of quantum mechanics? How does the quantum description of reality, which includes elements such as the \"[superposition](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle \"Superposition principle\") of states\" and \"[wave function collapse](https:\/\/en.wikipedia.org\/wiki\/Wave_function_collapse \"Wave function collapse\")\", give rise to the reality we perceive?\n\nSince its inception, the many counter-intuitive aspects and results of quantum mechanics have provoked strong [philosophical](https:\/\/en.wikipedia.org\/wiki\/Philosophical \"Philosophical\") debates and many [interpretations](https:\/\/en.wikipedia.org\/wiki\/Interpretations_of_quantum_mechanics \"Interpretations of quantum mechanics\"). The arguments centre on the probabilistic nature of quantum mechanics, the difficulties with [wavefunction collapse](https:\/\/en.wikipedia.org\/wiki\/Wavefunction_collapse \"Wavefunction collapse\") and the related [measurement problem](https:\/\/en.wikipedia.org\/wiki\/Measurement_problem \"Measurement problem\"), and [quantum nonlocality](https:\/\/en.wikipedia.org\/wiki\/Quantum_nonlocality \"Quantum nonlocality\"). Perhaps the only consensus that exists about these issues is that there is no consensus. [Richard Feynman](https:\/\/en.wikipedia.org\/wiki\/Richard_Feynman \"Richard Feynman\") once said, \"I think I can safely say that nobody understands quantum mechanics.\"[\\[52\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-55) According to [Steven Weinberg](https:\/\/en.wikipedia.org\/wiki\/Steven_Weinberg \"Steven Weinberg\"), \"There is now in my opinion no entirely satisfactory interpretation of quantum mechanics.\"[\\[53\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-56)\n\nThe views of [Niels Bohr](https:\/\/en.wikipedia.org\/wiki\/Niels_Bohr \"Niels Bohr\"), Werner Heisenberg and other physicists are often grouped together as the \"[Copenhagen interpretation](https:\/\/en.wikipedia.org\/wiki\/Copenhagen_interpretation \"Copenhagen interpretation\")\".[\\[54\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-57)[\\[55\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-58) According to these views, the probabilistic nature of quantum mechanics is not a *temporary* feature which will eventually be replaced by a deterministic theory, but is instead a *final* renunciation of the classical idea of \"causality\". Bohr in particular emphasized that any well-defined application of the quantum mechanical formalism must always make reference to the experimental arrangement, due to the [complementary](https:\/\/en.wikipedia.org\/wiki\/Complementarity_\\(physics\\) \"Complementarity (physics)\") nature of evidence obtained under different experimental situations. Copenhagen-type interpretations were adopted by Nobel laureates in quantum physics, including Bohr,[\\[56\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-BohrComo-59) Heisenberg,[\\[57\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-60) Schrödinger,[\\[58\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-61) Feynman,[\\[2\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Feynman-2) and [Zeilinger](https:\/\/en.wikipedia.org\/wiki\/Anton_Zeilinger \"Anton Zeilinger\")[\\[59\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-MaKoflerZeilinger-62) as well as 21st-century researchers in quantum foundations.[\\[60\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-:25-63)\n\n[Albert Einstein](https:\/\/en.wikipedia.org\/wiki\/Albert_Einstein \"Albert Einstein\"), himself one of the founders of [quantum theory](https:\/\/en.wikipedia.org\/wiki\/Old_quantum_theory \"Old quantum theory\"), was troubled by its apparent failure to respect some cherished metaphysical principles, such as [determinism](https:\/\/en.wikipedia.org\/wiki\/Determinism \"Determinism\") and [locality](https:\/\/en.wikipedia.org\/wiki\/Principle_of_locality \"Principle of locality\"). Einstein's long-running exchanges with Bohr about the meaning and status of quantum mechanics are now known as the [Bohr–Einstein debates](https:\/\/en.wikipedia.org\/wiki\/Bohr%E2%80%93Einstein_debates \"Bohr–Einstein debates\"). Einstein believed that underlying quantum mechanics must be a theory that explicitly forbids [action at a distance](https:\/\/en.wikipedia.org\/wiki\/Action_at_a_distance \"Action at a distance\"). He argued that quantum mechanics was incomplete, a theory that was valid but not fundamental, analogous to how [thermodynamics](https:\/\/en.wikipedia.org\/wiki\/Thermodynamics \"Thermodynamics\") is valid, but the fundamental theory behind it is [statistical mechanics](https:\/\/en.wikipedia.org\/wiki\/Statistical_mechanics \"Statistical mechanics\"). In 1935, Einstein and his collaborators [Boris Podolsky](https:\/\/en.wikipedia.org\/wiki\/Boris_Podolsky \"Boris Podolsky\") and [Nathan Rosen](https:\/\/en.wikipedia.org\/wiki\/Nathan_Rosen \"Nathan Rosen\") published an argument that the principle of locality implies the incompleteness of quantum mechanics, a [thought experiment](https:\/\/en.wikipedia.org\/wiki\/Thought_experiment \"Thought experiment\") later termed the [Einstein–Podolsky–Rosen paradox](https:\/\/en.wikipedia.org\/wiki\/Einstein%E2%80%93Podolsky%E2%80%93Rosen_paradox \"Einstein–Podolsky–Rosen paradox\").[\\[note 4\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-68) In 1964, [John Bell](https:\/\/en.wikipedia.org\/wiki\/John_Stewart_Bell \"John Stewart Bell\") showed that EPR's principle of locality, together with determinism, was actually incompatible with quantum mechanics: they implied constraints on the correlations produced by distance systems, now known as [Bell inequalities](https:\/\/en.wikipedia.org\/wiki\/Bell_inequalities \"Bell inequalities\"), that can be violated by entangled particles.[\\[65\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-69) Since then [several experiments](https:\/\/en.wikipedia.org\/wiki\/Bell_test \"Bell test\") have been performed to obtain these correlations, with the result that they do in fact violate Bell inequalities, and thus falsify the conjunction of locality with determinism.[\\[16\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-wiseman15-16)[\\[17\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-wolchover17-17)\n\n[Bohmian mechanics](https:\/\/en.wikipedia.org\/wiki\/Bohmian_mechanics \"Bohmian mechanics\") shows that it is possible to reformulate quantum mechanics to make it deterministic, at the price of making it explicitly nonlocal. It attributes not only a wave function to a physical system, but in addition a real position, that evolves deterministically under a nonlocal guiding equation. The evolution of a physical system is given at all times by the Schrödinger equation together with the guiding equation; there is never a collapse of the wave function. This solves the measurement problem.[\\[66\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-70)\n\n[![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/c\/c8\/Schroedingers_cat_film.svg\/250px-Schroedingers_cat_film.svg.png)](https:\/\/en.wikipedia.org\/wiki\/File:Schroedingers_cat_film.svg)\n\nThe [Schrödinger's cat](https:\/\/en.wikipedia.org\/wiki\/Schr%C3%B6dinger%27s_cat \"Schrödinger's cat\") thought experiment can be used to visualize the many-worlds interpretation of quantum mechanics, where a branching of the universe occurs through a superposition of two quantum mechanical states.\n\nEverett's [many-worlds interpretation](https:\/\/en.wikipedia.org\/wiki\/Many-worlds_interpretation \"Many-worlds interpretation\"), formulated in 1956, holds that *all* the possibilities described by quantum theory *simultaneously* occur in a multiverse composed of mostly independent parallel universes.[\\[67\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-71) This is a consequence of removing the axiom of the collapse of the wave packet. All possible states of the measured system and the measuring apparatus, together with the observer, are present in a real physical quantum superposition. While the multiverse is deterministic, we perceive non-deterministic behavior governed by probabilities, because we do not observe the multiverse as a whole, but only one parallel universe at a time. Exactly how this is supposed to work has been the subject of much debate. Several attempts have been made to make sense of this and derive the Born rule,[\\[68\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-dewitt73-72)[\\[69\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-wallace2003-73) with no consensus on whether they have been successful.[\\[70\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-ballentine1973-74)[\\[71\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-75)[\\[72\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-kent2009-76)\n\n[Relational quantum mechanics](https:\/\/en.wikipedia.org\/wiki\/Relational_quantum_mechanics \"Relational quantum mechanics\") appeared in the late 1990s as a modern derivative of Copenhagen-type ideas,[\\[73\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-77)[\\[74\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-78) and [QBism](https:\/\/en.wikipedia.org\/wiki\/QBism \"QBism\") was developed some years later.[\\[75\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-:23-79)[\\[76\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-80)\n\nHistory\n\nQuantum mechanics was developed in the early decades of the 20th century, driven by the need to explain phenomena that, in some cases, had been observed in earlier times. Scientific inquiry into the wave nature of light began in the 17th and 18th centuries, when scientists such as [Robert Hooke](https:\/\/en.wikipedia.org\/wiki\/Robert_Hooke \"Robert Hooke\"), [Christiaan Huygens](https:\/\/en.wikipedia.org\/wiki\/Christiaan_Huygens \"Christiaan Huygens\") and [Leonhard Euler](https:\/\/en.wikipedia.org\/wiki\/Leonhard_Euler \"Leonhard Euler\") proposed a wave theory of light based on experimental observations.[\\[77\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Born_&_Wolf-81) In 1803 English [polymath](https:\/\/en.wikipedia.org\/wiki\/Polymath \"Polymath\") [Thomas Young](https:\/\/en.wikipedia.org\/wiki\/Thomas_Young_\\(scientist\\) \"Thomas Young (scientist)\") described the famous [double-slit experiment](https:\/\/en.wikipedia.org\/wiki\/Young%27s_interference_experiment \"Young's interference experiment\").[\\[78\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-82) This experiment played a major role in the general acceptance of the [wave theory of light](https:\/\/en.wikipedia.org\/wiki\/Wave_theory_of_light \"Wave theory of light\").\n\nDuring the early 19th century, [chemical](https:\/\/en.wikipedia.org\/wiki\/Chemistry \"Chemistry\") research by [John Dalton](https:\/\/en.wikipedia.org\/wiki\/John_Dalton \"John Dalton\") and [Amedeo Avogadro](https:\/\/en.wikipedia.org\/wiki\/Amedeo_Avogadro \"Amedeo Avogadro\") lent weight to the [atomic theory](https:\/\/en.wikipedia.org\/wiki\/Atomic_theory \"Atomic theory\") of matter, an idea that [James Clerk Maxwell](https:\/\/en.wikipedia.org\/wiki\/James_Clerk_Maxwell \"James Clerk Maxwell\"), [Ludwig Boltzmann](https:\/\/en.wikipedia.org\/wiki\/Ludwig_Boltzmann \"Ludwig Boltzmann\") and others built upon to establish the [kinetic theory of gases](https:\/\/en.wikipedia.org\/wiki\/Kinetic_theory_of_gases \"Kinetic theory of gases\"). The successes of kinetic theory gave further credence to the idea that matter is composed of atoms, yet the theory also had shortcomings that would only be resolved by the development of quantum mechanics.[\\[79\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Feynman-kinetic-theory-83) While the early conception of atoms from [Greek philosophy](https:\/\/en.wikipedia.org\/wiki\/Greek_philosophy \"Greek philosophy\") had been that they were indivisible units – the word \"atom\" deriving from the [Greek](https:\/\/en.wikipedia.org\/wiki\/Greek_language \"Greek language\") for 'uncuttable' – the 19th century saw the formulation of hypotheses about subatomic structure. One important discovery in that regard was [Michael Faraday](https:\/\/en.wikipedia.org\/wiki\/Michael_Faraday \"Michael Faraday\")'s 1838 observation of a glow caused by an electrical discharge inside a glass tube containing gas at low pressure. [Julius Plücker](https:\/\/en.wikipedia.org\/wiki\/Julius_Pl%C3%BCcker \"Julius Plücker\"), [Johann Wilhelm Hittorf](https:\/\/en.wikipedia.org\/wiki\/Johann_Wilhelm_Hittorf \"Johann Wilhelm Hittorf\") and [Eugen Goldstein](https:\/\/en.wikipedia.org\/wiki\/Eugen_Goldstein \"Eugen Goldstein\") carried on and improved upon Faraday's work, leading to the identification of [cathode rays](https:\/\/en.wikipedia.org\/wiki\/Cathode_rays \"Cathode rays\"), which [J. J. Thomson](https:\/\/en.wikipedia.org\/wiki\/J._J._Thomson \"J. J. Thomson\") found to consist of subatomic particles that would be called electrons.[\\[80\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-84)[\\[81\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-85)\n\n[![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/a\/a7\/Max_Planck_%281858-1947%29.jpg\/250px-Max_Planck_%281858-1947%29.jpg)](https:\/\/en.wikipedia.org\/wiki\/File:Max_Planck_\\(1858-1947\\).jpg)\n\n[Max Planck](https:\/\/en.wikipedia.org\/wiki\/Max_Planck \"Max Planck\") is considered the father of the quantum theory.\n\nThe [black-body radiation](https:\/\/en.wikipedia.org\/wiki\/Black-body_radiation \"Black-body radiation\") problem was discovered by [Gustav Kirchhoff](https:\/\/en.wikipedia.org\/wiki\/Gustav_Kirchhoff \"Gustav Kirchhoff\") in 1859. In 1900, Max Planck proposed the hypothesis that energy is radiated and absorbed in discrete \"quanta\" (or energy packets), yielding a calculation that precisely matched the observed patterns of black-body radiation.[\\[82\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-86) The word *quantum* derives from the [Latin](https:\/\/en.wikipedia.org\/wiki\/Latin \"Latin\"), meaning \"how great\" or \"how much\".[\\[83\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-87) According to Planck, quantities of energy could be thought of as divided into \"elements\" whose size (*E*) would be proportional to their [frequency](https:\/\/en.wikipedia.org\/wiki\/Frequency \"Frequency\") (*ν*): ![{\\\\displaystyle E=h\\\\nu \\\\ }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/24767663bb931daecf1eb628be4fea46f32e3622), where *h* is the [Planck constant](https:\/\/en.wikipedia.org\/wiki\/Planck_constant \"Planck constant\"). Planck cautiously insisted that this was only an aspect of the processes of absorption and emission of radiation and was not the *physical reality* of the radiation.[\\[84\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-88) In fact, he considered his quantum hypothesis a mathematical trick to get the right answer rather than a sizable discovery.[\\[85\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Kragh-89) However, in 1905 Albert Einstein interpreted Planck's quantum hypothesis [realistically](https:\/\/en.wikipedia.org\/wiki\/Local_realism \"Local realism\") and used it to explain the [photoelectric effect](https:\/\/en.wikipedia.org\/wiki\/Photoelectric_effect \"Photoelectric effect\"), in which shining light on certain materials can eject electrons from the material. Niels Bohr then developed Planck's ideas about radiation into a [model of the hydrogen atom](https:\/\/en.wikipedia.org\/wiki\/Bohr_model \"Bohr model\") that successfully predicted the [spectral lines](https:\/\/en.wikipedia.org\/wiki\/Spectral_line \"Spectral line\") of hydrogen.[\\[86\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-90) Einstein further developed this idea to show that an [electromagnetic wave](https:\/\/en.wikipedia.org\/wiki\/Electromagnetic_wave \"Electromagnetic wave\") such as light could also be described as a particle (later called the photon), with a discrete amount of energy that depends on its frequency.[\\[87\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-91) In his paper \"On the Quantum Theory of Radiation\", Einstein expanded on the interaction between energy and matter to explain the absorption and emission of energy by atoms. Although overshadowed at the time by his general theory of relativity, this paper articulated the mechanism underlying the stimulated emission of radiation,[\\[88\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-92) which became the basis of the laser.[\\[89\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-93)\n\n[![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/6\/6e\/Solvay_conference_1927.jpg\/500px-Solvay_conference_1927.jpg)](https:\/\/en.wikipedia.org\/wiki\/File:Solvay_conference_1927.jpg)\n\nThe 1927 [Solvay Conference](https:\/\/en.wikipedia.org\/wiki\/Solvay_Conference \"Solvay Conference\") in [Brussels](https:\/\/en.wikipedia.org\/wiki\/Brussels \"Brussels\") was the fifth world physics conference.\n\nThis phase is known as the [old quantum theory](https:\/\/en.wikipedia.org\/wiki\/Old_quantum_theory \"Old quantum theory\"). Never complete or self-consistent, the old quantum theory was rather a set of [heuristic](https:\/\/en.wikipedia.org\/wiki\/Heuristic \"Heuristic\") corrections to classical mechanics.[\\[90\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-terHaar-94)[\\[91\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-95) The theory is now understood as a [semi-classical approximation](https:\/\/en.wikipedia.org\/wiki\/WKB_approximation#Application_to_the_Schr%C3%B6dinger_equation \"WKB approximation\") to modern quantum mechanics.[\\[92\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-96)[\\[93\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-97) Notable results from this period include, in addition to the work of Planck, Einstein and Bohr mentioned above, Einstein and [Peter Debye](https:\/\/en.wikipedia.org\/wiki\/Peter_Debye \"Peter Debye\")'s work on the [specific heat](https:\/\/en.wikipedia.org\/wiki\/Specific_heat \"Specific heat\") of solids, Bohr and [Hendrika Johanna van Leeuwen](https:\/\/en.wikipedia.org\/wiki\/Hendrika_Johanna_van_Leeuwen \"Hendrika Johanna van Leeuwen\")'s [proof](https:\/\/en.wikipedia.org\/wiki\/Bohr%E2%80%93Van_Leeuwen_theorem \"Bohr–Van Leeuwen theorem\") that classical physics cannot account for [diamagnetism](https:\/\/en.wikipedia.org\/wiki\/Diamagnetism \"Diamagnetism\"), and [Arnold Sommerfeld](https:\/\/en.wikipedia.org\/wiki\/Arnold_Sommerfeld \"Arnold Sommerfeld\")'s extension of the Bohr model to include special-relativistic effects.[\\[90\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-terHaar-94)[\\[94\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Aharoni-98)\n\nIn the mid-1920s quantum mechanics was developed to become the standard formulation for atomic physics. In 1923, the French physicist [Louis de Broglie](https:\/\/en.wikipedia.org\/wiki\/Louis_de_Broglie \"Louis de Broglie\") put forward his theory of matter waves by stating that particles can exhibit wave characteristics and vice versa. Building on de Broglie's approach, modern quantum mechanics was born in 1925, when the German physicists Werner Heisenberg, Max Born, and [Pascual Jordan](https:\/\/en.wikipedia.org\/wiki\/Pascual_Jordan \"Pascual Jordan\")[\\[95\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Edwards79-99)[\\[96\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Edwards81-100) developed [matrix mechanics](https:\/\/en.wikipedia.org\/wiki\/Matrix_mechanics \"Matrix mechanics\") and the Austrian physicist Erwin Schrödinger invented [wave mechanics](https:\/\/en.wikipedia.org\/wiki\/Schr%C3%B6dinger_equation \"Schrödinger equation\"). Born introduced the probabilistic interpretation of Schrödinger's wave function in July 1926.[\\[97\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-101) Thus, the entire field of quantum physics emerged, leading to its wider acceptance at the Fifth [Solvay Conference](https:\/\/en.wikipedia.org\/wiki\/Solvay_Conference \"Solvay Conference\") in 1927.[\\[98\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-pais1997-102)\n\nBy 1930, quantum mechanics had been further unified and formalized by [David Hilbert](https:\/\/en.wikipedia.org\/wiki\/David_Hilbert \"David Hilbert\"), Paul Dirac and [John von Neumann](https:\/\/en.wikipedia.org\/wiki\/John_von_Neumann \"John von Neumann\")[\\[99\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-103) with greater emphasis on [measurement](https:\/\/en.wikipedia.org\/wiki\/Measurement_in_quantum_mechanics \"Measurement in quantum mechanics\"), the statistical nature of our knowledge of reality, and [philosophical speculation about the 'observer'](https:\/\/en.wikipedia.org\/wiki\/Interpretations_of_quantum_mechanics \"Interpretations of quantum mechanics\"). It has since permeated many disciplines, including quantum chemistry, [quantum electronics](https:\/\/en.wikipedia.org\/wiki\/Quantum_electronics \"Quantum electronics\"), [quantum optics](https:\/\/en.wikipedia.org\/wiki\/Quantum_optics \"Quantum optics\"), and [quantum information science](https:\/\/en.wikipedia.org\/wiki\/Quantum_information_science \"Quantum information science\"). It also provides a useful framework for many features of the modern [periodic table of elements](https:\/\/en.wikipedia.org\/wiki\/Periodic_table_of_elements \"Periodic table of elements\"), and describes the behaviors of [atoms](https:\/\/en.wikipedia.org\/wiki\/Atoms \"Atoms\") during [chemical bonding](https:\/\/en.wikipedia.org\/wiki\/Chemical_bond \"Chemical bond\") and the flow of electrons in computer [semiconductors](https:\/\/en.wikipedia.org\/wiki\/Semiconductor \"Semiconductor\"), and therefore plays a crucial role in many modern technologies. While quantum mechanics was constructed to describe the world of the very small, it is also needed to explain some [macroscopic](https:\/\/en.wikipedia.org\/wiki\/Macroscopic \"Macroscopic\") phenomena such as [superconductors](https:\/\/en.wikipedia.org\/wiki\/Superconductors \"Superconductors\")[\\[100\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-feynman2015-104) and [superfluids](https:\/\/en.wikipedia.org\/wiki\/Superfluid \"Superfluid\").[\\[101\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-105)\n\nSee also\n\n- [Bra–ket notation](https:\/\/en.wikipedia.org\/wiki\/Bra%E2%80%93ket_notation \"Bra–ket notation\")\n- [Einstein's thought experiments](https:\/\/en.wikipedia.org\/wiki\/Einstein%27s_thought_experiments \"Einstein's thought experiments\")\n- [List of textbooks on classical mechanics and quantum mechanics](https:\/\/en.wikipedia.org\/wiki\/List_of_textbooks_on_classical_mechanics_and_quantum_mechanics \"List of textbooks on classical mechanics and quantum mechanics\")\n- [Macroscopic quantum phenomena](https:\/\/en.wikipedia.org\/wiki\/Macroscopic_quantum_phenomena \"Macroscopic quantum phenomena\")\n- [Phase-space formulation](https:\/\/en.wikipedia.org\/wiki\/Phase-space_formulation \"Phase-space formulation\")\n- [Regularization (physics)](https:\/\/en.wikipedia.org\/wiki\/Regularization_\\(physics\\) \"Regularization (physics)\")\n- [Two-state quantum system](https:\/\/en.wikipedia.org\/wiki\/Two-state_quantum_system \"Two-state quantum system\")\n\nExplanatory notes\n\n1. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-35)** A momentum eigenstate would be a perfectly monochromatic wave of infinite extent, which is not square-integrable. Likewise, a position eigenstate would be a [Dirac delta distribution](https:\/\/en.wikipedia.org\/wiki\/Dirac_delta_distribution \"Dirac delta distribution\"), not square-integrable and technically not a function at all. Consequently, neither can belong to the particle's Hilbert space. Physicists sometimes introduce fictitious \"bases\" for a Hilbert space comprising elements outside that space. These are invented for calculational convenience and do not represent physical states.[\\[26\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-Cohen-Tannoudji-26): 100–105\n2. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-feynmanIII_40-0)** See, for example, [the Feynman Lectures on Physics](https:\/\/en.wikipedia.org\/wiki\/The_Feynman_Lectures_on_Physics \"The Feynman Lectures on Physics\") for some of the technological applications which use quantum mechanics, e.g., [transistors](https:\/\/en.wikipedia.org\/wiki\/Transistor \"Transistor\") (vol **III**, pp. 14–11 ff), [integrated circuits](https:\/\/en.wikipedia.org\/wiki\/Integrated_circuit \"Integrated circuit\"), which are follow-on technology in solid-state physics (vol **II**, pp. 8–6), and [lasers](https:\/\/en.wikipedia.org\/wiki\/Laser \"Laser\") (vol **III**, pp. 9–13).\n3. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-46)** See *[Macroscopic quantum phenomena](https:\/\/en.wikipedia.org\/wiki\/Macroscopic_quantum_phenomena \"Macroscopic quantum phenomena\")*, *[Bose–Einstein condensate](https:\/\/en.wikipedia.org\/wiki\/Bose%E2%80%93Einstein_condensate \"Bose–Einstein condensate\")*, and *[Quantum machine](https:\/\/en.wikipedia.org\/wiki\/Quantum_machine \"Quantum machine\")*\n4. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-68)** The published form of the EPR argument was due to Podolsky, and Einstein himself was not satisfied with it. In his own publications and correspondence, Einstein used a different argument to insist that quantum mechanics is an incomplete theory.[\\[61\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-spekkens-64)[\\[62\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-howard-65)[\\[63\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-66)[\\[64\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_note-67)\n\nReferences\n\n1. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-Born1926_1-0)**\n   [Born, M.](https:\/\/en.wikipedia.org\/wiki\/Max_Born \"Max Born\") (1926). \"Zur Quantenmechanik der Stoßvorgänge\" \\[On the Quantum Mechanics of Collision Processes\\]. *Zeitschrift für Physik* (in German). **37** (12): 863–867\\. [Bibcode](https:\/\/en.wikipedia.org\/wiki\/Bibcode_\\(identifier\\) \"Bibcode (identifier)\"):[1926ZPhy...37..863B](https:\/\/ui.adsabs.harvard.edu\/abs\/1926ZPhy...37..863B). [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1007\/BF01397477](https:\/\/doi.org\/10.1007%2FBF01397477). [ISSN](https:\/\/en.wikipedia.org\/wiki\/ISSN_\\(identifier\\) \"ISSN (identifier)\") [1434-6001](https:\/\/search.worldcat.org\/issn\/1434-6001). [S2CID](https:\/\/en.wikipedia.org\/wiki\/S2CID_\\(identifier\\) \"S2CID (identifier)\") [119896026](https:\/\/api.semanticscholar.org\/CorpusID:119896026).\n2. ^ [***a***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-Feynman_2-0) [***b***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-Feynman_2-1) [***c***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-Feynman_2-2) [***d***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-Feynman_2-3)\n   Feynman, Richard; Leighton, Robert; Sands, Matthew (1964). [*The Feynman Lectures on Physics*](https:\/\/feynmanlectures.caltech.edu\/III_01.html). Vol. 3. California Institute of Technology. Retrieved 19 December 2020.\n   \n   Reprinted, Addison-Wesley, 1989, [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n   \n   [978-0-201-50064-6](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-201-50064-6 \"Special:BookSources\/978-0-201-50064-6\")\n3. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-3)**\n   Jaeger, Gregg (September 2014). \"What in the (quantum) world is macroscopic?\". *American Journal of Physics*. **82** (9): 896–905\\. [Bibcode](https:\/\/en.wikipedia.org\/wiki\/Bibcode_\\(identifier\\) \"Bibcode (identifier)\"):[2014AmJPh..82..896J](https:\/\/ui.adsabs.harvard.edu\/abs\/2014AmJPh..82..896J). [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1119\/1.4878358](https:\/\/doi.org\/10.1119%2F1.4878358).\n4. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-4)**\n   Fein, Yaakov Y.; Geyer, Philipp; Zwick, Patrick; Kiałka, Filip; Pedalino, Sebastian; Mayor, Marcel; Gerlich, Stefan; Arndt, Markus (September 2019). \"Quantum superposition of molecules beyond 25 kDa\". *Nature Physics*. **15** (12): 1242–1245\\. [Bibcode](https:\/\/en.wikipedia.org\/wiki\/Bibcode_\\(identifier\\) \"Bibcode (identifier)\"):[2019NatPh..15.1242F](https:\/\/ui.adsabs.harvard.edu\/abs\/2019NatPh..15.1242F). [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1038\/s41567-019-0663-9](https:\/\/doi.org\/10.1038%2Fs41567-019-0663-9). [S2CID](https:\/\/en.wikipedia.org\/wiki\/S2CID_\\(identifier\\) \"S2CID (identifier)\") [203638258](https:\/\/api.semanticscholar.org\/CorpusID:203638258).\n5. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-5)**\n   Bojowald, Martin (2015). \"Quantum cosmology: a review\". *Reports on Progress in Physics*. **78** (2) 023901. [arXiv](https:\/\/en.wikipedia.org\/wiki\/ArXiv_\\(identifier\\) \"ArXiv (identifier)\"):[1501\\.04899](https:\/\/arxiv.org\/abs\/1501.04899). [Bibcode](https:\/\/en.wikipedia.org\/wiki\/Bibcode_\\(identifier\\) \"Bibcode (identifier)\"):[2015RPPh...78b3901B](https:\/\/ui.adsabs.harvard.edu\/abs\/2015RPPh...78b3901B). [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1088\/0034-4885\/78\/2\/023901](https:\/\/doi.org\/10.1088%2F0034-4885%2F78%2F2%2F023901). [PMID](https:\/\/en.wikipedia.org\/wiki\/PMID_\\(identifier\\) \"PMID (identifier)\") [25582917](https:\/\/pubmed.ncbi.nlm.nih.gov\/25582917). [S2CID](https:\/\/en.wikipedia.org\/wiki\/S2CID_\\(identifier\\) \"S2CID (identifier)\") [18463042](https:\/\/api.semanticscholar.org\/CorpusID:18463042).\n6. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-6)**\n7. ^ [***a***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-Zwiebach2022_7-0) [***b***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-Zwiebach2022_7-1) [***c***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-Zwiebach2022_7-2) [***d***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-Zwiebach2022_7-3) [***e***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-Zwiebach2022_7-4) [***f***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-Zwiebach2022_7-5) [***g***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-Zwiebach2022_7-6) [***h***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-Zwiebach2022_7-7) [***i***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-Zwiebach2022_7-8) [***j***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-Zwiebach2022_7-9)\n   [Zwiebach, Barton](https:\/\/en.wikipedia.org\/wiki\/Barton_Zwiebach \"Barton Zwiebach\") (2022). *Mastering Quantum Mechanics: Essentials, Theory, and Applications*. MIT Press. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n   \n   [978-0-262-04613-8](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-262-04613-8 \"Special:BookSources\/978-0-262-04613-8\")\n   \n   .\n8. ^ [***a***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-Lederman_8-0) [***b***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-Lederman_8-1) [***c***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-Lederman_8-2)\n   Lederman, Leon M.; Hill, Christopher T. (2011). [*Quantum Physics for Poets*](https:\/\/books.google.com\/books?id=qY_yOwHg_WYC&pg=PA102). US: Prometheus Books. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n   \n   [978-1-61614-281-0](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-1-61614-281-0 \"Special:BookSources\/978-1-61614-281-0\")\n   \n   .\n9. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-M%C3%BCller-Kirsten_9-0)**\n   Müller-Kirsten, H. J. W. (2006). [*Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral*](https:\/\/books.google.com\/books?id=p1_Z81Le58MC&pg=PA14). US: World Scientific. p. 14. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n   \n   [978-981-256-691-1](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-981-256-691-1 \"Special:BookSources\/978-981-256-691-1\")\n   \n   .\n10. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-Plotnitsky_10-0)**\n    Plotnitsky, Arkady (2012). [*Niels Bohr and Complementarity: An Introduction*](https:\/\/books.google.com\/books?id=dmdUp97S4AYC&pg=PA75). US: Springer. pp. 75–76\\. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-1-4614-4517-3](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-1-4614-4517-3 \"Special:BookSources\/978-1-4614-4517-3\")\n    \n    .\n11. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-11)**\n    [Griffiths, David J.](https:\/\/en.wikipedia.org\/wiki\/David_J._Griffiths \"David J. Griffiths\") (1995). [*Introduction to Quantum Mechanics*](https:\/\/en.wikipedia.org\/wiki\/Introduction_to_Quantum_Mechanics_\\(book\\) \"Introduction to Quantum Mechanics (book)\"). Prentice Hall. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [0-13-124405-1](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/0-13-124405-1 \"Special:BookSources\/0-13-124405-1\")\n    \n    .\n12. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-Trixler2013_12-0)**\n    Trixler, F. (2013). [\"Quantum tunnelling to the origin and evolution of life\"](https:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC3768233). *Current Organic Chemistry*. **17** (16): 1758–1770\\. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.2174\/13852728113179990083](https:\/\/doi.org\/10.2174%2F13852728113179990083). [PMC](https:\/\/en.wikipedia.org\/wiki\/PMC_\\(identifier\\) \"PMC (identifier)\") [3768233](https:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC3768233). [PMID](https:\/\/en.wikipedia.org\/wiki\/PMID_\\(identifier\\) \"PMID (identifier)\") [24039543](https:\/\/pubmed.ncbi.nlm.nih.gov\/24039543).\n13. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-13)**\n    Phifer, Arnold (2012-03-27). [\"Developing more energy-efficient transistors through quantum tunneling\"](https:\/\/news.nd.edu\/news\/developing-more-energy-efficient-transistors-through-quantum-tunneling\/). *Notre Dame News*. Retrieved 2024-06-07.\n14. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-14)**\n    [Bub, Jeffrey](https:\/\/en.wikipedia.org\/wiki\/Jeffrey_Bub \"Jeffrey Bub\") (2019). [\"Quantum entanglement\"](https:\/\/plato.stanford.edu\/entries\/qt-entangle\/). In Zalta, Edward N. (ed.). *Stanford Encyclopedia of Philosophy*. Metaphysics Research Lab, Stanford University.\n15. ^ [***a***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-Caves_15-0) [***b***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-Caves_15-1)\n    [Caves, Carlton M.](https:\/\/en.wikipedia.org\/wiki\/Carlton_M._Caves \"Carlton M. Caves\") (2015). \"Quantum Information Science: Emerging No More\". In Kelley, Paul; Agrawal, Govind; Bass, Mike; Hecht, Jeff; Stroud, Carlos (eds.). *OSA Century of Optics*. [The Optical Society](https:\/\/en.wikipedia.org\/wiki\/The_Optical_Society \"The Optical Society\"). pp. 320–323\\. [arXiv](https:\/\/en.wikipedia.org\/wiki\/ArXiv_\\(identifier\\) \"ArXiv (identifier)\"):[1302\\.1864](https:\/\/arxiv.org\/abs\/1302.1864). [Bibcode](https:\/\/en.wikipedia.org\/wiki\/Bibcode_\\(identifier\\) \"Bibcode (identifier)\"):[2013arXiv1302.1864C](https:\/\/ui.adsabs.harvard.edu\/abs\/2013arXiv1302.1864C). [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-1-943580-04-0](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-1-943580-04-0 \"Special:BookSources\/978-1-943580-04-0\")\n    \n    .\n16. ^ [***a***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-wiseman15_16-0) [***b***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-wiseman15_16-1)\n    [Wiseman, Howard](https:\/\/en.wikipedia.org\/wiki\/Howard_M._Wiseman \"Howard M. Wiseman\") (October 2015). [\"Death by experiment for local realism\"](https:\/\/doi.org\/10.1038%2Fnature15631). *[Nature](https:\/\/en.wikipedia.org\/wiki\/Nature_\\(journal\\) \"Nature (journal)\")*. **526** (7575): 649–650\\. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1038\/nature15631](https:\/\/doi.org\/10.1038%2Fnature15631). [ISSN](https:\/\/en.wikipedia.org\/wiki\/ISSN_\\(identifier\\) \"ISSN (identifier)\") [0028-0836](https:\/\/search.worldcat.org\/issn\/0028-0836). [PMID](https:\/\/en.wikipedia.org\/wiki\/PMID_\\(identifier\\) \"PMID (identifier)\") [26503054](https:\/\/pubmed.ncbi.nlm.nih.gov\/26503054).\n17. ^ [***a***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-wolchover17_17-0) [***b***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-wolchover17_17-1)\n    [Wolchover, Natalie](https:\/\/en.wikipedia.org\/wiki\/Natalie_Wolchover \"Natalie Wolchover\") (7 February 2017). [\"Experiment Reaffirms Quantum Weirdness\"](https:\/\/www.quantamagazine.org\/20170207-bell-test-quantum-loophole\/). *[Quanta Magazine](https:\/\/en.wikipedia.org\/wiki\/Quanta_Magazine \"Quanta Magazine\")*. Retrieved 8 February 2020.\n18. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-18)**\n    [Baez, John C.](https:\/\/en.wikipedia.org\/wiki\/John_C._Baez \"John C. Baez\") (20 March 2020). [\"How to Learn Math and Physics\"](https:\/\/math.ucr.edu\/home\/baez\/books.html). *University of California, Riverside*. Retrieved 19 December 2020. \"there's no way to understand the interpretation of quantum mechanics without also being able to *solve quantum mechanics problems* – to understand the theory, you need to be able to use it (and vice versa)\"\n19. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-19)**\n    [Sagan, Carl](https:\/\/en.wikipedia.org\/wiki\/Carl_Sagan \"Carl Sagan\") (1996). [*The Demon-Haunted World: Science as a Candle in the Dark*](https:\/\/en.wikipedia.org\/wiki\/The_Demon-Haunted_World \"The Demon-Haunted World\"). Ballantine Books. p. 249. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [0-345-40946-9](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/0-345-40946-9 \"Special:BookSources\/0-345-40946-9\")\n    \n    . \"\"For most physics students, (the \"mathematical underpinning\" of quantum mechanics) might occupy them from, say, third grade to early graduate school – roughly 15 years. ... The job of the popularizer of science, trying to get across some idea of quantum mechanics to a general audience that has not gone through these initiation rites, is daunting. Indeed, there are no successful popularizations of quantum mechanics in my opinion – partly for this reason.\"\n20. ^ [***a***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-Holevo2001_20-0) [***b***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-Holevo2001_20-1) [***c***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-Holevo2001_20-2)\n    [Holevo, Alexander S.](https:\/\/en.wikipedia.org\/wiki\/Alexander_Holevo \"Alexander Holevo\") (2001). *Statistical Structure of Quantum Theory*. Lecture Notes in Physics. Springer. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [3-540-42082-7](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/3-540-42082-7 \"Special:BookSources\/3-540-42082-7\")\n    \n    . [OCLC](https:\/\/en.wikipedia.org\/wiki\/OCLC_\\(identifier\\) \"OCLC (identifier)\") [318268606](https:\/\/search.worldcat.org\/oclc\/318268606).\n21. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-21)**\n    Hall, Brian C. (2013). *Quantum Theory for Mathematicians*. [Graduate Texts in Mathematics](https:\/\/en.wikipedia.org\/wiki\/Graduate_Texts_in_Mathematics \"Graduate Texts in Mathematics\"). Vol. 267. Springer. p. 125. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-1-4614-7115-8](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-1-4614-7115-8 \"Special:BookSources\/978-1-4614-7115-8\")\n    \n    .\n22. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-22)**\n    Weinberg, Steven (2010). [*Dreams Of A Final Theory: The Search for The Fundamental Laws of Nature*](https:\/\/books.google.com\/books?id=OLrZkgPsZR0C). Random House. p. [82](https:\/\/books.google.com\/books?id=OLrZkgPsZR0C&pg=PT82). [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-1-4070-6396-6](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-1-4070-6396-6 \"Special:BookSources\/978-1-4070-6396-6\")\n    \n    .\n23. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-23)**\n    Zhang, Ruiqin; Deng, Conghao (1993). \"Exact solutions of the Schrödinger equation for some quantum-mechanical many-body systems\". *Physical Review A*. **47** (1): 71–77\\. [Bibcode](https:\/\/en.wikipedia.org\/wiki\/Bibcode_\\(identifier\\) \"Bibcode (identifier)\"):[1993PhRvA..47...71Z](https:\/\/ui.adsabs.harvard.edu\/abs\/1993PhRvA..47...71Z). [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1103\/PhysRevA.47.71](https:\/\/doi.org\/10.1103%2FPhysRevA.47.71). [ISSN](https:\/\/en.wikipedia.org\/wiki\/ISSN_\\(identifier\\) \"ISSN (identifier)\") [1050-2947](https:\/\/search.worldcat.org\/issn\/1050-2947). [PMID](https:\/\/en.wikipedia.org\/wiki\/PMID_\\(identifier\\) \"PMID (identifier)\") [9908895](https:\/\/pubmed.ncbi.nlm.nih.gov\/9908895).\n24. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-24)**\n    Li, Jing; Drummond, N. D.; Schuck, Peter; Olevano, Valerio (2019-04-01). [\"Comparing many-body approaches against the helium atom exact solution\"](https:\/\/doi.org\/10.21468%2FSciPostPhys.6.4.040). *SciPost Physics*. **6** (4): 40. [arXiv](https:\/\/en.wikipedia.org\/wiki\/ArXiv_\\(identifier\\) \"ArXiv (identifier)\"):[1801\\.09977](https:\/\/arxiv.org\/abs\/1801.09977). [Bibcode](https:\/\/en.wikipedia.org\/wiki\/Bibcode_\\(identifier\\) \"Bibcode (identifier)\"):[2019ScPP....6...40L](https:\/\/ui.adsabs.harvard.edu\/abs\/2019ScPP....6...40L). [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.21468\/SciPostPhys.6.4.040](https:\/\/doi.org\/10.21468%2FSciPostPhys.6.4.040). [ISSN](https:\/\/en.wikipedia.org\/wiki\/ISSN_\\(identifier\\) \"ISSN (identifier)\") [2542-4653](https:\/\/search.worldcat.org\/issn\/2542-4653).\n25. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-25)**\n    Drake, Gordon W. F. (2023). \"High Precision Calculations for Helium\". In Drake, Gordon W. F. (ed.). *Springer Handbook of Atomic, Molecular, and Optical Physics*. Springer Handbooks. Cham: Springer International Publishing. pp. 199–216\\. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1007\/978-3-030-73893-8\\_12](https:\/\/doi.org\/10.1007%2F978-3-030-73893-8_12). [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-3-030-73892-1](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-3-030-73892-1 \"Special:BookSources\/978-3-030-73892-1\")\n    \n    .\n26. ^ [***a***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-Cohen-Tannoudji_26-0) [***b***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-Cohen-Tannoudji_26-1) [***c***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-Cohen-Tannoudji_26-2) [***d***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-Cohen-Tannoudji_26-3)\n    [Cohen-Tannoudji, Claude](https:\/\/en.wikipedia.org\/wiki\/Claude_Cohen-Tannoudji \"Claude Cohen-Tannoudji\"); Diu, Bernard; Laloë, Franck (2005). *Quantum Mechanics*. Translated by Hemley, Susan Reid; Ostrowsky, Nicole; Ostrowsky, Dan. John Wiley & Sons. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [0-471-16433-X](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/0-471-16433-X \"Special:BookSources\/0-471-16433-X\")\n    \n    .\n27. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-L&L_27-0)**\n    [Landau, Lev D.](https:\/\/en.wikipedia.org\/wiki\/Lev_Landau \"Lev Landau\"); [Lifschitz, Evgeny M.](https:\/\/en.wikipedia.org\/wiki\/Evgeny_Lifshitz \"Evgeny Lifshitz\") (1977). [*Quantum Mechanics: Non-Relativistic Theory*](https:\/\/archive.org\/details\/QuantumMechanics_104). Vol. 3 (3rd ed.). [Pergamon Press](https:\/\/en.wikipedia.org\/wiki\/Pergamon_Press \"Pergamon Press\"). [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-0-08-020940-1](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-08-020940-1 \"Special:BookSources\/978-0-08-020940-1\")\n    \n    . [OCLC](https:\/\/en.wikipedia.org\/wiki\/OCLC_\\(identifier\\) \"OCLC (identifier)\") [2284121](https:\/\/search.worldcat.org\/oclc\/2284121).\n28. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-ballentine1970_28-0)**\n    Section 3.2 of\n    \n    Ballentine, Leslie E. (1970), \"The Statistical Interpretation of Quantum Mechanics\", *Reviews of Modern Physics*, **42** (4): 358–381, [Bibcode](https:\/\/en.wikipedia.org\/wiki\/Bibcode_\\(identifier\\) \"Bibcode (identifier)\"):[1970RvMP...42..358B](https:\/\/ui.adsabs.harvard.edu\/abs\/1970RvMP...42..358B), [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1103\/RevModPhys.42.358](https:\/\/doi.org\/10.1103%2FRevModPhys.42.358), [S2CID](https:\/\/en.wikipedia.org\/wiki\/S2CID_\\(identifier\\) \"S2CID (identifier)\") [120024263](https:\/\/api.semanticscholar.org\/CorpusID:120024263)\n    \n    . This fact is experimentally well-known for example in quantum optics; see e.g. chap. 2 and Fig. 2.1\n    \n    Leonhardt, Ulf (1997), [*Measuring the Quantum State of Light*](https:\/\/archive.org\/details\/measuringquantum0000leon), Cambridge: Cambridge University Press, [Bibcode](https:\/\/en.wikipedia.org\/wiki\/Bibcode_\\(identifier\\) \"Bibcode (identifier)\"):[1997mqsl.book.....L](https:\/\/ui.adsabs.harvard.edu\/abs\/1997mqsl.book.....L), [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [0-521-49730-2](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/0-521-49730-2 \"Special:BookSources\/0-521-49730-2\")\n    \n    .\n29. ^ [***a***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-:0_29-0) [***b***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-:0_29-1) [***c***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-:0_29-2)\n    [Nielsen, Michael A.](https:\/\/en.wikipedia.org\/wiki\/Michael_Nielsen \"Michael Nielsen\"); [Chuang, Isaac L.](https:\/\/en.wikipedia.org\/wiki\/Isaac_Chuang \"Isaac Chuang\") (2010). *Quantum Computation and Quantum Information* (2nd ed.). Cambridge University Press. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-1-107-00217-3](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-1-107-00217-3 \"Special:BookSources\/978-1-107-00217-3\")\n    \n    . [OCLC](https:\/\/en.wikipedia.org\/wiki\/OCLC_\\(identifier\\) \"OCLC (identifier)\") [844974180](https:\/\/search.worldcat.org\/oclc\/844974180).\n30. ^ [***a***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-:1_30-0) [***b***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-:1_30-1)\n    [Rieffel, Eleanor G.](https:\/\/en.wikipedia.org\/wiki\/Eleanor_Rieffel \"Eleanor Rieffel\"); Polak, Wolfgang H. (2011). [*Quantum Computing: A Gentle Introduction*](https:\/\/en.wikipedia.org\/wiki\/Quantum_Computing:_A_Gentle_Introduction \"Quantum Computing: A Gentle Introduction\"). MIT Press. 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[S2CID](https:\/\/en.wikipedia.org\/wiki\/S2CID_\\(identifier\\) \"S2CID (identifier)\") [55537196](https:\/\/api.semanticscholar.org\/CorpusID:55537196).\n61. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-spekkens_64-0)**\n    Harrigan, Nicholas; [Spekkens, Robert W.](https:\/\/en.wikipedia.org\/wiki\/Robert_Spekkens \"Robert Spekkens\") (2010). \"Einstein, incompleteness, and the epistemic view of quantum states\". *[Foundations of Physics](https:\/\/en.wikipedia.org\/wiki\/Foundations_of_Physics \"Foundations of Physics\")*. **40** (2): 125. [arXiv](https:\/\/en.wikipedia.org\/wiki\/ArXiv_\\(identifier\\) \"ArXiv (identifier)\"):[0706\\.2661](https:\/\/arxiv.org\/abs\/0706.2661). [Bibcode](https:\/\/en.wikipedia.org\/wiki\/Bibcode_\\(identifier\\) \"Bibcode (identifier)\"):[2010FoPh...40..125H](https:\/\/ui.adsabs.harvard.edu\/abs\/2010FoPh...40..125H). 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[doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1016\/0039-3681(85)90001-9](https:\/\/doi.org\/10.1016%2F0039-3681%2885%2990001-9).\n63. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-66)**\n    [Sauer, Tilman](https:\/\/en.wikipedia.org\/wiki\/Tilman_Sauer \"Tilman Sauer\") (1 December 2007). [\"An Einstein manuscript on the EPR paradox for spin observables\"](http:\/\/philsci-archive.pitt.edu\/3222\/). *Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics*. **38** (4): 879–887\\. [Bibcode](https:\/\/en.wikipedia.org\/wiki\/Bibcode_\\(identifier\\) \"Bibcode (identifier)\"):[2007SHPMP..38..879S](https:\/\/ui.adsabs.harvard.edu\/abs\/2007SHPMP..38..879S). [CiteSeerX](https:\/\/en.wikipedia.org\/wiki\/CiteSeerX_\\(identifier\\) \"CiteSeerX (identifier)\") [10\\.1.1.571.6089](https:\/\/citeseerx.ist.psu.edu\/viewdoc\/summary?doi=10.1.1.571.6089). [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1016\/j.shpsb.2007.03.002](https:\/\/doi.org\/10.1016%2Fj.shpsb.2007.03.002). [ISSN](https:\/\/en.wikipedia.org\/wiki\/ISSN_\\(identifier\\) \"ISSN (identifier)\") [1355-2198](https:\/\/search.worldcat.org\/issn\/1355-2198).\n64. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-67)**\n    Einstein, Albert (1949). \"Autobiographical Notes\". In Schilpp, Paul Arthur (ed.). *Albert Einstein: Philosopher-Scientist*. Open Court Publishing Company.\n65. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-69)**\n    [Bell, John Stewart](https:\/\/en.wikipedia.org\/wiki\/John_Stewart_Bell \"John Stewart Bell\") (1 November 1964). [\"On the Einstein Podolsky Rosen paradox\"](https:\/\/doi.org\/10.1103%2FPhysicsPhysiqueFizika.1.195). *[Physics Physique Fizika](https:\/\/en.wikipedia.org\/wiki\/Physics_Physique_Fizika \"Physics Physique Fizika\")*. **1** (3): 195–200\\. [Bibcode](https:\/\/en.wikipedia.org\/wiki\/Bibcode_\\(identifier\\) \"Bibcode (identifier)\"):[1964PhyNY...1..195B](https:\/\/ui.adsabs.harvard.edu\/abs\/1964PhyNY...1..195B). [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1103\/PhysicsPhysiqueFizika.1.195](https:\/\/doi.org\/10.1103%2FPhysicsPhysiqueFizika.1.195).\n66. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-70)**\n    Goldstein, Sheldon (2017). [\"Bohmian Mechanics\"](https:\/\/plato.stanford.edu\/entries\/qm-bohm\/). *Stanford Encyclopedia of Philosophy*. Metaphysics Research Lab, Stanford University.\n67. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-71)**\n    Barrett, Jeffrey (2018). [\"Everett's Relative-State Formulation of Quantum Mechanics\"](https:\/\/plato.stanford.edu\/entries\/qm-everett\/). In Zalta, Edward N. (ed.). *Stanford Encyclopedia of Philosophy*. Metaphysics Research Lab, Stanford University.\n68. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-dewitt73_72-0)**\n    [Everett, Hugh](https:\/\/en.wikipedia.org\/wiki\/Hugh_Everett_III \"Hugh Everett III\"); [Wheeler, J. A.](https:\/\/en.wikipedia.org\/wiki\/John_Archibald_Wheeler \"John Archibald Wheeler\"); [DeWitt, B. S.](https:\/\/en.wikipedia.org\/wiki\/Bryce_DeWitt \"Bryce DeWitt\"); [Cooper, L. N.](https:\/\/en.wikipedia.org\/wiki\/Leon_Cooper \"Leon Cooper\"); Van Vechten, D.; Graham, N. (1973). [DeWitt, Bryce](https:\/\/en.wikipedia.org\/wiki\/Bryce_DeWitt \"Bryce DeWitt\"); Graham, R. Neill (eds.). *The Many-Worlds Interpretation of Quantum Mechanics*. Princeton Series in Physics. Princeton, NJ: [Princeton University Press](https:\/\/en.wikipedia.org\/wiki\/Princeton_University_Press \"Princeton University Press\"). p. v. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [0-691-08131-X](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/0-691-08131-X \"Special:BookSources\/0-691-08131-X\")\n    \n    .\n69. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-wallace2003_73-0)**\n    Wallace, David (2003). \"Everettian Rationality: defending Deutsch's approach to probability in the Everett interpretation\". *Stud. Hist. Phil. Mod. Phys*. **34** (3): 415–438\\. [arXiv](https:\/\/en.wikipedia.org\/wiki\/ArXiv_\\(identifier\\) \"ArXiv (identifier)\"):[quant-ph\/0303050](https:\/\/arxiv.org\/abs\/quant-ph\/0303050). [Bibcode](https:\/\/en.wikipedia.org\/wiki\/Bibcode_\\(identifier\\) \"Bibcode (identifier)\"):[2003SHPMP..34..415W](https:\/\/ui.adsabs.harvard.edu\/abs\/2003SHPMP..34..415W). [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1016\/S1355-2198(03)00036-4](https:\/\/doi.org\/10.1016%2FS1355-2198%2803%2900036-4). [S2CID](https:\/\/en.wikipedia.org\/wiki\/S2CID_\\(identifier\\) \"S2CID (identifier)\") [1921913](https:\/\/api.semanticscholar.org\/CorpusID:1921913).\n70. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-ballentine1973_74-0)**\n    Ballentine, L. E. (1973). \"Can the statistical postulate of quantum theory be derived? – A critique of the many-universes interpretation\". *Foundations of Physics*. **3** (2): 229–240\\. [Bibcode](https:\/\/en.wikipedia.org\/wiki\/Bibcode_\\(identifier\\) \"Bibcode (identifier)\"):[1973FoPh....3..229B](https:\/\/ui.adsabs.harvard.edu\/abs\/1973FoPh....3..229B). [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1007\/BF00708440](https:\/\/doi.org\/10.1007%2FBF00708440). 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[ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-3-540-70622-9](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-3-540-70622-9 \"Special:BookSources\/978-3-540-70622-9\")\n    \n    . \"The conclusion seems to be that no generally accepted derivation of the Born rule has been given to date, but this does not imply that such a derivation is impossible in principle.\"\n72. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-kent2009_76-0)**\n    [Kent, Adrian](https:\/\/en.wikipedia.org\/wiki\/Adrian_Kent \"Adrian Kent\") (2010). \"One world versus many: The inadequacy of Everettian accounts of evolution, probability, and scientific confirmation\". In S. Saunders; J. Barrett; A. Kent; D. Wallace (eds.). *Many Worlds? Everett, Quantum Theory and Reality*. Oxford University Press. 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[doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1007\/s10701-009-9326-5](https:\/\/doi.org\/10.1007%2Fs10701-009-9326-5). [ISSN](https:\/\/en.wikipedia.org\/wiki\/ISSN_\\(identifier\\) \"ISSN (identifier)\") [0015-9018](https:\/\/search.worldcat.org\/issn\/0015-9018). [S2CID](https:\/\/en.wikipedia.org\/wiki\/S2CID_\\(identifier\\) \"S2CID (identifier)\") [17217776](https:\/\/api.semanticscholar.org\/CorpusID:17217776).\n74. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-78)**\n    [Calosi, Claudio](https:\/\/en.wikipedia.org\/wiki\/Claudio_Calosi_\\(philosopher\\) \"Claudio Calosi (philosopher)\"); Riedel, Timotheus (2024). [\"Relational Quantum Mechanics at the Crossroads\"](https:\/\/doi.org\/10.1007%2Fs10701-024-00810-5). *Foundations of Physics*. **54** (6): 74. 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Metaphysics Research Lab, Stanford University.\n76. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-80)**\n    Stacey, Blake C. (2023). \"Toward a World Game-Flavored as a Hawk's Wing\". In Berghofer, Philipp; Wiltsche, Harald A. (eds.). *Phenomenology and QBism: New Approaches to Quantum Mechanics*. Routledge. pp. 49–77\\. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.4324\/9781003259008-3](https:\/\/doi.org\/10.4324%2F9781003259008-3). [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-1-032-19181-2](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-1-032-19181-2 \"Special:BookSources\/978-1-032-19181-2\")\n    \n    .\n77. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-Born_&_Wolf_81-0)**\n    [Born, Max](https:\/\/en.wikipedia.org\/wiki\/Max_Born \"Max Born\"); [Wolf, Emil](https:\/\/en.wikipedia.org\/wiki\/Emil_Wolf \"Emil Wolf\") (1999). 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Retrieved 30 September 2021.\n    \n    Reprinted, Addison-Wesley, 1989, [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-0-201-50064-6](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-201-50064-6 \"Special:BookSources\/978-0-201-50064-6\")\n80. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-84)**\n    Martin, Andre (1986), \"Cathode Ray Tubes for Industrial and Military Applications\", in Hawkes, Peter (ed.), *Advances in Electronics and Electron Physics, Volume 67*, Academic Press, p. 183, [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-0-08-057733-3](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-08-057733-3 \"Special:BookSources\/978-0-08-057733-3\")\n    \n    , \"Evidence for the existence of \"cathode-rays\" was first found by Plücker and Hittorf ...\"\n81. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-85)**\n    Dahl, Per F. (1997). [*Flash of the Cathode Rays: A History of J. J. Thomson's Electron*](https:\/\/books.google.com\/books?id=xUzaWGocMdMC). CRC Press. pp. 47–57\\. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-0-7503-0453-5](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-7503-0453-5 \"Special:BookSources\/978-0-7503-0453-5\")\n    \n    .\n82. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-86)**\n    [Mehra, J.](https:\/\/en.wikipedia.org\/wiki\/Jagdish_Mehra \"Jagdish Mehra\"); Rechenberg, H. (1982). *The Historical Development of Quantum Theory, Vol. 1: The Quantum Theory of Planck, Einstein, Bohr and Sommerfeld. Its Foundation and the Rise of Its Difficulties (1900–1925)*. New York: Springer-Verlag. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-0-387-90642-3](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-387-90642-3 \"Special:BookSources\/978-0-387-90642-3\")\n    \n    .\n83. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-87)**\n    [\"Quantum\"](http:\/\/www.merriam-webster.com\/dictionary\/quantum). *Merriam-Webster Dictionary*. [Archived](https:\/\/web.archive.org\/web\/20121026104456\/http:\/\/www.merriam-webster.com\/dictionary\/quantum) from the original on Oct 26, 2012. Retrieved 18 August 2012.\n84. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-88)**\n    [Kuhn, T. S.](https:\/\/en.wikipedia.org\/wiki\/Thomas_Samuel_Kuhn \"Thomas Samuel Kuhn\") (1978). *Black-body theory and the quantum discontinuity 1894–1912*. Oxford: Clarendon Press. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-0-19-502383-1](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-19-502383-1 \"Special:BookSources\/978-0-19-502383-1\")\n    \n    .\n85. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-Kragh_89-0)**\n    [Kragh, Helge](https:\/\/en.wikipedia.org\/wiki\/Helge_Kragh \"Helge Kragh\") (1 December 2000). [\"Max Planck: the reluctant revolutionary\"](https:\/\/physicsworld.com\/a\/max-planck-the-reluctant-revolutionary\/). *[Physics World](https:\/\/en.wikipedia.org\/wiki\/Physics_World \"Physics World\")*. Retrieved 12 December 2020.\n86. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-90)**\n    [Stachel, John](https:\/\/en.wikipedia.org\/wiki\/John_Stachel \"John Stachel\") (2009). \"Bohr and the Photon\". *Quantum Reality, Relativistic Causality, and Closing the Epistemic Circle*. The Western Ontario Series in Philosophy of Science. Vol. 73. Dordrecht: Springer. pp. 69–83\\. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1007\/978-1-4020-9107-0\\_5](https:\/\/doi.org\/10.1007%2F978-1-4020-9107-0_5). [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-1-4020-9106-3](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-1-4020-9106-3 \"Special:BookSources\/978-1-4020-9106-3\")\n    \n    .\n87. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-91)**\n    Einstein, Albert (1905). [\"Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt\"](https:\/\/doi.org\/10.1002%2Fandp.19053220607) \\[On a heuristic point of view concerning the production and transformation of light\\]. *[Annalen der Physik](https:\/\/en.wikipedia.org\/wiki\/Annalen_der_Physik \"Annalen der Physik\")* (in German). **17** (6): 132–148\\. [Bibcode](https:\/\/en.wikipedia.org\/wiki\/Bibcode_\\(identifier\\) \"Bibcode (identifier)\"):[1905AnP...322..132E](https:\/\/ui.adsabs.harvard.edu\/abs\/1905AnP...322..132E). [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1002\/andp.19053220607](https:\/\/doi.org\/10.1002%2Fandp.19053220607).\n    \n    Reprinted in\n    \n    [Stachel, John](https:\/\/en.wikipedia.org\/wiki\/John_Stachel \"John Stachel\"), ed. (1989). *The Collected Papers of Albert Einstein* (in German). Vol. 2. Princeton University Press. pp. 149–166\\.\n    \n    See also \"Einstein's early work on the quantum hypothesis\", ibid. pp. 134–148.\n88. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-92)**\n    [Einstein, Albert](https:\/\/en.wikipedia.org\/wiki\/Albert_Einstein \"Albert Einstein\") (1917). \"Zur Quantentheorie der Strahlung\" \\[On the Quantum Theory of Radiation\\]. *[Physikalische Zeitschrift](https:\/\/en.wikipedia.org\/wiki\/Physikalische_Zeitschrift \"Physikalische Zeitschrift\")* (in German). **18**: 121–128\\. [Bibcode](https:\/\/en.wikipedia.org\/wiki\/Bibcode_\\(identifier\\) \"Bibcode (identifier)\"):[1917PhyZ...18..121E](https:\/\/ui.adsabs.harvard.edu\/abs\/1917PhyZ...18..121E).\n    \n    Translated in\n    \n    Einstein, A. (1967). \"On the Quantum Theory of Radiation\". *The Old Quantum Theory*. Elsevier. pp. 167–183\\. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1016\/b978-0-08-012102-4.50018-8](https:\/\/doi.org\/10.1016%2Fb978-0-08-012102-4.50018-8). [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-0-08-012102-4](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-08-012102-4 \"Special:BookSources\/978-0-08-012102-4\")\n    \n    .\n89. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-93)**\n    [Ball, Philip](https:\/\/en.wikipedia.org\/wiki\/Philip_Ball \"Philip Ball\") (2017-08-31). [\"A century ago Einstein sparked the notion of the laser\"](https:\/\/physicsworld.com\/a\/a-century-ago-einstein-sparked-the-notion-of-the-laser\/). *[Physics World](https:\/\/en.wikipedia.org\/wiki\/Physics_World \"Physics World\")*. Retrieved 2024-03-23.\n90. ^ [***a***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-terHaar_94-0) [***b***](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-terHaar_94-1)\n    ter Haar, D. (1967). [*The Old Quantum Theory*](https:\/\/archive.org\/details\/oldquantumtheory0000haar). Pergamon Press. pp. 3–75\\. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-0-08-012101-7](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-08-012101-7 \"Special:BookSources\/978-0-08-012101-7\")\n    \n    . [LCCN](https:\/\/en.wikipedia.org\/wiki\/LCCN_\\(identifier\\) \"LCCN (identifier)\") [66-29628](https:\/\/lccn.loc.gov\/66-29628).\n91. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-95)**\n    Bokulich, Alisa; Bokulich, Peter (2020-08-13). [\"Bohr's Correspondence Principle\"](https:\/\/plato.stanford.edu\/entries\/bohr-correspondence\/). In [Zalta, Edward N.](https:\/\/en.wikipedia.org\/wiki\/Edward_N._Zalta \"Edward N. Zalta\") (ed.). *[Stanford Encyclopedia of Philosophy](https:\/\/en.wikipedia.org\/wiki\/Stanford_Encyclopedia_of_Philosophy \"Stanford Encyclopedia of Philosophy\")*. [ISSN](https:\/\/en.wikipedia.org\/wiki\/ISSN_\\(identifier\\) \"ISSN (identifier)\") [1095-5054](https:\/\/search.worldcat.org\/issn\/1095-5054). [OCLC](https:\/\/en.wikipedia.org\/wiki\/OCLC_\\(identifier\\) \"OCLC (identifier)\") [429049174](https:\/\/search.worldcat.org\/oclc\/429049174).\n92. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-96)**\n    [\"Semi-classical approximation\"](https:\/\/web.archive.org\/web\/20221007190530\/https:\/\/encyclopediaofmath.org\/index.php?title=Semi-classical_approximation). *Encyclopedia of Mathematics*. Archived from [the original](https:\/\/www.encyclopediaofmath.org\/index.php?title=Semi-classical_approximation) on 7 October 2022. Retrieved 1 February 2020.\n93. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-97)**\n    [Sakurai, J. J.](https:\/\/en.wikipedia.org\/wiki\/J._J._Sakurai \"J. J. Sakurai\"); Napolitano, J. (2014). \"Quantum Dynamics\". [*Modern Quantum Mechanics*](https:\/\/en.wikipedia.org\/wiki\/Modern_Quantum_Mechanics \"Modern Quantum Mechanics\"). Pearson. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-1-292-02410-3](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-1-292-02410-3 \"Special:BookSources\/978-1-292-02410-3\")\n    \n    . [OCLC](https:\/\/en.wikipedia.org\/wiki\/OCLC_\\(identifier\\) \"OCLC (identifier)\") [929609283](https:\/\/search.worldcat.org\/oclc\/929609283).\n94. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-Aharoni_98-0)**\n    [Aharoni, Amikam](https:\/\/en.wikipedia.org\/wiki\/Amikam_Aharoni \"Amikam Aharoni\") (1996). [*Introduction to the Theory of Ferromagnetism*](https:\/\/archive.org\/details\/introductiontoth00ahar\/page\/6). [Clarendon Press](https:\/\/en.wikipedia.org\/wiki\/Clarendon_Press \"Clarendon Press\"). pp. [6–7](https:\/\/archive.org\/details\/introductiontoth00ahar\/page\/6). [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [0-19-851791-2](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/0-19-851791-2 \"Special:BookSources\/0-19-851791-2\")\n    \n    .\n95. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-Edwards79_99-0)** David Edwards, \"The Mathematical Foundations of Quantum Mechanics\", *Synthese*, Volume 42, Number 1\/September, 1979, pp. 1–70.\n96. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-Edwards81_100-0)** David Edwards, \"The Mathematical Foundations of Quantum Field Theory: Fermions, Gauge Fields, and Super-symmetry, Part I: Lattice Field Theories\", *International Journal of Theoretical Physics*, Vol. 20, No. 7 (1981).\n97. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-101)**\n    [Bernstein, Jeremy](https:\/\/en.wikipedia.org\/wiki\/Jeremy_Bernstein \"Jeremy Bernstein\") (November 2005). [\"Max Born and the quantum theory\"](https:\/\/doi.org\/10.1119%2F1.2060717). *[American Journal of Physics](https:\/\/en.wikipedia.org\/wiki\/American_Journal_of_Physics \"American Journal of Physics\")*. **73** (11): 999–1008\\. [Bibcode](https:\/\/en.wikipedia.org\/wiki\/Bibcode_\\(identifier\\) \"Bibcode (identifier)\"):[2005AmJPh..73..999B](https:\/\/ui.adsabs.harvard.edu\/abs\/2005AmJPh..73..999B). [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1119\/1.2060717](https:\/\/doi.org\/10.1119%2F1.2060717). [ISSN](https:\/\/en.wikipedia.org\/wiki\/ISSN_\\(identifier\\) \"ISSN (identifier)\") [0002-9505](https:\/\/search.worldcat.org\/issn\/0002-9505).\n98. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-pais1997_102-0)**\n    [Pais, Abraham](https:\/\/en.wikipedia.org\/wiki\/Abraham_Pais \"Abraham Pais\") (1997). [*A Tale of Two Continents: A Physicist's Life in a Turbulent World*](https:\/\/archive.org\/details\/taleoftwocontine00pais). Princeton, New Jersey: Princeton University Press. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [0-691-01243-1](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/0-691-01243-1 \"Special:BookSources\/0-691-01243-1\")\n    \n    .\n99. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-103)**\n    Van Hove, Leon (1958). [\"Von Neumann's contributions to quantum mechanics\"](https:\/\/www.ams.org\/journals\/bull\/1958-64-03\/S0002-9904-1958-10206-2\/S0002-9904-1958-10206-2.pdf) (PDF). *[Bulletin of the American Mathematical Society](https:\/\/en.wikipedia.org\/wiki\/Bulletin_of_the_American_Mathematical_Society \"Bulletin of the American Mathematical Society\")*. **64** (3): Part 2:95–99. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1090\/s0002-9904-1958-10206-2](https:\/\/doi.org\/10.1090%2Fs0002-9904-1958-10206-2). [Archived](https:\/\/web.archive.org\/web\/20240120073106\/https:\/\/www.ams.org\/journals\/bull\/1958-64-03\/S0002-9904-1958-10206-2\/S0002-9904-1958-10206-2.pdf) (PDF) from the original on Jan 20, 2024.\n100. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-feynman2015_104-0)**\n     [Feynman, Richard](https:\/\/en.wikipedia.org\/wiki\/Richard_Feynman \"Richard Feynman\"). [\"The Feynman Lectures on Physics Vol. III Ch. 21: The Schrödinger Equation in a Classical Context: A Seminar on Superconductivity, 21-4\"](https:\/\/feynmanlectures.caltech.edu\/III_21.html#Ch21-S5). [California Institute of Technology](https:\/\/en.wikipedia.org\/wiki\/California_Institute_of_Technology \"California Institute of Technology\"). Retrieved 24 November 2015. \"...it was long believed that the wave function of the Schrödinger equation would never have a macroscopic representation analogous to the macroscopic representation of the amplitude for photons. On the other hand, it is now realized that the phenomena of superconductivity presents us with just this situation.\"\n     \n     `{{cite web}}`: CS1 maint: url-status ([link](https:\/\/en.wikipedia.org\/wiki\/Category:CS1_maint:_url-status \"Category:CS1 maint: url-status\"))\n101. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#cite_ref-105)**\n     Packard, Richard (2006). [\"Berkeley Experiments on Superfluid Macroscopic Quantum Effects\"](https:\/\/web.archive.org\/web\/20151125112132\/http:\/\/research.physics.berkeley.edu\/packard\/publications\/Articles\/LT24_Berk_expts_on_macro_sup_effects.pdf) (PDF). Physics Department, University of California, Berkeley. Archived from [the original](http:\/\/physics.berkeley.edu\/sites\/default\/files\/_\/lt24_berk_expts_on_macro_sup_effects.pdf) (PDF) on 25 November 2015. Retrieved 24 November 2015.\n\nFurther reading\n\nThe following titles, all by working physicists, attempt to communicate quantum theory to lay people, using a minimum of technical apparatus:\n\n- [Chester, Marvin](https:\/\/en.wikipedia.org\/wiki\/Marvin_Chester \"Marvin Chester\") (1987). *Primer of Quantum Mechanics*. John Wiley. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  [0-486-42878-8](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/0-486-42878-8 \"Special:BookSources\/0-486-42878-8\")\n- [Cox, Brian](https:\/\/en.wikipedia.org\/wiki\/Brian_Cox_\\(physicist\\) \"Brian Cox (physicist)\"); [Forshaw, Jeff](https:\/\/en.wikipedia.org\/wiki\/Jeff_Forshaw \"Jeff Forshaw\") (2011). [*The Quantum Universe: Everything That Can Happen Does Happen*](https:\/\/en.wikipedia.org\/wiki\/The_Quantum_Universe \"The Quantum Universe\"). Allen Lane. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  \n  [978-1-84614-432-5](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-1-84614-432-5 \"Special:BookSources\/978-1-84614-432-5\")\n  \n  .\n- [Richard Feynman](https:\/\/en.wikipedia.org\/wiki\/Richard_Feynman \"Richard Feynman\"), 1985. *[QED: The Strange Theory of Light and Matter](https:\/\/en.wikipedia.org\/wiki\/QED:_The_Strange_Theory_of_Light_and_Matter \"QED: The Strange Theory of Light and Matter\")*, Princeton University Press. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  [0-691-08388-6](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/0-691-08388-6 \"Special:BookSources\/0-691-08388-6\")\n  . Four elementary lectures on quantum electrodynamics and [quantum field theory](https:\/\/en.wikipedia.org\/wiki\/Quantum_field_theory \"Quantum field theory\"), yet containing many insights for the expert.\n- [Ghirardi, GianCarlo](https:\/\/en.wikipedia.org\/wiki\/Giancarlo_Ghirardi \"Giancarlo Ghirardi\"), 2004. *Sneaking a Look at God's Cards*, Gerald Malsbary, trans. Princeton Univ. Press. The most technical of the works cited here. Passages using [algebra](https:\/\/en.wikipedia.org\/wiki\/Algebra \"Algebra\"), [trigonometry](https:\/\/en.wikipedia.org\/wiki\/Trigonometry \"Trigonometry\"), and [bra–ket notation](https:\/\/en.wikipedia.org\/wiki\/Bra%E2%80%93ket_notation \"Bra–ket notation\") can be passed over on a first reading.\n- [N. David Mermin](https:\/\/en.wikipedia.org\/wiki\/N._David_Mermin \"N. David Mermin\"), 1990, \"Spooky actions at a distance: mysteries of the QT\" in his *Boojums All the Way Through*. Cambridge University Press: 110–76.\n- [Victor Stenger](https:\/\/en.wikipedia.org\/wiki\/Victor_Stenger \"Victor Stenger\"), 2000. *Timeless Reality: Symmetry, Simplicity, and Multiple Universes*. Buffalo, New York: Prometheus Books. Chpts. 5–8. Includes cosmological and philosophical considerations.\n\nMore technical:\n\n- [Bernstein, Jeremy](https:\/\/en.wikipedia.org\/wiki\/Jeremy_Bernstein \"Jeremy Bernstein\") (2009). [*Quantum Leaps*](https:\/\/books.google.com\/books?id=j0Me3brYOL0C). Cambridge, Massachusetts: Belknap Press of Harvard University Press. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  \n  [978-0-674-03541-6](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-674-03541-6 \"Special:BookSources\/978-0-674-03541-6\")\n  \n  .\n- [Bohm, David](https:\/\/en.wikipedia.org\/wiki\/David_Bohm \"David Bohm\") (1989). [*Quantum Theory*](https:\/\/archive.org\/details\/quantumtheory0000bohm). Dover Publications. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  \n  [978-0-486-65969-5](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-486-65969-5 \"Special:BookSources\/978-0-486-65969-5\")\n  \n  .\n- [Binney, James](https:\/\/en.wikipedia.org\/wiki\/James_Binney \"James Binney\"); Skinner, David (2008). *The Physics of Quantum Mechanics*. Oxford University Press. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  \n  [978-0-19-968857-9](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-19-968857-9 \"Special:BookSources\/978-0-19-968857-9\")\n  \n  .\n- Eisberg, Robert; [Resnick, Robert](https:\/\/en.wikipedia.org\/wiki\/Robert_Resnick \"Robert Resnick\") (1985). [*Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles*](https:\/\/archive.org\/details\/quantumphysicsof00eisb) (2nd ed.). Wiley. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  \n  [978-0-471-87373-0](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-471-87373-0 \"Special:BookSources\/978-0-471-87373-0\")\n  \n  .\n- [Bryce DeWitt](https:\/\/en.wikipedia.org\/wiki\/Bryce_DeWitt \"Bryce DeWitt\"), R. Neill Graham, eds., 1973. *The Many-Worlds Interpretation of Quantum Mechanics*, Princeton Series in Physics, Princeton University Press. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  [0-691-08131-X](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/0-691-08131-X \"Special:BookSources\/0-691-08131-X\")\n- [Everett, Hugh](https:\/\/en.wikipedia.org\/wiki\/Hugh_Everett \"Hugh Everett\") (1957). \"Relative State Formulation of Quantum Mechanics\". *Reviews of Modern Physics*. **29** (3): 454–462\\. [Bibcode](https:\/\/en.wikipedia.org\/wiki\/Bibcode_\\(identifier\\) \"Bibcode (identifier)\"):[1957RvMP...29..454E](https:\/\/ui.adsabs.harvard.edu\/abs\/1957RvMP...29..454E). [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1103\/RevModPhys.29.454](https:\/\/doi.org\/10.1103%2FRevModPhys.29.454). [S2CID](https:\/\/en.wikipedia.org\/wiki\/S2CID_\\(identifier\\) \"S2CID (identifier)\") [17178479](https:\/\/api.semanticscholar.org\/CorpusID:17178479).\n- [Feynman, Richard P.](https:\/\/en.wikipedia.org\/wiki\/Richard_Feynman \"Richard Feynman\"); [Leighton, Robert B.](https:\/\/en.wikipedia.org\/wiki\/Robert_B._Leighton \"Robert B. Leighton\"); Sands, Matthew (1965). [*The Feynman Lectures on Physics*](https:\/\/en.wikipedia.org\/wiki\/The_Feynman_Lectures_on_Physics \"The Feynman Lectures on Physics\"). Vol. 1–3\\. Addison-Wesley. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  \n  [978-0-7382-0008-8](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-7382-0008-8 \"Special:BookSources\/978-0-7382-0008-8\")\n  \n  .\n- [D. Greenberger](https:\/\/en.wikipedia.org\/wiki\/Daniel_Greenberger \"Daniel Greenberger\"), [K. Hentschel](https:\/\/en.wikipedia.org\/wiki\/Klaus_Hentschel \"Klaus Hentschel\"), F. Weinert, eds., 2009. *Compendium of quantum physics, Concepts, experiments, history and philosophy*, Springer-Verlag, Berlin, Heidelberg. Short articles on many QM topics.\n- [Griffiths, David J.](https:\/\/en.wikipedia.org\/wiki\/David_J._Griffiths \"David J. Griffiths\") (2004). *[Introduction to Quantum Mechanics](https:\/\/en.wikipedia.org\/wiki\/Introduction_to_Quantum_Mechanics_\\(book\\) \"Introduction to Quantum Mechanics (book)\")* (2nd ed.). Prentice Hall. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  \n  [978-0-13-111892-8](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-13-111892-8 \"Special:BookSources\/978-0-13-111892-8\")\n  \n  . [OCLC](https:\/\/en.wikipedia.org\/wiki\/OCLC_\\(identifier\\) \"OCLC (identifier)\") [40251748](https:\/\/search.worldcat.org\/oclc\/40251748).\n  A standard undergraduate text.\n- [Max Jammer](https:\/\/en.wikipedia.org\/wiki\/Max_Jammer \"Max Jammer\"), 1966. *The Conceptual Development of Quantum Mechanics*. McGraw Hill.\n- [Hagen Kleinert](https:\/\/en.wikipedia.org\/wiki\/Hagen_Kleinert \"Hagen Kleinert\"), 2004. *Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets*, 3rd ed. Singapore: World Scientific. [Draft of 4th edition.](http:\/\/www.physik.fu-berlin.de\/~kleinert\/b5) [Archived](https:\/\/web.archive.org\/web\/20080615134934\/http:\/\/www.physik.fu-berlin.de\/~kleinert\/b5) 2008-06-15 at the [Wayback Machine](https:\/\/en.wikipedia.org\/wiki\/Wayback_Machine \"Wayback Machine\")\n- Landau, L. D.; Lifshitz, E. M. (1977). *Quantum Mechanics: Non-Relativistic Theory*. Vol. 3 (3rd ed.). [Pergamon Press](https:\/\/en.wikipedia.org\/wiki\/Pergamon_Press \"Pergamon Press\"). [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  \n  [978-0-08-020940-1](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-08-020940-1 \"Special:BookSources\/978-0-08-020940-1\")\n  \n  .\n  [Online copy](https:\/\/archive.org\/details\/QuantumMechanics_104)\n- [Liboff, Richard L.](https:\/\/en.wikipedia.org\/wiki\/Liboff,_Richard \"Liboff, Richard\") (2002). *Introductory Quantum Mechanics*. Addison-Wesley. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  \n  [978-0-8053-8714-8](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-8053-8714-8 \"Special:BookSources\/978-0-8053-8714-8\")\n  \n  .\n- Gunther Ludwig, 1968. *Wave Mechanics*. London: Pergamon Press. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  [0-08-203204-1](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/0-08-203204-1 \"Special:BookSources\/0-08-203204-1\")\n- [George Mackey](https:\/\/en.wikipedia.org\/wiki\/George_Mackey \"George Mackey\") (2004). *The mathematical foundations of quantum mechanics*. Dover Publications. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  [0-486-43517-2](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/0-486-43517-2 \"Special:BookSources\/0-486-43517-2\")\n  .\n- [Merzbacher, Eugen](https:\/\/en.wikipedia.org\/wiki\/Eugen_Merzbacher \"Eugen Merzbacher\") (1998). *Quantum Mechanics*. Wiley, John & Sons, Inc. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  \n  [978-0-471-88702-7](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-471-88702-7 \"Special:BookSources\/978-0-471-88702-7\")\n  \n  .\n- [Albert Messiah](https:\/\/en.wikipedia.org\/wiki\/Albert_Messiah \"Albert Messiah\"), 1966. *Quantum Mechanics* (Vol. I), English translation from French by G. M. Temmer. North Holland, John Wiley & Sons. Cf. chpt. IV, section III.\n- [Omnès, Roland](https:\/\/en.wikipedia.org\/wiki\/Roland_Omn%C3%A8s \"Roland Omnès\") (1999). [*Understanding Quantum Mechanics*](https:\/\/archive.org\/details\/understandingqua00omne). Princeton University Press. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  \n  [978-0-691-00435-8](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-691-00435-8 \"Special:BookSources\/978-0-691-00435-8\")\n  \n  . [OCLC](https:\/\/en.wikipedia.org\/wiki\/OCLC_\\(identifier\\) \"OCLC (identifier)\") [39849482](https:\/\/search.worldcat.org\/oclc\/39849482).\n- [Scerri, Eric. R.](https:\/\/en.wikipedia.org\/wiki\/Eric_R._Scerri \"Eric R. Scerri\") (2006). *The Periodic Table: Its Story and Its Significance*. Oxford University Press. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  \n  [0-19-530573-6](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/0-19-530573-6 \"Special:BookSources\/0-19-530573-6\")\n  \n  .\n  Considers the extent to which chemistry and the periodic system have been reduced to quantum mechanics.\n- [Schiff, Leonard I.](https:\/\/en.wikipedia.org\/wiki\/Leonard_I._Schiff \"Leonard I. Schiff\") (1955). *Quantum Mechanics*. McGraw Hill.\n- [Shankar, R.](https:\/\/en.wikipedia.org\/wiki\/Ramamurti_Shankar \"Ramamurti Shankar\") (1994). *Principles of Quantum Mechanics*. Springer. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  \n  [978-0-306-44790-7](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-306-44790-7 \"Special:BookSources\/978-0-306-44790-7\")\n  \n  .\n- [Stone, A. Douglas](https:\/\/en.wikipedia.org\/wiki\/A._Douglas_Stone \"A. Douglas Stone\") (2013). [*Einstein and the Quantum*](https:\/\/archive.org\/details\/einsteinquantumq0000ston). Princeton University Press. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  \n  [978-0-691-13968-5](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-691-13968-5 \"Special:BookSources\/978-0-691-13968-5\")\n  \n  .\n- *What is Quantum Mechanics? A Physics Adventure*. Boston: [Transnational College](https:\/\/en.wikipedia.org\/w\/index.php?title=Transnational_College&action=edit&redlink=1 \"Transnational College (page does not exist)\"), Language Research Foundation. 1996. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  \n  [978-0-9643504-1-0](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-9643504-1-0 \"Special:BookSources\/978-0-9643504-1-0\")\n  \n  . [OCLC](https:\/\/en.wikipedia.org\/wiki\/OCLC_\\(identifier\\) \"OCLC (identifier)\") [34661512](https:\/\/search.worldcat.org\/oclc\/34661512).\n- [Veltman, Martinus J. G.](https:\/\/en.wikipedia.org\/wiki\/Martinus_J._G._Veltman \"Martinus J. G. Veltman\") (2003), *Facts and Mysteries in Elementary Particle Physics*.\n\nExternal links\n\n- [Introduction to quantum mechanics by Timon Idema](https:\/\/interactivetextbooks.tudelft.nl\/introduction-to-quantum-mechanics\/)\n- [Quantum Physics Made Relatively Simple](https:\/\/bethe.cornell.edu\/): three video lectures by [Hans Bethe](https:\/\/en.wikipedia.org\/wiki\/Hans_Bethe \"Hans Bethe\").\n\n**Course material**\n\n- [Quantum Cook Book](http:\/\/oyc.yale.edu\/sites\/default\/files\/notes_quantum_cookbook.pdf) and [PHYS 201: Fundamentals of Physics II](http:\/\/oyc.yale.edu\/physics\/phys-201#sessions) by [Ramamurti Shankar](https:\/\/en.wikipedia.org\/wiki\/Ramamurti_Shankar \"Ramamurti Shankar\"), Yale OpenCourseware.\n- *[Modern Physics: With waves, thermodynamics, and optics](https:\/\/lightandmatter.com\/mod\/)* – an online textbook.\n- [MIT OpenCourseWare](https:\/\/en.wikipedia.org\/wiki\/MIT_OpenCourseWare \"MIT OpenCourseWare\"): [Chemistry](https:\/\/ocw.mit.edu\/courses\/chemistry\/) and [Physics](https:\/\/ocw.mit.edu\/courses\/physics\/). See [8\\.04](https:\/\/ocw.mit.edu\/courses\/physics\/8-04-quantum-physics-i-spring-2016\/), [8\\.05](https:\/\/ocw.mit.edu\/courses\/physics\/8-05-quantum-physics-ii-fall-2013\/index.htm) and [8\\.06](https:\/\/ocw.mit.edu\/courses\/physics\/8-06-quantum-physics-iii-spring-2018\/index.htm).\n- [⁠5\\+1\/2⁠ Examples in Quantum Mechanics](http:\/\/physics.csbsju.edu\/QM\/).\n\n**Philosophy**\n\n- Ismael, Jenann. [\"Quantum Mechanics\"](https:\/\/plato.stanford.edu\/entries\/qm\/). In [Zalta, Edward N.](https:\/\/en.wikipedia.org\/wiki\/Edward_N._Zalta \"Edward N. Zalta\") (ed.). *[Stanford Encyclopedia of Philosophy](https:\/\/en.wikipedia.org\/wiki\/Stanford_Encyclopedia_of_Philosophy \"Stanford Encyclopedia of Philosophy\")*. [ISSN](https:\/\/en.wikipedia.org\/wiki\/ISSN_\\(identifier\\) \"ISSN (identifier)\") [1095-5054](https:\/\/search.worldcat.org\/issn\/1095-5054). [OCLC](https:\/\/en.wikipedia.org\/wiki\/OCLC_\\(identifier\\) \"OCLC (identifier)\") [429049174](https:\/\/search.worldcat.org\/oclc\/429049174).\n- [Zalta, Edward N.](https:\/\/en.wikipedia.org\/wiki\/Edward_N._Zalta \"Edward N. Zalta\") (ed.). [\"Philosophical Issues in Quantum Theory\"](https:\/\/plato.stanford.edu\/entries\/qt-issues\/). *[Stanford Encyclopedia of Philosophy](https:\/\/en.wikipedia.org\/wiki\/Stanford_Encyclopedia_of_Philosophy \"Stanford Encyclopedia of Philosophy\")*. [ISSN](https:\/\/en.wikipedia.org\/wiki\/ISSN_\\(identifier\\) \"ISSN (identifier)\") [1095-5054](https:\/\/search.worldcat.org\/issn\/1095-5054). [OCLC](https:\/\/en.wikipedia.org\/wiki\/OCLC_\\(identifier\\) \"OCLC (identifier)\") [429049174](https:\/\/search.worldcat.org\/oclc\/429049174).","meta_canonical":null,"ml_categories_json":"{\"\/Science\":975,\"\/Science\/Physics\":930}","ml_types_json":"{\"\/Article\":999,\"\/Article\/Wiki\":911}","ml_intent_types_json":"{\"Informational\":999}","meta_language":"en","attrs_author":null,"attrs_publish_time":0,"attrs_original_publish_time":1375978975,"attrs_is_republished":0,"attrs_nr_words":"14757","attrs_boilerpipe_nr_words":"11777","body_ext_links_number":424,"body_int_links_number":967,"meta_nofollow":0,"meta_noarchive":0,"props_was_rendered":0,"src_redirect":"","download_time_msec":65,"download_ttfb_msec":34,"download_size":113727}

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"Quantum systems" redirects here. For the company, see Quantum-Systems . Wave functions of the electron in a hydrogen atom at different energy levels. Quantum mechanics cannot predict the exact location of a particle in space, only the probability of finding it at different locations. [ 1 ] The brighter areas represent a higher probability of finding the electron. Quantum mechanics is the fundamental physical theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms . [ 2 ] : 1.1 It is the foundation of all quantum physics , which includes quantum chemistry , quantum biology , quantum field theory , quantum technology , and quantum information science . Quantum mechanics can describe many systems that classical physics cannot. Classical physics can describe many aspects of nature at an ordinary ( macroscopic and (optical) microscopic ) scale, however is insufficient for describing them at very small submicroscopic (atomic and subatomic ) scales. Classical mechanics can be derived from quantum mechanics as an approximation that is valid at ordinary scales. [ 3 ] Quantum systems have bound states that are quantized to discrete values of energy , momentum , angular momentum , and other quantities, in contrast to classical systems where these quantities can be measured continuously. Measurements of quantum systems show characteristics of both particles and waves ( wave–particle duality ), and there are limits to how accurately the value of a physical quantity can be predicted prior to its measurement, given a complete set of initial conditions (the uncertainty principle ). Quantum mechanics arose gradually from theories to explain observations that could not be reconciled with classical physics , such as Max Planck 's solution in 1900 to the black-body radiation problem, and the correspondence between energy and frequency in Albert Einstein 's 1905 paper , which explained the photoelectric effect . These early attempts to understand microscopic phenomena, now known as the " old quantum theory ", led to the full development of quantum mechanics in the mid-1920s by Niels Bohr , Erwin Schrödinger , Werner Heisenberg , Max Born , Paul Dirac and others. The modern theory is formulated in various specially developed mathematical formalisms . In one of them, a mathematical entity called the wave function provides information, in the form of probability amplitudes , about what measurements of a particle's energy, momentum, and other physical properties may yield. Overview and fundamental concepts Quantum mechanics allows the calculation of properties and behaviour of physical systems . It is typically applied to microscopic systems: molecules , atoms and subatomic particles . It has been demonstrated to hold for complex molecules with thousands of atoms, [ 4 ] but its application to human beings raises philosophical problems, such as Wigner's friend , and its application to the universe as a whole remains speculative. [ 5 ] Predictions of quantum mechanics have been verified experimentally to an extremely high degree of accuracy . For example, the refinement of quantum mechanics for the interaction of light and matter, known as quantum electrodynamics (QED), has been shown to agree with experiment to within 1 part in 10 12 when predicting the magnetic properties of an electron. [ 6 ] A fundamental feature of the theory is that it usually cannot predict with certainty what will happen, but only gives probabilities. Mathematically, a probability is found by taking the square of the absolute value of a complex number , known as a probability amplitude. This is known as the Born rule , named after physicist Max Born . For example, a quantum particle like an electron can be described by a wave function, which associates to each point in space a probability amplitude. Applying the Born rule to these amplitudes gives a probability density function for the position that the electron will be found to have when an experiment is performed to measure it. This is the best the theory can do; it cannot say for certain where the electron will be found. The Schrödinger equation relates the collection of probability amplitudes that pertain to one moment of time to the collection of probability amplitudes that pertain to another. [ 7 ] : 67–87 One consequence of the mathematical rules of quantum mechanics is a tradeoff in predictability between measurable quantities. The most famous form of this uncertainty principle says that no matter how a quantum particle is prepared or how carefully experiments upon it are arranged, it is impossible to have a precise prediction for a measurement of its position and also at the same time for a measurement of its momentum . [ 7 ] : 427–435 An illustration of the double-slit experiment Another consequence of the mathematical rules of quantum mechanics is the phenomenon of quantum interference , which is often illustrated with the double-slit experiment . In the basic version of this experiment, a coherent light source , such as a laser beam, illuminates a plate pierced by two parallel slits, and the light passing through the slits is observed on a screen behind the plate. [ 8 ] : 102–111 [ 2 ] : 1.1–1.8 The wave nature of light causes the light waves passing through the two slits to interfere , producing bright and dark bands on the screen – a result that would not be expected if light consisted of classical particles. [ 8 ] However, the light is always found to be absorbed at the screen at discrete points, as individual particles rather than waves; the interference pattern appears via the varying density of these particle hits on the screen. Furthermore, versions of the experiment that include detectors at the slits find that each detected photon passes through one slit (as would a classical particle), and not through both slits (as would a wave). [ 8 ] : 109 [ 9 ] [ 10 ] However, such experiments demonstrate that particles do not form the interference pattern if one detects which slit they pass through. This behavior is known as wave–particle duality . In addition to light, electrons , atoms , and molecules are all found to exhibit the same dual behavior when fired towards a double slit. [ 2 ] A simplified diagram of quantum tunneling , a phenomenon by which a particle may move through a barrier which would be impossible under classical mechanics Another non-classical phenomenon predicted by quantum mechanics is quantum tunnelling : a particle that goes up against a potential barrier can cross it, even if its kinetic energy is smaller than the maximum of the potential. [ 11 ] In classical mechanics this particle would be trapped. Quantum tunnelling has several important consequences, enabling radioactive decay , nuclear fusion in stars, and applications such as scanning tunnelling microscopy , tunnel diode and tunnel field-effect transistor . [ 12 ] [ 13 ] When quantum systems interact, the result can be the creation of quantum entanglement : their properties become so intertwined that a description of the whole solely in terms of the individual parts is no longer possible. Erwin Schrödinger called entanglement "... the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought". [ 14 ] Quantum entanglement enables quantum computing and is part of quantum communication protocols, such as quantum key distribution and superdense coding . [ 15 ] Contrary to popular misconception, entanglement does not allow sending signals faster than light , as demonstrated by the no-communication theorem . [ 15 ] Another possibility opened by entanglement is testing for " hidden variables ", hypothetical properties more fundamental than the quantities addressed in quantum theory itself, knowledge of which would allow more exact predictions than quantum theory provides. A collection of results, most significantly Bell's theorem , have demonstrated that broad classes of such hidden-variable theories are in fact incompatible with quantum physics. According to Bell's theorem, if nature actually operates in accord with any theory of local hidden variables, then the results of a Bell test will be constrained in a particular, quantifiable way. Many Bell tests have been performed and they have shown results incompatible with the constraints imposed by local hidden variables. [ 16 ] [ 17 ] It is not possible to present these concepts in more than a superficial way without introducing the mathematics involved; understanding quantum mechanics requires not only manipulating complex numbers, but also linear algebra , differential equations , group theory , and other more advanced subjects. [ 18 ] [ 19 ] Accordingly, this article will present a mathematical formulation of quantum mechanics and survey its application to some useful and oft-studied examples. Mathematical formulation In the mathematically rigorous formulation of quantum mechanics, the state of a quantum mechanical system is a vector belonging to a ( separable ) complex Hilbert space . This vector is postulated to be normalized under the Hilbert space inner product, that is, it obeys , and it is well-defined up to a complex number of modulus 1 (the global phase), that is, and represent the same physical system. In other words, the possible states are points in the projective space of a Hilbert space, usually called the complex projective space . The exact nature of this Hilbert space is dependent on the system – for example, for describing position and momentum the Hilbert space is the space of complex square-integrable functions , [ 20 ] : 13 while the Hilbert space for the spin of a single proton is simply the space of two-dimensional complex vectors with the usual inner product. [ 20 ] : 20 Physical quantities of interest – position, momentum, energy, spin – are represented by observables, which are Hermitian (more precisely, self-adjoint ) linear operators acting on the Hilbert space. [ 20 ] : 17 A quantum state can be an eigenvector of an observable, in which case it is called an eigenstate , and the associated eigenvalue corresponds to the value of the observable in that eigenstate. More generally, a quantum state will be a linear combination of the eigenstates, known as a quantum superposition . When an observable is measured, the result will be one of its eigenvalues with probability given by the Born rule : in the simplest case the eigenvalue is non-degenerate and the probability is given by , where is its associated unit-length eigenvector. More generally, the eigenvalue is degenerate and the probability is given by , where is the projector onto its associated eigenspace. [ 21 ] In the continuous case, these formulas give instead the probability density . After the measurement , if result was obtained, the quantum state is postulated to collapse to , in the non-degenerate case, or to , in the general case. The probabilistic nature of quantum mechanics thus stems from the act of measurement. This is one of the most debated aspects of quantum theory, with different interpretations of quantum mechanics giving radically different answers to questions regarding quantum-state collapse, as discussed below . Time evolution of a quantum state The time evolution of a quantum state is described by the Schrödinger equation: Here denotes the Hamiltonian , the observable corresponding to the total energy of the system, and is the reduced Planck constant . The constant is introduced so that the Hamiltonian is reduced to the classical Hamiltonian in cases where the quantum system can be approximated by a classical system; the ability to make such an approximation in certain limits is called the correspondence principle . The solution of this differential equation is given by The operator is known as the time-evolution operator, and has the crucial property that it is unitary . This time evolution is deterministic in the sense that – given an initial quantum state – it makes a definite prediction of what the quantum state will be at any later time. [ 22 ] Fig. 1: Probability densities corresponding to the wave functions of an electron in a hydrogen atom possessing definite energy levels (increasing from the top of the image to the bottom: n = 1, 2, 3, ...) and angular momenta (increasing across from left to right: s , p , d , ...). Denser areas correspond to higher probability density in a position measurement. Such wave functions are directly comparable to Chladni's figures of acoustic modes of vibration in classical physics and are modes of oscillation as well, possessing a sharp energy and thus, a definite frequency. The angular momentum and energy are quantized and take only discrete values like those shown – as is the case for resonant frequencies in acoustics. Some wave functions produce probability distributions that are independent of time, such as eigenstates of the Hamiltonian. [ 7 ] : 133–137 Many systems that are treated dynamically in classical mechanics are described by such "static" wave functions. For example, a single electron in an unexcited atom is pictured classically as a particle moving in a circular trajectory around the atomic nucleus , whereas in quantum mechanics, it is described by a static wave function surrounding the nucleus. For example, the electron wave function for an unexcited hydrogen atom is a spherically symmetric function known as an s orbital ( Fig. 1 ). Analytic solutions of the Schrödinger equation are known for very few relatively simple model Hamiltonians including the quantum harmonic oscillator , the particle in a box , the dihydrogen cation , and the hydrogen atom . Even the helium atom – which contains just two electrons – has defied all attempts at a fully analytic treatment, admitting no solution in closed form . [ 23 ] [ 24 ] [ 25 ] However, there are techniques for finding approximate solutions. One method, called perturbation theory , uses the analytic result for a simple quantum mechanical model to create a result for a related but more complicated model by (for example) the addition of a weak potential energy . [ 7 ] : 793 Another approximation method applies to systems for which quantum mechanics produces only small deviations from classical behavior. These deviations can then be computed based on the classical motion. [ 7 ] : 849 Uncertainty principle One consequence of the basic quantum formalism is the uncertainty principle. In its most familiar form, this states that no preparation of a quantum particle can imply simultaneously precise predictions both for a measurement of its position and for a measurement of its momentum. [ 26 ] [ 27 ] Both position and momentum are observables, meaning that they are represented by Hermitian operators . The position operator and momentum operator do not commute, but rather satisfy the canonical commutation relation : Given a quantum state, the Born rule lets us compute expectation values for both and , and moreover for powers of them. Defining the uncertainty for an observable by a standard deviation , we have and likewise for the momentum: The uncertainty principle states that Either standard deviation can in principle be made arbitrarily small, but not both simultaneously. [ 28 ] This inequality generalizes to arbitrary pairs of self-adjoint operators and . The commutator of these two operators is and this provides the lower bound on the product of standard deviations: Another consequence of the canonical commutation relation is that the position and momentum operators are Fourier transforms of each other, so that a description of an object according to its momentum is the Fourier transform of its description according to its position. The fact that dependence in momentum is the Fourier transform of the dependence in position means that the momentum operator is equivalent (up to an factor) to taking the derivative according to the position, since in Fourier analysis differentiation corresponds to multiplication in the dual space . This is why in quantum equations in position space, the momentum is replaced by , and in particular in the non-relativistic Schrödinger equation in position space the momentum-squared term is replaced with a Laplacian times . [ 26 ] Composite systems and entanglement When two different quantum systems are considered together, the Hilbert space of the combined system is the tensor product of the Hilbert spaces of the two components. For example, let A and B be two quantum systems, with Hilbert spaces and , respectively. The Hilbert space of the composite system is then If the state for the first system is the vector and the state for the second system is , then the state of the composite system is Not all states in the joint Hilbert space can be written in this form, however, because the superposition principle implies that linear combinations of these "separable" or "product states" are also valid. For example, if and are both possible states for system , and likewise and are both possible states for system , then is a valid joint state that is not separable. States that are not separable are called entangled . [ 29 ] [ 30 ] If the state for a composite system is entangled, it is impossible to describe either component system A or system B by a state vector. One can instead define reduced density matrices that describe the statistics that can be obtained by making measurements on either component system alone. This necessarily causes a loss of information, though: knowing the reduced density matrices of the individual systems is not enough to reconstruct the state of the composite system. [ 29 ] [ 30 ] Just as density matrices specify the state of a subsystem of a larger system, analogously, positive operator-valued measures (POVMs) describe the effect on a subsystem of a measurement performed on a larger system. POVMs are extensively used in quantum information theory. [ 29 ] [ 31 ] As described above, entanglement is a key feature of models of measurement processes in which an apparatus becomes entangled with the system being measured. Systems interacting with the environment in which they reside generally become entangled with that environment, a phenomenon known as quantum decoherence . This can explain why, in practice, quantum effects are difficult to observe in systems larger than microscopic. [ 32 ] Equivalence between formulations There are many mathematically equivalent formulations of quantum mechanics. One of the oldest and most common is the " transformation theory " proposed by Paul Dirac , which unifies and generalizes the two earliest formulations of quantum mechanics – matrix mechanics (invented by Werner Heisenberg ) and wave mechanics (invented by Erwin Schrödinger ). [ 33 ] An alternative formulation of quantum mechanics is Feynman 's path integral formulation , in which a quantum-mechanical amplitude is considered as a sum over all possible classical and non-classical paths between the initial and final states. This is the quantum-mechanical counterpart of the action principle in classical mechanics. [ 34 ] Symmetries and conservation laws The Hamiltonian is known as the generator of time evolution, since it defines a unitary time-evolution operator for each value of . From this relation between and , it follows that any observable that commutes with will be conserved : its expectation value will not change over time. [ 7 ] : 471 This statement generalizes, as mathematically, any Hermitian operator can generate a family of unitary operators parameterized by a variable . Under the evolution generated by , any observable that commutes with will be conserved. Moreover, if is conserved by evolution under , then is conserved under the evolution generated by . This implies a quantum version of the result proven by Emmy Noether in classical ( Lagrangian ) mechanics: for every differentiable symmetry of a Hamiltonian, there exists a corresponding conservation law . Examples Free particle Position space probability density of a Gaussian wave packet moving in one dimension in free space The simplest example of a quantum system with a position degree of freedom is a free particle in a single spatial dimension. A free particle is one which is not subject to external influences, so that its Hamiltonian consists only of its kinetic energy: The general solution of the Schrödinger equation is given by which is a superposition of all possible plane waves , which are eigenstates of the momentum operator with momentum . The coefficients of the superposition are , which is the Fourier transform of the initial quantum state . It is not possible for the solution to be a single momentum eigenstate, or a single position eigenstate, as these are not normalizable quantum states. [ note 1 ] Instead, we can consider a Gaussian wave packet : which has Fourier transform, and therefore momentum distribution We see that as we make smaller the spread in position gets smaller, but the spread in momentum gets larger. Conversely, by making larger we make the spread in momentum smaller, but the spread in position gets larger. This illustrates the uncertainty principle. As we let the Gaussian wave packet evolve in time, we see that its center moves through space at a constant velocity (like a classical particle with no forces acting on it). However, the wave packet will also spread out as time progresses, which means that the position becomes more and more uncertain. The uncertainty in momentum, however, stays constant. [ 35 ] Particle in a box 1-dimensional potential energy box (or infinite potential well) The particle in a one-dimensional potential energy box is the most mathematically simple example where restraints lead to the quantization of energy levels. The box is defined as having zero potential energy everywhere inside a certain region, and therefore infinite potential energy everywhere outside that region. [ 26 ] : 77–78 For the one-dimensional case in the direction, the time-independent Schrödinger equation may be written With the differential operator defined by the previous equation is evocative of the classic kinetic energy analogue , with state in this case having energy coincident with the kinetic energy of the particle. The general solutions of the Schrödinger equation for the particle in a box are or, from Euler's formula , The infinite potential walls of the box determine the values of and at and where must be zero. Thus, at , and . At , in which cannot be zero as this would conflict with the postulate that has norm 1. Therefore, since , must be an integer multiple of , This constraint on implies a constraint on the energy levels, yielding A finite potential well is the generalization of the infinite potential well problem to potential wells having finite depth. The finite potential well problem is mathematically more complicated than the infinite particle-in-a-box problem as the wave function is not pinned to zero at the walls of the well. Instead, the wave function must satisfy more complicated mathematical boundary conditions as it is nonzero in regions outside the well. Another related problem is that of the rectangular potential barrier , which furnishes a model for the quantum tunneling effect that plays an important role in the performance of modern technologies such as flash memory and scanning tunneling microscopy . Harmonic oscillator Some trajectories of a harmonic oscillator (i.e. a ball attached to a spring ) in classical mechanics (A-B) and quantum mechanics (C-H). In quantum mechanics, the position of the ball is represented by a wave (called the wave function), with the real part shown in blue and the imaginary part shown in red. Some of the trajectories (such as C, D, E, and F) are standing waves (or " stationary states "). Each standing-wave frequency is proportional to a possible energy level of the oscillator. This "energy quantization" does not occur in classical physics, where the oscillator can have any energy. As in the classical case, the potential for the quantum harmonic oscillator is given by [ 7 ] : 234 This problem can either be treated by directly solving the Schrödinger equation, which is not trivial, or by using the more elegant "ladder method" first proposed by Paul Dirac. The eigenstates are given by where H n are the Hermite polynomials and the corresponding energy levels are This is another example illustrating the discretization of energy for bound states . Mach–Zehnder interferometer Schematic of a Mach–Zehnder interferometer The Mach–Zehnder interferometer (MZI) illustrates the concepts of superposition and interference with linear algebra in dimension 2, rather than differential equations. It can be seen as a simplified version of the double-slit experiment, but it is of interest in its own right, for example in the delayed choice quantum eraser , the Elitzur–Vaidman bomb tester , and in studies of quantum entanglement. [ 36 ] [ 37 ] We can model a photon going through the interferometer by considering that at each point it can be in a superposition of only two paths: the "lower" path which starts from the left, goes straight through both beam splitters, and ends at the top, and the "upper" path which starts from the bottom, goes straight through both beam splitters, and ends at the right. The quantum state of the photon is therefore a vector that is a superposition of the "lower" path and the "upper" path , that is, for complex . In order to respect the postulate that we require that . Both beam splitters are modelled as the unitary matrix , which means that when a photon meets the beam splitter it will either stay on the same path with a probability amplitude of , or be reflected to the other path with a probability amplitude of . The phase shifter on the upper arm is modelled as the unitary matrix , which means that if the photon is on the "upper" path it will gain a relative phase of , and it will stay unchanged if it is in the lower path. A photon that enters the interferometer from the left will then be acted upon with a beam splitter , a phase shifter , and another beam splitter , and so end up in the state and the probabilities that it will be detected at the right or at the top are given respectively by One can therefore use the Mach–Zehnder interferometer to estimate the phase shift by estimating these probabilities. It is interesting to consider what would happen if the photon were definitely in either the "lower" or "upper" paths between the beam splitters. This can be accomplished by blocking one of the paths, or equivalently by removing the first beam splitter (and feeding the photon from the left or the bottom, as desired). In both cases, there will be no interference between the paths anymore, and the probabilities are given by , independently of the phase . From this we can conclude that the photon does not take one path or another after the first beam splitter, but rather that it is in a genuine quantum superposition of the two paths. [ 38 ] Applications Quantum mechanics has had enormous success in explaining many of the features of our universe, with regard to small-scale and discrete quantities and interactions which cannot be explained by classical methods . [ note 2 ] Quantum mechanics is often the only theory that can reveal the individual behaviors of the subatomic particles that make up all forms of matter (electrons, protons , neutrons , photons , and others). Solid-state physics and materials science are dependent upon quantum mechanics. [ 39 ] In many aspects, modern technology operates at a scale where quantum effects are significant. Important applications of quantum theory include quantum chemistry , quantum optics , quantum computing , superconducting magnets , light-emitting diodes , the optical amplifier and the laser, the transistor and semiconductors such as the microprocessor , medical and research imaging such as magnetic resonance imaging and electron microscopy . [ 40 ] Explanations for many biological and physical phenomena are rooted in the nature of the chemical bond, most notably the macro-molecule DNA . Relation to other scientific theories Classical mechanics The rules of quantum mechanics assert that the state space of a system is a Hilbert space and that observables of the system are Hermitian operators acting on vectors in that space – although they do not tell us which Hilbert space or which operators. These can be chosen appropriately in order to obtain a quantitative description of a quantum system, a necessary step in making physical predictions. An important guide for making these choices is the correspondence principle , a heuristic which states that the predictions of quantum mechanics reduce to those of classical mechanics in the regime of large quantum numbers . [ 41 ] One can also start from an established classical model of a particular system, and then try to guess the underlying quantum model that would give rise to the classical model in the correspondence limit. This approach is known as quantization . [ 42 ] : 299 [ 43 ] When quantum mechanics was originally formulated, it was applied to models whose correspondence limit was non-relativistic classical mechanics. For instance, the well-known model of the quantum harmonic oscillator uses an explicitly non-relativistic expression for the kinetic energy of the oscillator, and is thus a quantum version of the classical harmonic oscillator . [ 7 ] : 234 Complications arise with chaotic systems , which do not have good quantum numbers, and quantum chaos studies the relationship between classical and quantum descriptions in these systems. [ 42 ] : 353 Quantum decoherence is a mechanism through which quantum systems lose coherence , and thus become incapable of displaying many typically quantum effects: quantum superpositions become simply probabilistic mixtures, and quantum entanglement becomes simply classical correlations. [ 7 ] : 687–730 Quantum coherence is not typically evident at macroscopic scales, though at temperatures approaching absolute zero quantum behavior may manifest macroscopically. [ note 3 ] Many macroscopic properties of a classical system are a direct consequence of the quantum behavior of its parts. For example, the stability of bulk matter (consisting of atoms and molecules which would quickly collapse under electric forces alone), the rigidity of solids, and the mechanical, thermal, chemical, optical and magnetic properties of matter are all results of the interaction of electric charges under the rules of quantum mechanics. [ 44 ] Special relativity and electrodynamics Early attempts to merge quantum mechanics with special relativity involved the replacement of the Schrödinger equation with a covariant equation such as the Klein–Gordon equation or the Dirac equation . While these theories were successful in explaining many experimental results, they had certain unsatisfactory qualities stemming from their neglect of the relativistic creation and annihilation of particles. A fully relativistic quantum theory required the development of quantum field theory, which applies quantization to a field (rather than a fixed set of particles). The first complete quantum field theory, quantum electrodynamics , provides a fully quantum description of the electromagnetic interaction . Quantum electrodynamics is, along with general relativity , one of the most accurate physical theories ever devised. [ 45 ] [ 46 ] The full apparatus of quantum field theory is often unnecessary for describing electrodynamic systems. A simpler approach, one that has been used since the inception of quantum mechanics, is to treat charged particles as quantum mechanical objects being acted on by a classical electromagnetic field . For example, the elementary quantum model of the hydrogen atom describes the electric field of the hydrogen atom using a classical Coulomb potential . [ 7 ] : 285 Likewise, in a Stern–Gerlach experiment , a charged particle is modeled as a quantum system, while the background magnetic field is described classically. [ 42 ] : 26 This "semi-classical" approach fails if quantum fluctuations in the electromagnetic field play an important role, such as in the emission of photons by charged particles . Quantum field theories for the strong nuclear force and the weak nuclear force have also been developed. The quantum field theory of the strong nuclear force is called quantum chromodynamics , and describes the interactions of subnuclear particles such as quarks and gluons . The weak nuclear force and the electromagnetic force were unified, in their quantized forms, into a single quantum field theory (known as electroweak theory ), by the physicists Abdus Salam , Sheldon Glashow and Steven Weinberg . [ 47 ] Relation to general relativity Even though the predictions of both quantum theory and general relativity have been supported by rigorous and repeated empirical evidence , their abstract formalisms contradict each other and they have proven extremely difficult to incorporate into one consistent, cohesive model. Gravity is negligible in many areas of particle physics, so that unification between general relativity and quantum mechanics is not an urgent issue in those particular applications. However, the lack of a correct theory of quantum gravity is an important issue in physical cosmology and the search by physicists for an elegant " Theory of Everything " (TOE). Consequently, resolving the inconsistencies between both theories has been a major goal of 20th- and 21st-century physics. This TOE would combine not only the models of subatomic physics but also derive the four fundamental forces of nature from a single force or phenomenon. [ 48 ] One proposal for doing so is string theory , which posits that the point-like particles of particle physics are replaced by one-dimensional objects called strings . String theory describes how these strings propagate through space and interact with each other. On distance scales larger than the string scale, a string looks just like an ordinary particle, with its mass , charge , and other properties determined by the vibrational state of the string. In string theory, one of the many vibrational states of the string corresponds to the graviton , a quantum mechanical particle that carries gravitational force. [ 49 ] [ 50 ] Another popular theory is loop quantum gravity (LQG), which describes quantum properties of gravity and is thus a theory of quantum spacetime . LQG is an attempt to merge and adapt standard quantum mechanics and standard general relativity. This theory describes space as an extremely fine fabric "woven" of finite loops called spin networks . The evolution of a spin network over time is called a spin foam . The characteristic length scale of a spin foam is the Planck length , approximately 1.616×10 −35 m, and so lengths shorter than the Planck length are not physically meaningful in LQG. [ 51 ] Philosophical implications Unsolved problem in physics Is there a preferred interpretation of quantum mechanics? How does the quantum description of reality, which includes elements such as the " superposition of states" and " wave function collapse ", give rise to the reality we perceive? Since its inception, the many counter-intuitive aspects and results of quantum mechanics have provoked strong philosophical debates and many interpretations . The arguments centre on the probabilistic nature of quantum mechanics, the difficulties with wavefunction collapse and the related measurement problem , and quantum nonlocality . Perhaps the only consensus that exists about these issues is that there is no consensus. Richard Feynman once said, "I think I can safely say that nobody understands quantum mechanics." [ 52 ] According to Steven Weinberg , "There is now in my opinion no entirely satisfactory interpretation of quantum mechanics." [ 53 ] The views of Niels Bohr , Werner Heisenberg and other physicists are often grouped together as the " Copenhagen interpretation ". [ 54 ] [ 55 ] According to these views, the probabilistic nature of quantum mechanics is not a temporary feature which will eventually be replaced by a deterministic theory, but is instead a final renunciation of the classical idea of "causality". Bohr in particular emphasized that any well-defined application of the quantum mechanical formalism must always make reference to the experimental arrangement, due to the complementary nature of evidence obtained under different experimental situations. Copenhagen-type interpretations were adopted by Nobel laureates in quantum physics, including Bohr, [ 56 ] Heisenberg, [ 57 ] Schrödinger, [ 58 ] Feynman, [ 2 ] and Zeilinger [ 59 ] as well as 21st-century researchers in quantum foundations. [ 60 ] Albert Einstein , himself one of the founders of quantum theory , was troubled by its apparent failure to respect some cherished metaphysical principles, such as determinism and locality . Einstein's long-running exchanges with Bohr about the meaning and status of quantum mechanics are now known as the Bohr–Einstein debates . Einstein believed that underlying quantum mechanics must be a theory that explicitly forbids action at a distance . He argued that quantum mechanics was incomplete, a theory that was valid but not fundamental, analogous to how thermodynamics is valid, but the fundamental theory behind it is statistical mechanics . In 1935, Einstein and his collaborators Boris Podolsky and Nathan Rosen published an argument that the principle of locality implies the incompleteness of quantum mechanics, a thought experiment later termed the Einstein–Podolsky–Rosen paradox . [ note 4 ] In 1964, John Bell showed that EPR's principle of locality, together with determinism, was actually incompatible with quantum mechanics: they implied constraints on the correlations produced by distance systems, now known as Bell inequalities , that can be violated by entangled particles. [ 65 ] Since then several experiments have been performed to obtain these correlations, with the result that they do in fact violate Bell inequalities, and thus falsify the conjunction of locality with determinism. [ 16 ] [ 17 ] Bohmian mechanics shows that it is possible to reformulate quantum mechanics to make it deterministic, at the price of making it explicitly nonlocal. It attributes not only a wave function to a physical system, but in addition a real position, that evolves deterministically under a nonlocal guiding equation. The evolution of a physical system is given at all times by the Schrödinger equation together with the guiding equation; there is never a collapse of the wave function. This solves the measurement problem. [ 66 ] The Schrödinger's cat thought experiment can be used to visualize the many-worlds interpretation of quantum mechanics, where a branching of the universe occurs through a superposition of two quantum mechanical states. Everett's many-worlds interpretation , formulated in 1956, holds that all the possibilities described by quantum theory simultaneously occur in a multiverse composed of mostly independent parallel universes. [ 67 ] This is a consequence of removing the axiom of the collapse of the wave packet. All possible states of the measured system and the measuring apparatus, together with the observer, are present in a real physical quantum superposition. While the multiverse is deterministic, we perceive non-deterministic behavior governed by probabilities, because we do not observe the multiverse as a whole, but only one parallel universe at a time. Exactly how this is supposed to work has been the subject of much debate. Several attempts have been made to make sense of this and derive the Born rule, [ 68 ] [ 69 ] with no consensus on whether they have been successful. [ 70 ] [ 71 ] [ 72 ] Relational quantum mechanics appeared in the late 1990s as a modern derivative of Copenhagen-type ideas, [ 73 ] [ 74 ] and QBism was developed some years later. [ 75 ] [ 76 ] History Quantum mechanics was developed in the early decades of the 20th century, driven by the need to explain phenomena that, in some cases, had been observed in earlier times. Scientific inquiry into the wave nature of light began in the 17th and 18th centuries, when scientists such as Robert Hooke , Christiaan Huygens and Leonhard Euler proposed a wave theory of light based on experimental observations. [ 77 ] In 1803 English polymath Thomas Young described the famous double-slit experiment . [ 78 ] This experiment played a major role in the general acceptance of the wave theory of light . During the early 19th century, chemical research by John Dalton and Amedeo Avogadro lent weight to the atomic theory of matter, an idea that James Clerk Maxwell , Ludwig Boltzmann and others built upon to establish the kinetic theory of gases . The successes of kinetic theory gave further credence to the idea that matter is composed of atoms, yet the theory also had shortcomings that would only be resolved by the development of quantum mechanics. [ 79 ] While the early conception of atoms from Greek philosophy had been that they were indivisible units – the word "atom" deriving from the Greek for 'uncuttable' – the 19th century saw the formulation of hypotheses about subatomic structure. One important discovery in that regard was Michael Faraday 's 1838 observation of a glow caused by an electrical discharge inside a glass tube containing gas at low pressure. Julius Plücker , Johann Wilhelm Hittorf and Eugen Goldstein carried on and improved upon Faraday's work, leading to the identification of cathode rays , which J. J. Thomson found to consist of subatomic particles that would be called electrons. [ 80 ] [ 81 ] Max Planck is considered the father of the quantum theory. The black-body radiation problem was discovered by Gustav Kirchhoff in 1859. In 1900, Max Planck proposed the hypothesis that energy is radiated and absorbed in discrete "quanta" (or energy packets), yielding a calculation that precisely matched the observed patterns of black-body radiation. [ 82 ] The word quantum derives from the Latin , meaning "how great" or "how much". [ 83 ] According to Planck, quantities of energy could be thought of as divided into "elements" whose size ( E ) would be proportional to their frequency ( ν ): , where h is the Planck constant . Planck cautiously insisted that this was only an aspect of the processes of absorption and emission of radiation and was not the physical reality of the radiation. [ 84 ] In fact, he considered his quantum hypothesis a mathematical trick to get the right answer rather than a sizable discovery. [ 85 ] However, in 1905 Albert Einstein interpreted Planck's quantum hypothesis realistically and used it to explain the photoelectric effect , in which shining light on certain materials can eject electrons from the material. Niels Bohr then developed Planck's ideas about radiation into a model of the hydrogen atom that successfully predicted the spectral lines of hydrogen. [ 86 ] Einstein further developed this idea to show that an electromagnetic wave such as light could also be described as a particle (later called the photon), with a discrete amount of energy that depends on its frequency. [ 87 ] In his paper "On the Quantum Theory of Radiation", Einstein expanded on the interaction between energy and matter to explain the absorption and emission of energy by atoms. Although overshadowed at the time by his general theory of relativity, this paper articulated the mechanism underlying the stimulated emission of radiation, [ 88 ] which became the basis of the laser. [ 89 ] The 1927 Solvay Conference in Brussels was the fifth world physics conference. This phase is known as the old quantum theory . Never complete or self-consistent, the old quantum theory was rather a set of heuristic corrections to classical mechanics. [ 90 ] [ 91 ] The theory is now understood as a semi-classical approximation to modern quantum mechanics. [ 92 ] [ 93 ] Notable results from this period include, in addition to the work of Planck, Einstein and Bohr mentioned above, Einstein and Peter Debye 's work on the specific heat of solids, Bohr and Hendrika Johanna van Leeuwen 's proof that classical physics cannot account for diamagnetism , and Arnold Sommerfeld 's extension of the Bohr model to include special-relativistic effects. [ 90 ] [ 94 ] In the mid-1920s quantum mechanics was developed to become the standard formulation for atomic physics. In 1923, the French physicist Louis de Broglie put forward his theory of matter waves by stating that particles can exhibit wave characteristics and vice versa. Building on de Broglie's approach, modern quantum mechanics was born in 1925, when the German physicists Werner Heisenberg, Max Born, and Pascual Jordan [ 95 ] [ 96 ] developed matrix mechanics and the Austrian physicist Erwin Schrödinger invented wave mechanics . Born introduced the probabilistic interpretation of Schrödinger's wave function in July 1926. [ 97 ] Thus, the entire field of quantum physics emerged, leading to its wider acceptance at the Fifth Solvay Conference in 1927. [ 98 ] By 1930, quantum mechanics had been further unified and formalized by David Hilbert , Paul Dirac and John von Neumann [ 99 ] with greater emphasis on measurement , the statistical nature of our knowledge of reality, and philosophical speculation about the 'observer' . It has since permeated many disciplines, including quantum chemistry, quantum electronics , quantum optics , and quantum information science . It also provides a useful framework for many features of the modern periodic table of elements , and describes the behaviors of atoms during chemical bonding and the flow of electrons in computer semiconductors , and therefore plays a crucial role in many modern technologies. While quantum mechanics was constructed to describe the world of the very small, it is also needed to explain some macroscopic phenomena such as superconductors [ 100 ] and superfluids . [ 101 ] See also Bra–ket notation Einstein's thought experiments List of textbooks on classical mechanics and quantum mechanics Macroscopic quantum phenomena Phase-space formulation Regularization (physics) Two-state quantum system Explanatory notes ^ A momentum eigenstate would be a perfectly monochromatic wave of infinite extent, which is not square-integrable. Likewise, a position eigenstate would be a Dirac delta distribution , not square-integrable and technically not a function at all. Consequently, neither can belong to the particle's Hilbert space. Physicists sometimes introduce fictitious "bases" for a Hilbert space comprising elements outside that space. These are invented for calculational convenience and do not represent physical states. [ 26 ] : 100–105 ^ See, for example, the Feynman Lectures on Physics for some of the technological applications which use quantum mechanics, e.g., transistors (vol III , pp. 14–11 ff), integrated circuits , which are follow-on technology in solid-state physics (vol II , pp. 8–6), and lasers (vol III , pp. 9–13). ^ See Macroscopic quantum phenomena , Bose–Einstein condensate , and Quantum machine ^ The published form of the EPR argument was due to Podolsky, and Einstein himself was not satisfied with it. 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S2CID 1921913 . ^ Ballentine, L. E. (1973). "Can the statistical postulate of quantum theory be derived? – A critique of the many-universes interpretation". Foundations of Physics . 3 (2): 229– 240. Bibcode : 1973FoPh....3..229B . doi : 10.1007/BF00708440 . S2CID 121747282 . ^ Landsman, N. P. (2008). "The Born rule and its interpretation" (PDF) . In Weinert, F.; Hentschel, K.; Greenberger, D.; Falkenburg, B. (eds.). Compendium of Quantum Physics . Springer. ISBN 978-3-540-70622-9 . The conclusion seems to be that no generally accepted derivation of the Born rule has been given to date, but this does not imply that such a derivation is impossible in principle. ^ Kent, Adrian (2010). "One world versus many: The inadequacy of Everettian accounts of evolution, probability, and scientific confirmation". In S. Saunders; J. Barrett; A. Kent; D. Wallace (eds.). Many Worlds? Everett, Quantum Theory and Reality . Oxford University Press. arXiv : 0905.0624 . Bibcode : 2009arXiv0905.0624K . ^ Van Fraassen, Bas C. (April 2010). "Rovelli's World" . Foundations of Physics . 40 (4): 390– 417. Bibcode : 2010FoPh...40..390V . doi : 10.1007/s10701-009-9326-5 . ISSN 0015-9018 . S2CID 17217776 . ^ Calosi, Claudio ; Riedel, Timotheus (2024). "Relational Quantum Mechanics at the Crossroads" . Foundations of Physics . 54 (6): 74. Bibcode : 2024FoPh...54...74C . doi : 10.1007/s10701-024-00810-5 . hdl : 10278/5097808 . ^ Healey, Richard (2016). "Quantum-Bayesian and Pragmatist Views of Quantum Theory" . In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy . Metaphysics Research Lab, Stanford University. ^ Stacey, Blake C. (2023). "Toward a World Game-Flavored as a Hawk's Wing". In Berghofer, Philipp; Wiltsche, Harald A. (eds.). Phenomenology and QBism: New Approaches to Quantum Mechanics . Routledge. pp. 49– 77. doi : 10.4324/9781003259008-3 . ISBN 978-1-032-19181-2 . ^ Born, Max ; Wolf, Emil (1999). Principles of Optics . Cambridge University Press. ISBN 0-521-64222-1 . OCLC 1151058062 . ^ Scheider, Walter (April 1986). "Bringing one of the great moments of science to the classroom" . The Physics Teacher . 24 (4): 217– 219. Bibcode : 1986PhTea..24..217S . doi : 10.1119/1.2341987 . ISSN 0031-921X . ^ Feynman, Richard; Leighton, Robert; Sands, Matthew (1964). The Feynman Lectures on Physics . Vol. 1. California Institute of Technology . Retrieved 30 September 2021 . Reprinted, Addison-Wesley, 1989, ISBN 978-0-201-50064-6 ^ Martin, Andre (1986), "Cathode Ray Tubes for Industrial and Military Applications", in Hawkes, Peter (ed.), Advances in Electronics and Electron Physics, Volume 67 , Academic Press, p. 183, ISBN 978-0-08-057733-3 , Evidence for the existence of "cathode-rays" was first found by Plücker and Hittorf ... ^ Dahl, Per F. (1997). Flash of the Cathode Rays: A History of J. J. Thomson's Electron . CRC Press. pp. 47– 57. ISBN 978-0-7503-0453-5 . ^ Mehra, J. ; Rechenberg, H. (1982). The Historical Development of Quantum Theory, Vol. 1: The Quantum Theory of Planck, Einstein, Bohr and Sommerfeld. Its Foundation and the Rise of Its Difficulties (1900–1925) . New York: Springer-Verlag. ISBN 978-0-387-90642-3 . ^ "Quantum" . Merriam-Webster Dictionary . Archived from the original on Oct 26, 2012 . Retrieved 18 August 2012 . ^ Kuhn, T. S. (1978). Black-body theory and the quantum discontinuity 1894–1912 . Oxford: Clarendon Press. ISBN 978-0-19-502383-1 . ^ Kragh, Helge (1 December 2000). "Max Planck: the reluctant revolutionary" . Physics World . Retrieved 12 December 2020 . ^ Stachel, John (2009). "Bohr and the Photon". Quantum Reality, Relativistic Causality, and Closing the Epistemic Circle . The Western Ontario Series in Philosophy of Science. Vol. 73. Dordrecht: Springer. pp. 69– 83. doi : 10.1007/978-1-4020-9107-0_5 . ISBN 978-1-4020-9106-3 . ^ Einstein, Albert (1905). "Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt" [On a heuristic point of view concerning the production and transformation of light]. Annalen der Physik (in German). 17 (6): 132– 148. Bibcode : 1905AnP...322..132E . doi : 10.1002/andp.19053220607 . Reprinted in Stachel, John , ed. (1989). The Collected Papers of Albert Einstein (in German). Vol. 2. Princeton University Press. pp. 149– 166. See also "Einstein's early work on the quantum hypothesis", ibid. pp. 134–148. ^ Einstein, Albert (1917). "Zur Quantentheorie der Strahlung" [On the Quantum Theory of Radiation]. Physikalische Zeitschrift (in German). 18 : 121– 128. Bibcode : 1917PhyZ...18..121E . Translated in Einstein, A. (1967). "On the Quantum Theory of Radiation". The Old Quantum Theory . Elsevier. pp. 167– 183. doi : 10.1016/b978-0-08-012102-4.50018-8 . ISBN 978-0-08-012102-4 . ^ Ball, Philip (2017-08-31). "A century ago Einstein sparked the notion of the laser" . Physics World . Retrieved 2024-03-23 . ^ a b ter Haar, D. (1967). The Old Quantum Theory . Pergamon Press. pp. 3– 75. ISBN 978-0-08-012101-7 . LCCN 66-29628 . ^ Bokulich, Alisa; Bokulich, Peter (2020-08-13). "Bohr's Correspondence Principle" . In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy . ISSN 1095-5054 . OCLC 429049174 . ^ "Semi-classical approximation" . Encyclopedia of Mathematics . Archived from the original on 7 October 2022 . Retrieved 1 February 2020 . ^ Sakurai, J. J. ; Napolitano, J. (2014). "Quantum Dynamics". Modern Quantum Mechanics . Pearson. ISBN 978-1-292-02410-3 . OCLC 929609283 . ^ Aharoni, Amikam (1996). Introduction to the Theory of Ferromagnetism . Clarendon Press . pp. 6–7 . ISBN 0-19-851791-2 . ^ David Edwards, "The Mathematical Foundations of Quantum Mechanics", Synthese , Volume 42, Number 1/September, 1979, pp. 1–70. ^ David Edwards, "The Mathematical Foundations of Quantum Field Theory: Fermions, Gauge Fields, and Super-symmetry, Part I: Lattice Field Theories", International Journal of Theoretical Physics , Vol. 20, No. 7 (1981). ^ Bernstein, Jeremy (November 2005). "Max Born and the quantum theory" . American Journal of Physics . 73 (11): 999– 1008. Bibcode : 2005AmJPh..73..999B . doi : 10.1119/1.2060717 . ISSN 0002-9505 . ^ Pais, Abraham (1997). A Tale of Two Continents: A Physicist's Life in a Turbulent World . Princeton, New Jersey: Princeton University Press. ISBN 0-691-01243-1 . ^ Van Hove, Leon (1958). "Von Neumann's contributions to quantum mechanics" (PDF) . Bulletin of the American Mathematical Society . 64 (3): Part 2:95–99. doi : 10.1090/s0002-9904-1958-10206-2 . Archived (PDF) from the original on Jan 20, 2024. ^ Feynman, Richard . "The Feynman Lectures on Physics Vol. III Ch. 21: The Schrödinger Equation in a Classical Context: A Seminar on Superconductivity, 21-4" . California Institute of Technology . Retrieved 24 November 2015 . ...it was long believed that the wave function of the Schrödinger equation would never have a macroscopic representation analogous to the macroscopic representation of the amplitude for photons. On the other hand, it is now realized that the phenomena of superconductivity presents us with just this situation. {{ cite web }} : CS1 maint: url-status ( link ) ^ Packard, Richard (2006). "Berkeley Experiments on Superfluid Macroscopic Quantum Effects" (PDF) . Physics Department, University of California, Berkeley. Archived from the original (PDF) on 25 November 2015 . Retrieved 24 November 2015 . Further reading The following titles, all by working physicists, attempt to communicate quantum theory to lay people, using a minimum of technical apparatus: Chester, Marvin (1987). Primer of Quantum Mechanics . John Wiley. ISBN 0-486-42878-8 Cox, Brian ; Forshaw, Jeff (2011). The Quantum Universe: Everything That Can Happen Does Happen . Allen Lane. ISBN 978-1-84614-432-5 . Richard Feynman , 1985. QED: The Strange Theory of Light and Matter , Princeton University Press. ISBN 0-691-08388-6 . Four elementary lectures on quantum electrodynamics and quantum field theory , yet containing many insights for the expert. Ghirardi, GianCarlo , 2004. Sneaking a Look at God's Cards , Gerald Malsbary, trans. Princeton Univ. Press. The most technical of the works cited here. Passages using algebra , trigonometry , and bra–ket notation can be passed over on a first reading. N. David Mermin , 1990, "Spooky actions at a distance: mysteries of the QT" in his Boojums All the Way Through . Cambridge University Press: 110–76. Victor Stenger , 2000. Timeless Reality: Symmetry, Simplicity, and Multiple Universes . Buffalo, New York: Prometheus Books. Chpts. 5–8. Includes cosmological and philosophical considerations. More technical: Bernstein, Jeremy (2009). Quantum Leaps . Cambridge, Massachusetts: Belknap Press of Harvard University Press. ISBN 978-0-674-03541-6 . Bohm, David (1989). Quantum Theory . Dover Publications. ISBN 978-0-486-65969-5 . Binney, James ; Skinner, David (2008). The Physics of Quantum Mechanics . Oxford University Press. ISBN 978-0-19-968857-9 . Eisberg, Robert; Resnick, Robert (1985). Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles (2nd ed.). Wiley. ISBN 978-0-471-87373-0 . Bryce DeWitt , R. Neill Graham, eds., 1973. The Many-Worlds Interpretation of Quantum Mechanics , Princeton Series in Physics, Princeton University Press. ISBN 0-691-08131-X Everett, Hugh (1957). "Relative State Formulation of Quantum Mechanics". Reviews of Modern Physics . 29 (3): 454– 462. Bibcode : 1957RvMP...29..454E . doi : 10.1103/RevModPhys.29.454 . S2CID 17178479 . Feynman, Richard P. ; Leighton, Robert B. ; Sands, Matthew (1965). The Feynman Lectures on Physics . Vol. 1– 3. Addison-Wesley. ISBN 978-0-7382-0008-8 . D. Greenberger , K. Hentschel , F. Weinert, eds., 2009. Compendium of quantum physics, Concepts, experiments, history and philosophy , Springer-Verlag, Berlin, Heidelberg. Short articles on many QM topics. Griffiths, David J. (2004). Introduction to Quantum Mechanics (2nd ed.). Prentice Hall. ISBN 978-0-13-111892-8 . OCLC 40251748 . A standard undergraduate text. Max Jammer , 1966. The Conceptual Development of Quantum Mechanics . McGraw Hill. Hagen Kleinert , 2004. Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets , 3rd ed. Singapore: World Scientific. Draft of 4th edition. Archived 2008-06-15 at the Wayback Machine Landau, L. D.; Lifshitz, E. M. (1977). Quantum Mechanics: Non-Relativistic Theory . Vol. 3 (3rd ed.). Pergamon Press . ISBN 978-0-08-020940-1 . Online copy Liboff, Richard L. (2002). Introductory Quantum Mechanics . Addison-Wesley. ISBN 978-0-8053-8714-8 . Gunther Ludwig, 1968. Wave Mechanics . London: Pergamon Press. ISBN 0-08-203204-1 George Mackey (2004). The mathematical foundations of quantum mechanics . Dover Publications. ISBN 0-486-43517-2 . Merzbacher, Eugen (1998). Quantum Mechanics . Wiley, John & Sons, Inc. ISBN 978-0-471-88702-7 . Albert Messiah , 1966. Quantum Mechanics (Vol. I), English translation from French by G. M. Temmer. North Holland, John Wiley & Sons. Cf. chpt. IV, section III. Omnès, Roland (1999). Understanding Quantum Mechanics . Princeton University Press. ISBN 978-0-691-00435-8 . OCLC 39849482 . Scerri, Eric. R. (2006). The Periodic Table: Its Story and Its Significance . Oxford University Press. ISBN 0-19-530573-6 . Considers the extent to which chemistry and the periodic system have been reduced to quantum mechanics. Schiff, Leonard I. (1955). Quantum Mechanics . McGraw Hill. Shankar, R. (1994). Principles of Quantum Mechanics . Springer. ISBN 978-0-306-44790-7 . Stone, A. Douglas (2013). Einstein and the Quantum . Princeton University Press. ISBN 978-0-691-13968-5 . What is Quantum Mechanics? A Physics Adventure . Boston: Transnational College , Language Research Foundation. 1996. ISBN 978-0-9643504-1-0 . OCLC 34661512 . Veltman, Martinus J. G. (2003), Facts and Mysteries in Elementary Particle Physics . External links Introduction to quantum mechanics by Timon Idema Quantum Physics Made Relatively Simple : three video lectures by Hans Bethe . Course material Quantum Cook Book and PHYS 201: Fundamentals of Physics II by Ramamurti Shankar , Yale OpenCourseware. Modern Physics: With waves, thermodynamics, and optics – an online textbook. MIT OpenCourseWare : Chemistry and Physics . See 8.04 , 8.05 and 8.06 . ⁠5 + 1 / 2 ⁠ Examples in Quantum Mechanics . Philosophy Ismael, Jenann. "Quantum Mechanics" . In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy . ISSN 1095-5054 . OCLC 429049174 . Zalta, Edward N. (ed.). "Philosophical Issues in Quantum Theory" . Stanford Encyclopedia of Philosophy . ISSN 1095-5054 . OCLC 429049174 .

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[o]") ## Contents move to sidebar hide - [(Top)](https://en.wikipedia.org/wiki/Quantum_mechanics) - [1 Overview and fundamental concepts](https://en.wikipedia.org/wiki/Quantum_mechanics#Overview_and_fundamental_concepts) - [2 Mathematical formulation](https://en.wikipedia.org/wiki/Quantum_mechanics#Mathematical_formulation) Toggle Mathematical formulation subsection - [2\.1 Time evolution of a quantum state](https://en.wikipedia.org/wiki/Quantum_mechanics#Time_evolution_of_a_quantum_state) - [2\.2 Uncertainty principle](https://en.wikipedia.org/wiki/Quantum_mechanics#Uncertainty_principle) - [2\.3 Composite systems and entanglement](https://en.wikipedia.org/wiki/Quantum_mechanics#Composite_systems_and_entanglement) - [2\.4 Equivalence between formulations](https://en.wikipedia.org/wiki/Quantum_mechanics#Equivalence_between_formulations) - [2\.5 Symmetries and conservation laws](https://en.wikipedia.org/wiki/Quantum_mechanics#Symmetries_and_conservation_laws) - [3 Examples](https://en.wikipedia.org/wiki/Quantum_mechanics#Examples) Toggle Examples subsection - [3\.1 Free particle](https://en.wikipedia.org/wiki/Quantum_mechanics#Free_particle) - [3\.2 Particle in a box](https://en.wikipedia.org/wiki/Quantum_mechanics#Particle_in_a_box) - [3\.3 Harmonic oscillator](https://en.wikipedia.org/wiki/Quantum_mechanics#Harmonic_oscillator) - [3\.4 Mach–Zehnder interferometer](https://en.wikipedia.org/wiki/Quantum_mechanics#Mach%E2%80%93Zehnder_interferometer) - [4 Applications](https://en.wikipedia.org/wiki/Quantum_mechanics#Applications) - [5 Relation to other scientific theories](https://en.wikipedia.org/wiki/Quantum_mechanics#Relation_to_other_scientific_theories) Toggle Relation to other scientific theories subsection - [5\.1 Classical mechanics](https://en.wikipedia.org/wiki/Quantum_mechanics#Classical_mechanics) - [5\.2 Special relativity and electrodynamics](https://en.wikipedia.org/wiki/Quantum_mechanics#Special_relativity_and_electrodynamics) - [5\.3 Relation to general relativity](https://en.wikipedia.org/wiki/Quantum_mechanics#Relation_to_general_relativity) - [6 Philosophical implications](https://en.wikipedia.org/wiki/Quantum_mechanics#Philosophical_implications) - [7 History](https://en.wikipedia.org/wiki/Quantum_mechanics#History) - [8 See also](https://en.wikipedia.org/wiki/Quantum_mechanics#See_also) - [9 Explanatory notes](https://en.wikipedia.org/wiki/Quantum_mechanics#Explanatory_notes) - [10 References](https://en.wikipedia.org/wiki/Quantum_mechanics#References) - [11 Further reading](https://en.wikipedia.org/wiki/Quantum_mechanics#Further_reading) - [12 External links](https://en.wikipedia.org/wiki/Quantum_mechanics#External_links) Toggle the table of contents # Quantum mechanics 142 languages - [Afrikaans](https://af.wikipedia.org/wiki/Kwantummeganika "Kwantummeganika – Afrikaans") - [Alemannisch](https://als.wikipedia.org/wiki/Quantenmechanik "Quantenmechanik – Alemannic") - [Aragonés](https://an.wikipedia.org/wiki/Mecanica_cuantica "Mecanica cuantica – Aragonese") - [अंगिका](https://anp.wikipedia.org/wiki/%E0%A4%95%E0%A5%8D%E0%A4%B5%E0%A4%BE%E0%A4%82%E0%A4%9F%E0%A4%AE_%E0%A4%B8%E0%A4%BF%E0%A4%A6%E0%A5%8D%E0%A4%A7%E0%A4%BE%E0%A4%82%E0%A4%A4 "क्वांटम सिद्धांत – Angika") - [العربية](https://ar.wikipedia.org/wiki/%D9%85%D9%8A%D9%83%D8%A7%D9%86%D9%8A%D9%83%D8%A7_%D8%A7%D9%84%D9%83%D9%85 "ميكانيكا الكم – Arabic") - [مصرى](https://arz.wikipedia.org/wiki/%D8%A7%D9%84%D9%85%D9%8A%D9%83%D8%A7%D9%86%D9%8A%D9%83%D8%A7_%D8%A7%D9%84%D9%83%D9%85%D9%8A%D9%87 "الميكانيكا الكميه – Egyptian Arabic") - [অসমীয়া](https://as.wikipedia.org/wiki/%E0%A6%95%E0%A7%8B%E0%A7%B1%E0%A6%BE%E0%A6%A3%E0%A7%8D%E0%A6%9F%E0%A6%BE%E0%A6%AE_%E0%A6%AC%E0%A6%B2%E0%A6%AC%E0%A6%BF%E0%A6%9C%E0%A7%8D%E0%A6%9E%E0%A6%BE%E0%A6%A8 "কোৱাণ্টাম বলবিজ্ঞান – Assamese") - [Asturianu](https://ast.wikipedia.org/wiki/Mec%C3%A1nica_cu%C3%A1ntica "Mecánica cuántica – Asturian") - [Azərbaycanca](https://az.wikipedia.org/wiki/Kvant_mexanikas%C4%B1 "Kvant mexanikası – Azerbaijani") - [تۆرکجه](https://azb.wikipedia.org/wiki/%DA%A9%D9%88%D8%A7%D9%86%D8%AA%D9%88%D9%85_%D9%85%DA%A9%D8%A7%D9%86%DB%8C%DA%A9%DB%8C "کوانتوم مکانیکی – South Azerbaijani") - [Башҡортса](https://ba.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0%D2%BB%D1%8B "Квант механикаһы – Bashkir") - [Basa Bali](https://ban.wikipedia.org/wiki/M%C3%A9kanika_kuantum "Mékanika kuantum – Balinese") - [Boarisch](https://bar.wikipedia.org/wiki/Fuzalmechanik "Fuzalmechanik – Bavarian") - [Žemaitėška](https://bat-smg.wikipedia.org/wiki/Kvant%C4%97n%C4%97_mekan%C4%97ka "Kvantėnė mekanėka – Samogitian") - [Bikol Central](https://bcl.wikipedia.org/wiki/Mekanikang_kwantum "Mekanikang kwantum – Central Bikol") - [Беларуская (тарашкевіца)](https://be-tarask.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D0%B0%D0%B2%D0%B0%D1%8F_%D0%BC%D1%8D%D1%85%D0%B0%D0%BD%D1%96%D0%BA%D0%B0 "Квантавая мэханіка – Belarusian (Taraškievica orthography)") - [Беларуская](https://be.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D0%B0%D0%B2%D0%B0%D1%8F_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D1%96%D0%BA%D0%B0 "Квантавая механіка – Belarusian") - [Български](https://bg.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D0%BE%D0%B2%D0%B0_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0 "Квантова механика – Bulgarian") - [भोजपुरी](https://bh.wikipedia.org/wiki/%E0%A4%95%E0%A5%8D%E0%A4%B5%E0%A4%BE%E0%A4%82%E0%A4%9F%E0%A4%AE_%E0%A4%AE%E0%A5%88%E0%A4%95%E0%A5%87%E0%A4%A8%E0%A4%BF%E0%A4%95%E0%A5%8D%E0%A4%B8 "क्वांटम मैकेनिक्स – Bhojpuri") - [ပအိုဝ်ႏဘာႏသာႏ](https://blk.wikipedia.org/wiki/%E1%80%80%E1%80%BD%E1%80%99%E1%80%BA%E1%80%90%E1%80%99%E1%80%BA%E1%80%99%E1%80%80%E1%80%B9%E1%80%80%E1%80%94%E1%80%85%E1%80%BA "ကွမ်တမ်မက္ကနစ် – Pa'O") - [বাংলা](https://bn.wikipedia.org/wiki/%E0%A6%95%E0%A7%8B%E0%A6%AF%E0%A6%BC%E0%A6%BE%E0%A6%A8%E0%A7%8D%E0%A6%9F%E0%A6%BE%E0%A6%AE_%E0%A6%AC%E0%A6%B2%E0%A6%AC%E0%A6%BF%E0%A6%9C%E0%A7%8D%E0%A6%9E%E0%A6%BE%E0%A6%A8 "কোয়ান্টাম বলবিজ্ঞান – Bangla") - [Brezhoneg](https://br.wikipedia.org/wiki/Mekanikerezh_kwantek "Mekanikerezh kwantek – Breton") - [Bosanski](https://bs.wikipedia.org/wiki/Kvantna_mehanika "Kvantna mehanika – Bosnian") - [Буряад](https://bxr.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D1%8B%D0%BD_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0 "Квантын механика – Russia Buriat") - [Català](https://ca.wikipedia.org/wiki/Mec%C3%A0nica_qu%C3%A0ntica "Mecànica quàntica – Catalan") - [閩東語 / Mìng-dĕ̤ng-ngṳ̄](https://cdo.wikipedia.org/wiki/Li%C3%B4ng-c%E1%B9%B3%CC%84_l%C4%ADk-h%C5%8Fk "Liông-cṳ̄ lĭk-hŏk – Mindong") - [کوردی](https://ckb.wikipedia.org/wiki/%D9%85%DB%8C%DA%A9%D8%A7%D9%86%DB%8C%DA%A9%DB%8C_%DA%A9%D9%88%D8%A7%D9%86%D8%AA%DB%86%D9%85 "میکانیکی کوانتۆم – Central Kurdish") - [Čeština](https://cs.wikipedia.org/wiki/Kvantov%C3%A1_mechanika "Kvantová mechanika – Czech") - [Чӑвашла](https://cv.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D0%BB%D0%B0_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0 "Квантла механика – Chuvash") - [Cymraeg](https://cy.wikipedia.org/wiki/Mecaneg_cwantwm "Mecaneg cwantwm – Welsh") - [Dansk](https://da.wikipedia.org/wiki/Kvantemekanik "Kvantemekanik – Danish") - [Deutsch](https://de.wikipedia.org/wiki/Quantenmechanik "Quantenmechanik – German") - [Ελληνικά](https://el.wikipedia.org/wiki/%CE%9A%CE%B2%CE%B1%CE%BD%CF%84%CE%B9%CE%BA%CE%AE_%CE%BC%CE%B7%CF%87%CE%B1%CE%BD%CE%B9%CE%BA%CE%AE "Κβαντική μηχανική – Greek") - [Esperanto](https://eo.wikipedia.org/wiki/Kvantuma_mekaniko "Kvantuma mekaniko – Esperanto") - [Español](https://es.wikipedia.org/wiki/Mec%C3%A1nica_cu%C3%A1ntica "Mecánica cuántica – Spanish") - [Eesti](https://et.wikipedia.org/wiki/Kvantmehaanika "Kvantmehaanika – Estonian") - [Euskara](https://eu.wikipedia.org/wiki/Mekanika_kuantiko "Mekanika kuantiko – Basque") - [Estremeñu](https://ext.wikipedia.org/wiki/Mec%C3%A1nica_cu%C3%A1ntica "Mecánica cuántica – Extremaduran") - [فارسی](https://fa.wikipedia.org/wiki/%D9%85%DA%A9%D8%A7%D9%86%DB%8C%DA%A9_%DA%A9%D9%88%D8%A7%D9%86%D8%AA%D9%88%D9%85%DB%8C "مکانیک کوانتومی – Persian") - [Suomi](https://fi.wikipedia.org/wiki/Kvanttimekaniikka "Kvanttimekaniikka – Finnish") - [Võro](https://fiu-vro.wikipedia.org/wiki/Kvantmekaaniga "Kvantmekaaniga – Võro") - [Français](https://fr.wikipedia.org/wiki/M%C3%A9canique_quantique "Mécanique quantique – French") - [Nordfriisk](https://frr.wikipedia.org/wiki/Kwantenmechaanik "Kwantenmechaanik – Northern Frisian") - [Gaeilge](https://ga.wikipedia.org/wiki/Meicnic_chandamach "Meicnic chandamach – Irish") - [Kriyòl gwiyannen](https://gcr.wikipedia.org/wiki/M%C3%A9kanik_kantik "Mékanik kantik – Guianan Creole") - [Gàidhlig](https://gd.wikipedia.org/wiki/Meacanaigs_quantumach "Meacanaigs quantumach – Scottish Gaelic") - [Galego](https://gl.wikipedia.org/wiki/Mec%C3%A1nica_cu%C3%A1ntica "Mecánica cuántica – Galician") - [Avañe'ẽ](https://gn.wikipedia.org/wiki/Mek%C3%A1nika_ku%C3%A1ntika "Mekánika kuántika – Guarani") - [Hausa](https://ha.wikipedia.org/wiki/Kimiyyar_kwantom "Kimiyyar kwantom – Hausa") - [עברית](https://he.wikipedia.org/wiki/%D7%9E%D7%9B%D7%A0%D7%99%D7%A7%D7%AA_%D7%94%D7%A7%D7%95%D7%95%D7%A0%D7%98%D7%99%D7%9D "מכניקת הקוונטים – Hebrew") - [हिन्दी](https://hi.wikipedia.org/wiki/%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%AE%E0%A4%BE%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%BE_%E0%A4%AF%E0%A4%BE%E0%A4%A8%E0%A5%8D%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%BF%E0%A4%95%E0%A5%80 "प्रमात्रा यान्त्रिकी – Hindi") - [Fiji Hindi](https://hif.wikipedia.org/wiki/Quantum_mechanics "Quantum mechanics – Fiji Hindi") - [Hrvatski](https://hr.wikipedia.org/wiki/Kvantna_mehanika "Kvantna mehanika – Croatian") - [Kreyòl ayisyen](https://ht.wikipedia.org/wiki/Mekanik_kantik "Mekanik kantik – Haitian Creole") - [Magyar](https://hu.wikipedia.org/wiki/Kvantummechanika "Kvantummechanika – Hungarian") - [Հայերեն](https://hy.wikipedia.org/wiki/%D5%94%D5%BE%D5%A1%D5%B6%D5%BF%D5%A1%D5%B5%D5%AB%D5%B6_%D5%B4%D5%A5%D5%AD%D5%A1%D5%B6%D5%AB%D5%AF%D5%A1 "Քվանտային մեխանիկա – Armenian") - [Արեւմտահայերէն](https://hyw.wikipedia.org/wiki/%D5%94%D5%B8%D6%82%D5%A1%D5%B6%D5%BF%D5%A1%D5%B5%D5%AB%D5%B6_%D5%B4%D5%A5%D5%A3%D5%A1%D5%B6%D5%AB%D5%AF "Քուանտային մեգանիկ – Western Armenian") - [Interlingua](https://ia.wikipedia.org/wiki/Mechanica_quantic "Mechanica quantic – Interlingua") - [Bahasa Indonesia](https://id.wikipedia.org/wiki/Mekanika_kuantum "Mekanika kuantum – Indonesian") - [Igbo](https://ig.wikipedia.org/wiki/Quantum_mechanics "Quantum mechanics – Igbo") - [Ido](https://io.wikipedia.org/wiki/Quantumala_mekaniko "Quantumala mekaniko – Ido") - [Íslenska](https://is.wikipedia.org/wiki/Skammtafr%C3%A6%C3%B0i "Skammtafræði – Icelandic") - [Italiano](https://it.wikipedia.org/wiki/Meccanica_quantistica "Meccanica quantistica – Italian") - [日本語](https://ja.wikipedia.org/wiki/%E9%87%8F%E5%AD%90%E5%8A%9B%E5%AD%A6 "量子力学 – Japanese") - [Patois](https://jam.wikipedia.org/wiki/Kuantom_mikianix "Kuantom mikianix – Jamaican Creole English") - [ქართული](https://ka.wikipedia.org/wiki/%E1%83%99%E1%83%95%E1%83%90%E1%83%9C%E1%83%A2%E1%83%A3%E1%83%A0%E1%83%98_%E1%83%9B%E1%83%94%E1%83%A5%E1%83%90%E1%83%9C%E1%83%98%E1%83%99%E1%83%90 "კვანტური მექანიკა – Georgian") - [Qaraqalpaqsha](https://kaa.wikipedia.org/wiki/Kvant_mexanika "Kvant mexanika – Kara-Kalpak") - [Kabɩyɛ](https://kbp.wikipedia.org/wiki/%C3%91%CA%8B%C5%8B_ho%C9%96e "Ñʋŋ hoɖe – Kabiye") - [Қазақша](https://kk.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D1%82%D1%8B%D2%9B_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0 "Кванттық механика – Kazakh") - [ಕನ್ನಡ](https://kn.wikipedia.org/wiki/%E0%B2%95%E0%B3%8D%E0%B2%B5%E0%B2%BE%E0%B2%82%E0%B2%9F%E0%B2%AE%E0%B3%8D_%E0%B2%AD%E0%B3%8C%E0%B2%A4%E0%B2%B6%E0%B2%BE%E0%B2%B8%E0%B3%8D%E0%B2%A4%E0%B3%8D%E0%B2%B0 "ಕ್ವಾಂಟಮ್ ಭೌತಶಾಸ್ತ್ರ – Kannada") - [한국어](https://ko.wikipedia.org/wiki/%EC%96%91%EC%9E%90%EC%97%AD%ED%95%99 "양자역학 – Korean") - [Кыргызча](https://ky.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D1%82%D1%8B%D0%BA_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0 "Кванттык механика – Kyrgyz") - [Latina](https://la.wikipedia.org/wiki/Mechanica_quantica "Mechanica quantica – Latin") - [Limburgs](https://li.wikipedia.org/wiki/Kwantummechanica "Kwantummechanica – Limburgish") - [Lombard](https://lmo.wikipedia.org/wiki/Mec%C3%A0nega_quant%C3%ACstega "Mecànega quantìstega – Lombard") - [Lietuvių](https://lt.wikipedia.org/wiki/Kvantin%C4%97_mechanika "Kvantinė mechanika – Lithuanian") - [Latviešu](https://lv.wikipedia.org/wiki/Kvantu_meh%C4%81nika "Kvantu mehānika – Latvian") - [Македонски](https://mk.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D0%BD%D0%B0_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0 "Квантна механика – Macedonian") - [മലയാളം](https://ml.wikipedia.org/wiki/%E0%B4%95%E0%B5%8D%E0%B4%B5%E0%B4%BE%E0%B4%A3%E0%B5%8D%E0%B4%9F%E0%B4%82_%E0%B4%AC%E0%B4%B2%E0%B4%A4%E0%B4%A8%E0%B5%8D%E0%B4%A4%E0%B5%8D%E0%B4%B0%E0%B4%82 "ക്വാണ്ടം ബലതന്ത്രം – Malayalam") - [Монгол](https://mn.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA "Квант механик – Mongolian") - [मराठी](https://mr.wikipedia.org/wiki/%E0%A4%AA%E0%A5%81%E0%A4%82%E0%A4%9C_%E0%A4%AF%E0%A4%BE%E0%A4%AE%E0%A4%BF%E0%A4%95%E0%A5%80 "पुंज यामिकी – Marathi") - [Bahasa Melayu](https://ms.wikipedia.org/wiki/Mekanik_kuantum "Mekanik kuantum – Malay") - [Malti](https://mt.wikipedia.org/wiki/Mekkanika_kwantistika "Mekkanika kwantistika – Maltese") - [မြန်မာဘာသာ](https://my.wikipedia.org/wiki/%E1%80%80%E1%80%BD%E1%80%99%E1%80%BA%E1%80%90%E1%80%99%E1%80%BA%E1%80%99%E1%80%80%E1%80%B9%E1%80%80%E1%80%84%E1%80%BA%E1%80%B8%E1%80%94%E1%80%85%E1%80%BA "ကွမ်တမ်မက္ကင်းနစ် – Burmese") - [مازِرونی](https://mzn.wikipedia.org/wiki/%DA%A9%D9%88%D8%A7%D9%86%D8%AA%D9%88%D9%85%DB%8C_%D9%81%DB%8C%D8%B2%DB%8C%DA%A9 "کوانتومی فیزیک – Mazanderani") - [Plattdüütsch](https://nds.wikipedia.org/wiki/Quantenmechanik "Quantenmechanik – Low German") - [नेपाली](https://ne.wikipedia.org/wiki/%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%AE%E0%A4%BE%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%BE_%E0%A4%AF%E0%A4%BE%E0%A4%A8%E0%A5%8D%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%BF%E0%A4%95%E0%A5%80 "प्रमात्रा यान्त्रिकी – Nepali") - [नेपाल भाषा](https://new.wikipedia.org/wiki/%E0%A4%95%E0%A5%8D%E0%A4%B5%E0%A4%BE%E0%A4%A8%E0%A5%8D%E0%A4%9F%E0%A4%AE_%E0%A4%AE%E0%A5%87%E0%A4%95%E0%A4%BE%E0%A4%A8%E0%A4%BF%E0%A4%95%E0%A5%8D%E0%A4%B8%E0%A5%8D "क्वान्टम मेकानिक्स् – Newari") - [Nederlands](https://nl.wikipedia.org/wiki/Kwantummechanica "Kwantummechanica – Dutch") - [Norsk nynorsk](https://nn.wikipedia.org/wiki/Kvantemekanikk "Kvantemekanikk – Norwegian Nynorsk") - [Norsk bokmål](https://no.wikipedia.org/wiki/Kvantemekanikk "Kvantemekanikk – Norwegian Bokmål") - [Occitan](https://oc.wikipedia.org/wiki/Mecanica_quantica "Mecanica quantica – Occitan") - [ਪੰਜਾਬੀ](https://pa.wikipedia.org/wiki/%E0%A8%95%E0%A9%81%E0%A8%86%E0%A8%82%E0%A8%9F%E0%A8%AE_%E0%A8%AE%E0%A8%95%E0%A9%88%E0%A8%A8%E0%A8%BF%E0%A8%95%E0%A8%B8 "ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ – Punjabi") - [Polski](https://pl.wikipedia.org/wiki/Mechanika_kwantowa "Mechanika kwantowa – Polish") - [Piemontèis](https://pms.wikipedia.org/wiki/Mec%C3%A0nica_qu%C3%A0ntica "Mecànica quàntica – Piedmontese") - [پنجابی](https://pnb.wikipedia.org/wiki/%DA%A9%D9%88%D8%A7%D9%86%D9%B9%D9%85_%D9%85%DA%A9%DB%8C%D9%86%DA%A9%D8%B3 "کوانٹم مکینکس – Western Punjabi") - [پښتو](https://ps.wikipedia.org/wiki/%DA%A9%D9%88%D8%A7%D9%86%D9%BC%D9%88%D9%85_%D9%85%DB%8C%D8%AE%D8%A7%D9%86%DB%8C%DA%A9 "کوانټوم میخانیک – Pashto") - [Português](https://pt.wikipedia.org/wiki/Mec%C3%A2nica_qu%C3%A2ntica "Mecânica quântica – Portuguese") - [ရခိုင်](https://rki.wikipedia.org/wiki/%E1%80%80%E1%80%BD%E1%80%99%E1%80%BA%E1%80%90%E1%80%99%E1%80%BA%E1%80%99%E1%80%80%E1%80%B9%E1%80%80%E1%80%84%E1%80%BA%E1%80%B8%E1%80%94%E1%80%85%E1%80%BA "ကွမ်တမ်မက္ကင်းနစ် – Arakanese") - [Română](https://ro.wikipedia.org/wiki/Mecanic%C4%83_cuantic%C4%83 "Mecanică cuantică – Romanian") - [Русский](https://ru.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D0%BE%D0%B2%D0%B0%D1%8F_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0 "Квантовая механика – Russian") - [Русиньскый](https://rue.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D0%BE%D0%B2%D0%B0_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D1%96%D0%BA%D0%B0 "Квантова механіка – Rusyn") - [संस्कृतम्](https://sa.wikipedia.org/wiki/%E0%A4%B8%E0%A5%82%E0%A4%95%E0%A5%8D%E0%A4%B7%E0%A5%8D%E0%A4%AE%E0%A4%AD%E0%A5%8C%E0%A4%A4%E0%A4%BF%E0%A4%95%E0%A4%B6%E0%A4%BE%E0%A4%B8%E0%A5%8D%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%AE%E0%A5%8D "सूक्ष्मभौतिकशास्त्रम् – Sanskrit") - [Саха тыла](https://sah.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D0%BE%D0%B2%D0%B0%D0%B9_%D1%84%D0%B8%D0%B7%D0%B8%D0%BA%D0%B0 "Квантовай физика – Yakut") - [Sicilianu](https://scn.wikipedia.org/wiki/Micc%C3%A0nica_quant%C3%ACstica "Miccànica quantìstica – Sicilian") - [Scots](https://sco.wikipedia.org/wiki/Quantum_mechanics "Quantum mechanics – Scots") - [سنڌي](https://sd.wikipedia.org/wiki/%DA%AA%D9%88%D8%A7%D9%86%D9%BD%D9%85_%D9%85%DA%AA%D9%8A%D9%86%DA%AA%D8%B3 "ڪوانٽم مڪينڪس – Sindhi") - [Srpskohrvatski / српскохрватски](https://sh.wikipedia.org/wiki/Kvantna_mehanika "Kvantna mehanika – Serbo-Croatian") - [Taclḥit](https://shi.wikipedia.org/wiki/Tamikanikt_tasmktant "Tamikanikt tasmktant – Tachelhit") - [සිංහල](https://si.wikipedia.org/wiki/%E0%B6%9A%E0%B7%8A%E0%B7%80%E0%B7%9C%E0%B6%B1%E0%B7%8A%E0%B6%A7%E0%B6%B8%E0%B7%8A_%E0%B6%BA%E0%B7%8F%E0%B6%B1%E0%B7%8A%E0%B6%AD%E0%B7%8A%E2%80%8D%E0%B6%BB_%E0%B7%80%E0%B7%92%E0%B6%AF%E0%B7%8A%E2%80%8D%E0%B6%BA%E0%B7%8F%E2%80%8D%E0%B7%80 "ක්වොන්ටම් යාන්ත්‍ර විද්‍යා‍ව – Sinhala") - [Simple English](https://simple.wikipedia.org/wiki/Quantum_mechanics "Quantum mechanics – Simple English") - [Slovenčina](https://sk.wikipedia.org/wiki/Kvantov%C3%A1_mechanika "Kvantová mechanika – Slovak") - [Slovenščina](https://sl.wikipedia.org/wiki/Kvantna_mehanika "Kvantna mehanika – Slovenian") - [Shqip](https://sq.wikipedia.org/wiki/Mekanika_kuantike "Mekanika kuantike – Albanian") - [Српски / srpski](https://sr.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D0%BD%D0%B0_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0 "Квантна механика – Serbian") - [Sunda](https://su.wikipedia.org/wiki/M%C3%A9kanika_kuantum "Mékanika kuantum – Sundanese") - [Svenska](https://sv.wikipedia.org/wiki/Kvantmekanik "Kvantmekanik – Swedish") - [Kiswahili](https://sw.wikipedia.org/wiki/Umakanika_kwanta "Umakanika kwanta – Swahili") - [Ślůnski](https://szl.wikipedia.org/wiki/Kwantow%C5%8F_mechanika "Kwantowŏ mechanika – Silesian") - [தமிழ்](https://ta.wikipedia.org/wiki/%E0%AE%95%E0%AF%81%E0%AE%B5%E0%AE%BE%E0%AE%A3%E0%AF%8D%E0%AE%9F%E0%AE%AE%E0%AF%8D_%E0%AE%87%E0%AE%AF%E0%AE%99%E0%AF%8D%E0%AE%95%E0%AE%BF%E0%AE%AF%E0%AE%B2%E0%AF%8D "குவாண்டம் இயங்கியல் – Tamil") - [తెలుగు](https://te.wikipedia.org/wiki/%E0%B0%95%E0%B1%8D%E0%B0%B5%E0%B0%BE%E0%B0%82%E0%B0%9F%E0%B0%82_%E0%B0%AF%E0%B0%BE%E0%B0%82%E0%B0%A4%E0%B1%8D%E0%B0%B0%E0%B0%BF%E0%B0%95_%E0%B0%B6%E0%B0%BE%E0%B0%B8%E0%B1%8D%E0%B0%A4%E0%B1%8D%E0%B0%B0%E0%B0%82 "క్వాంటం యాంత్రిక శాస్త్రం – Telugu") - [Тоҷикӣ](https://tg.wikipedia.org/wiki/%D0%9C%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0%D0%B8_%D0%BA%D0%B2%D0%B0%D0%BD%D1%82%D3%A3 "Механикаи квантӣ – Tajik") - [ไทย](https://th.wikipedia.org/wiki/%E0%B8%81%E0%B8%A5%E0%B8%A8%E0%B8%B2%E0%B8%AA%E0%B8%95%E0%B8%A3%E0%B9%8C%E0%B8%84%E0%B8%A7%E0%B8%AD%E0%B8%99%E0%B8%95%E0%B8%B1%E0%B8%A1 "กลศาสตร์ควอนตัม – Thai") - [Tagalog](https://tl.wikipedia.org/wiki/Mekanikang_quantum "Mekanikang quantum – Tagalog") - [Toki pona](https://tok.wikipedia.org/wiki/sona_tawa_pi_wan_lili "sona tawa pi wan lili – Toki Pona") - [Türkçe](https://tr.wikipedia.org/wiki/Kuantum_mekani%C4%9Fi "Kuantum mekaniği – Turkish") - [Татарча / tatarça](https://tt.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0%D1%81%D1%8B "Квант механикасы – Tatar") - [Українська](https://uk.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D0%BE%D0%B2%D0%B0_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D1%96%D0%BA%D0%B0 "Квантова механіка – Ukrainian") - [اردو](https://ur.wikipedia.org/wiki/%D9%82%D8%AF%D8%B1%DB%8C_%D9%85%DB%8C%DA%A9%D8%A7%D9%86%DB%8C%D8%A7%D8%AA "قدری میکانیات – Urdu") - [Oʻzbekcha / ўзбекча](https://uz.wikipedia.org/wiki/Kvant_mexanika "Kvant mexanika – Uzbek") - [Vèneto](https://vec.wikipedia.org/wiki/Mec%C3%A0nega_cuant%C3%ACstega "Mecànega cuantìstega – Venetian") - [Vepsän kel’](https://vep.wikipedia.org/wiki/Kvantmehanik "Kvantmehanik – Veps") - [Tiếng Việt](https://vi.wikipedia.org/wiki/C%C6%A1_h%E1%BB%8Dc_l%C6%B0%E1%BB%A3ng_t%E1%BB%AD "Cơ học lượng tử – Vietnamese") - [Winaray](https://war.wikipedia.org/wiki/Mekanika_kwantum "Mekanika kwantum – Waray") - [吴语](https://wuu.wikipedia.org/wiki/%E9%87%8F%E5%AD%90%E5%8A%9B%E5%AD%A6 "量子力学 – Wu") - [ייִדיש](https://yi.wikipedia.org/wiki/%D7%A7%D7%95%D7%95%D7%90%D7%A0%D7%98%D7%9F-%D7%9E%D7%A2%D7%9B%D7%90%D7%A0%D7%99%D7%A7 "קוואנטן-מעכאניק – Yiddish") - [ⵜⴰⵎⴰⵣⵉⵖⵜ ⵜⴰⵏⴰⵡⴰⵢⵜ](https://zgh.wikipedia.org/wiki/%E2%B5%9C%E2%B4%B0%E2%B5%99%E2%B5%8F%E2%B4%B7%E2%B4%B7%E2%B5%93%E2%B5%9C_%E2%B5%9C%E2%B4%B0%E2%B5%99%E2%B5%8E%E2%B4%BD%E2%B5%9C%E2%B4%B0%E2%B5%A2%E2%B5%9C "ⵜⴰⵙⵏⴷⴷⵓⵜ ⵜⴰⵙⵎⴽⵜⴰⵢⵜ – Standard Moroccan Tamazight") - [文言](https://zh-classical.wikipedia.org/wiki/%E9%87%8F%E5%AD%90%E5%8A%9B%E5%AD%B8 "量子力學 – Literary Chinese") - [閩南語 / Bân-lâm-gí](https://zh-min-nan.wikipedia.org/wiki/Li%C5%8Dng-ch%C3%BA_le%CC%8Dk-ha%CC%8Dk "Liōng-chú le̍k-ha̍k – Minnan") - [粵語](https://zh-yue.wikipedia.org/wiki/%E9%87%8F%E5%AD%90%E5%8A%9B%E5%AD%B8 "量子力學 – Cantonese") - [中文](https://zh.wikipedia.org/wiki/%E9%87%8F%E5%AD%90%E5%8A%9B%E5%AD%A6 "量子力学 – Chinese") - [IsiZulu](https://zu.wikipedia.org/wiki/Ukuguxazela_kohoyana "Ukuguxazela kohoyana – Zulu") [Edit links](https://www.wikidata.org/wiki/Special:EntityPage/Q944#sitelinks-wikipedia "Edit interlanguage links") - [Article](https://en.wikipedia.org/wiki/Quantum_mechanics "View the content page [c]") - [Talk](https://en.wikipedia.org/wiki/Talk:Quantum_mechanics "Discuss improvements to the content page [t]") English - [Read](https://en.wikipedia.org/wiki/Quantum_mechanics) - [View source](https://en.wikipedia.org/w/index.php?title=Quantum_mechanics&action=edit "This page is protected. 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Click here for more information.](https://upload.wikimedia.org/wikipedia/en/thumb/9/94/Symbol_support_vote.svg/20px-Symbol_support_vote.svg.png)](https://en.wikipedia.org/wiki/Wikipedia:Good_articles* "This is a good article. Click here for more information.") [![Page semi-protected](https://upload.wikimedia.org/wikipedia/en/thumb/1/1b/Semi-protection-shackle.svg/20px-Semi-protection-shackle.svg.png)](https://en.wikipedia.org/wiki/Wikipedia:Protection_policy#semi "This article is semi-protected.") From Wikipedia, the free encyclopedia Description of physical properties at the atomic and subatomic scale "Quantum systems" redirects here. For the company, see [Quantum-Systems](https://en.wikipedia.org/wiki/Quantum-Systems "Quantum-Systems"). For a more accessible and less technical introduction to this topic, see [Introduction to quantum mechanics](https://en.wikipedia.org/wiki/Introduction_to_quantum_mechanics "Introduction to quantum mechanics"). [![](https://upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Hydrogen_Density_Plots.png/500px-Hydrogen_Density_Plots.png)](https://en.wikipedia.org/wiki/File:Hydrogen_Density_Plots.png) [Wave functions](https://en.wikipedia.org/wiki/Wave_function "Wave function") of the [electron](https://en.wikipedia.org/wiki/Electron "Electron") in a hydrogen atom at different energy levels. Quantum mechanics cannot predict the exact location of a particle in space, only the probability of finding it at different locations.[\[1\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Born1926-1) The brighter areas represent a higher probability of finding the electron. | | |---| | Part of a series of articles about | | [Quantum mechanics]() | | i ℏ d d t \| Ψ ⟩ \= H ^ \| Ψ ⟩ {\\displaystyle i\\hbar {\\frac {d}{dt}}\|\\Psi \\rangle ={\\hat {H}}\|\\Psi \\rangle } ![{\\displaystyle i\\hbar {\\frac {d}{dt}}\|\\Psi \\rangle ={\\hat {H}}\|\\Psi \\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/1799e4a910c7d26396922a20ef5ceec25ca1871c)[Schrödinger equation](https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation "Schrödinger equation") | **Quantum mechanics** is the fundamental physical [theory](https://en.wikipedia.org/wiki/Scientific_theory "Scientific theory") that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of [atoms](https://en.wikipedia.org/wiki/Atom "Atom").[\[2\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Feynman-2): 1.1 It is the foundation of all **quantum physics**, which includes [quantum chemistry](https://en.wikipedia.org/wiki/Quantum_chemistry "Quantum chemistry"), [quantum biology](https://en.wikipedia.org/wiki/Quantum_biology "Quantum biology"), [quantum field theory](https://en.wikipedia.org/wiki/Quantum_field_theory "Quantum field theory"), [quantum technology](https://en.wikipedia.org/wiki/Quantum_technology "Quantum technology"), and [quantum information science](https://en.wikipedia.org/wiki/Quantum_information_science "Quantum information science"). Quantum mechanics can describe many systems that [classical physics](https://en.wikipedia.org/wiki/Classical_physics "Classical physics") cannot. Classical physics can describe many aspects of nature at an ordinary ([macroscopic](https://en.wikipedia.org/wiki/Macroscopic "Macroscopic") and [(optical) microscopic](https://en.wikipedia.org/wiki/Microscopic_scale "Microscopic scale")) scale, however is insufficient for describing them at very small [submicroscopic](https://en.wikipedia.org/wiki/Submicroscopic "Submicroscopic") (atomic and [subatomic](https://en.wikipedia.org/wiki/Subatomic "Subatomic")) scales. Classical mechanics can be derived from quantum mechanics as an approximation that is valid at ordinary scales.[\[3\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-3) **Quantum systems** have [bound](https://en.wikipedia.org/wiki/Bound_state "Bound state") states that are [quantized](https://en.wikipedia.org/wiki/Quantization_\(physics\) "Quantization (physics)") to [discrete values](https://en.wikipedia.org/wiki/Discrete_mathematics "Discrete mathematics") of [energy](https://en.wikipedia.org/wiki/Energy "Energy"), [momentum](https://en.wikipedia.org/wiki/Momentum "Momentum"), [angular momentum](https://en.wikipedia.org/wiki/Angular_momentum "Angular momentum"), and other quantities, in contrast to classical systems where these quantities can be measured continuously. Measurements of quantum systems show characteristics of both [particles](https://en.wikipedia.org/wiki/Particle "Particle") and [waves](https://en.wikipedia.org/wiki/Wave "Wave") ([wave–particle duality](https://en.wikipedia.org/wiki/Wave%E2%80%93particle_duality "Wave–particle duality")), and there are limits to how accurately the value of a physical quantity can be predicted prior to its measurement, given a complete set of initial conditions (the [uncertainty principle](https://en.wikipedia.org/wiki/Uncertainty_principle "Uncertainty principle")). Quantum mechanics arose gradually from theories to explain observations that could not be reconciled with [classical physics](https://en.wikipedia.org/wiki/Classical_physics "Classical physics"), such as [Max Planck](https://en.wikipedia.org/wiki/Max_Planck "Max Planck")'s solution in 1900 to the [black-body radiation](https://en.wikipedia.org/wiki/Black-body_radiation "Black-body radiation") problem, and the correspondence between energy and frequency in [Albert Einstein](https://en.wikipedia.org/wiki/Albert_Einstein "Albert Einstein")'s [1905 paper](https://en.wikipedia.org/wiki/Annus_Mirabilis_papers#Photoelectric_effect "Annus Mirabilis papers"), which explained the [photoelectric effect](https://en.wikipedia.org/wiki/Photoelectric_effect "Photoelectric effect"). These early attempts to understand microscopic phenomena, now known as the "[old quantum theory](https://en.wikipedia.org/wiki/Old_quantum_theory "Old quantum theory")", led to the full development of quantum mechanics in the mid-1920s by [Niels Bohr](https://en.wikipedia.org/wiki/Niels_Bohr "Niels Bohr"), [Erwin Schrödinger](https://en.wikipedia.org/wiki/Erwin_Schr%C3%B6dinger "Erwin Schrödinger"), [Werner Heisenberg](https://en.wikipedia.org/wiki/Werner_Heisenberg "Werner Heisenberg"), [Max Born](https://en.wikipedia.org/wiki/Max_Born "Max Born"), [Paul Dirac](https://en.wikipedia.org/wiki/Paul_Dirac "Paul Dirac") and others. The modern theory is formulated in various [specially developed mathematical formalisms](https://en.wikipedia.org/wiki/Mathematical_formulations_of_quantum_mechanics "Mathematical formulations of quantum mechanics"). In one of them, a mathematical entity called the [wave function](https://en.wikipedia.org/wiki/Wave_function "Wave function") provides information, in the form of [probability amplitudes](https://en.wikipedia.org/wiki/Probability_amplitude "Probability amplitude"), about what measurements of a particle's energy, momentum, and other physical properties may yield. ## Overview and fundamental concepts Quantum mechanics allows the calculation of properties and behaviour of [physical systems](https://en.wikipedia.org/wiki/Physical_systems "Physical systems"). It is typically applied to microscopic systems: [molecules](https://en.wikipedia.org/wiki/Molecules "Molecules"), [atoms](https://en.wikipedia.org/wiki/Atoms "Atoms") and [subatomic particles](https://en.wikipedia.org/wiki/Subatomic_particle "Subatomic particle"). It has been demonstrated to hold for complex molecules with thousands of atoms,[\[4\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-4) but its application to human beings raises philosophical problems, such as [Wigner's friend](https://en.wikipedia.org/wiki/Wigner%27s_friend "Wigner's friend"), and its application to the universe as a whole remains speculative.[\[5\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-5) Predictions of quantum mechanics have been verified experimentally to an extremely high degree of [accuracy](https://en.wikipedia.org/wiki/Accuracy "Accuracy"). For example, the refinement of quantum mechanics for the interaction of light and matter, known as [quantum electrodynamics](https://en.wikipedia.org/wiki/Quantum_electrodynamics "Quantum electrodynamics") (QED), has been [shown to agree with experiment](https://en.wikipedia.org/wiki/Precision_tests_of_QED "Precision tests of QED") to within 1 part in 1012 when predicting the magnetic properties of an electron.[\[6\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-6) A fundamental feature of the theory is that it usually cannot predict with certainty what will happen, but only gives probabilities. Mathematically, a probability is found by taking the square of the absolute value of a [complex number](https://en.wikipedia.org/wiki/Complex_number "Complex number"), known as a probability amplitude. This is known as the [Born rule](https://en.wikipedia.org/wiki/Born_rule "Born rule"), named after physicist [Max Born](https://en.wikipedia.org/wiki/Max_Born "Max Born"). For example, a quantum particle like an [electron](https://en.wikipedia.org/wiki/Electron "Electron") can be described by a wave function, which associates to each point in space a probability amplitude. Applying the Born rule to these amplitudes gives a [probability density function](https://en.wikipedia.org/wiki/Probability_density_function "Probability density function") for the position that the electron will be found to have when an experiment is performed to measure it. This is the best the theory can do; it cannot say for certain where the electron will be found. The [Schrödinger equation](https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation "Schrödinger equation") relates the collection of probability amplitudes that pertain to one moment of time to the collection of probability amplitudes that pertain to another.[\[7\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Zwiebach2022-7): 67–87 One consequence of the mathematical rules of quantum mechanics is a tradeoff in predictability between measurable quantities. The most famous form of this [uncertainty principle](https://en.wikipedia.org/wiki/Uncertainty_principle "Uncertainty principle") says that no matter how a quantum particle is prepared or how carefully experiments upon it are arranged, it is impossible to have a precise prediction for a measurement of its position and also at the same time for a measurement of its [momentum](https://en.wikipedia.org/wiki/Momentum "Momentum").[\[7\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Zwiebach2022-7): 427–435 [![](https://upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Double-slit.svg/330px-Double-slit.svg.png)](https://en.wikipedia.org/wiki/File:Double-slit.svg) An illustration of the [double-slit experiment](https://en.wikipedia.org/wiki/Double-slit_experiment "Double-slit experiment") Another consequence of the mathematical rules of quantum mechanics is the phenomenon of [quantum interference](https://en.wikipedia.org/wiki/Quantum_interference "Quantum interference"), which is often illustrated with the [double-slit experiment](https://en.wikipedia.org/wiki/Double-slit_experiment "Double-slit experiment"). In the basic version of this experiment, a [coherent light source](https://en.wikipedia.org/wiki/Coherence_\(physics\) "Coherence (physics)"), such as a [laser](https://en.wikipedia.org/wiki/Laser "Laser") beam, illuminates a plate pierced by two parallel slits, and the light passing through the slits is observed on a screen behind the plate.[\[8\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Lederman-8): 102–111 [\[2\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Feynman-2): 1.1–1.8 The wave nature of light causes the light waves passing through the two slits to [interfere](https://en.wikipedia.org/wiki/Interference_\(wave_propagation\) "Interference (wave propagation)"), producing bright and dark bands on the screen – a result that would not be expected if light consisted of classical particles.[\[8\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Lederman-8) However, the light is always found to be absorbed at the screen at discrete points, as individual particles rather than waves; the interference pattern appears via the varying density of these particle hits on the screen. Furthermore, versions of the experiment that include detectors at the slits find that each detected [photon](https://en.wikipedia.org/wiki/Photon "Photon") passes through one slit (as would a classical particle), and not through both slits (as would a wave).[\[8\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Lederman-8): 109 [\[9\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-M%C3%BCller-Kirsten-9)[\[10\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Plotnitsky-10) However, [such experiments](https://en.wikipedia.org/wiki/Double-slit_experiment#Which_way "Double-slit experiment") demonstrate that particles do not form the interference pattern if one detects which slit they pass through. This behavior is known as [wave–particle duality](https://en.wikipedia.org/wiki/Wave%E2%80%93particle_duality "Wave–particle duality"). In addition to light, [electrons](https://en.wikipedia.org/wiki/Electrons "Electrons"), [atoms](https://en.wikipedia.org/wiki/Atoms "Atoms"), and [molecules](https://en.wikipedia.org/wiki/Molecules "Molecules") are all found to exhibit the same dual behavior when fired towards a double slit.[\[2\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Feynman-2) [![](https://upload.wikimedia.org/wikipedia/commons/thumb/5/5f/QuantumTunnel.jpg/330px-QuantumTunnel.jpg)](https://en.wikipedia.org/wiki/File:QuantumTunnel.jpg) A simplified diagram of [quantum tunneling](https://en.wikipedia.org/wiki/Quantum_tunneling "Quantum tunneling"), a phenomenon by which a particle may move through a barrier which would be impossible under classical mechanics Another non-classical phenomenon predicted by quantum mechanics is [quantum tunnelling](https://en.wikipedia.org/wiki/Quantum_tunnelling "Quantum tunnelling"): a particle that goes up against a [potential barrier](https://en.wikipedia.org/wiki/Potential_barrier "Potential barrier") can cross it, even if its kinetic energy is smaller than the maximum of the potential.[\[11\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-11) In classical mechanics this particle would be trapped. Quantum tunnelling has several important consequences, enabling [radioactive decay](https://en.wikipedia.org/wiki/Radioactive_decay "Radioactive decay"), [nuclear fusion](https://en.wikipedia.org/wiki/Nuclear_fusion "Nuclear fusion") in stars, and applications such as [scanning tunnelling microscopy](https://en.wikipedia.org/wiki/Scanning_tunnelling_microscopy "Scanning tunnelling microscopy"), [tunnel diode](https://en.wikipedia.org/wiki/Tunnel_diode "Tunnel diode") and [tunnel field-effect transistor](https://en.wikipedia.org/wiki/Tunnel_field-effect_transistor "Tunnel field-effect transistor").[\[12\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Trixler2013-12)[\[13\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-13) When quantum systems interact, the result can be the creation of [quantum entanglement](https://en.wikipedia.org/wiki/Quantum_entanglement "Quantum entanglement"): their properties become so intertwined that a description of the whole solely in terms of the individual parts is no longer possible. Erwin Schrödinger called entanglement "...*the* characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought".[\[14\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-14) Quantum entanglement enables [quantum computing](https://en.wikipedia.org/wiki/Quantum_computing "Quantum computing") and is part of quantum communication protocols, such as [quantum key distribution](https://en.wikipedia.org/wiki/Quantum_key_distribution "Quantum key distribution") and [superdense coding](https://en.wikipedia.org/wiki/Superdense_coding "Superdense coding").[\[15\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Caves-15) Contrary to popular misconception, entanglement does not allow sending signals [faster than light](https://en.wikipedia.org/wiki/Faster_than_light "Faster than light"), as demonstrated by the [no-communication theorem](https://en.wikipedia.org/wiki/No-communication_theorem "No-communication theorem").[\[15\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Caves-15) Another possibility opened by entanglement is testing for "[hidden variables](https://en.wikipedia.org/wiki/Hidden_variable_theory "Hidden variable theory")", hypothetical properties more fundamental than the quantities addressed in quantum theory itself, knowledge of which would allow more exact predictions than quantum theory provides. A collection of results, most significantly [Bell's theorem](https://en.wikipedia.org/wiki/Bell%27s_theorem "Bell's theorem"), have demonstrated that broad classes of such hidden-variable theories are in fact incompatible with quantum physics. According to Bell's theorem, if nature actually operates in accord with any theory of *local* hidden variables, then the results of a [Bell test](https://en.wikipedia.org/wiki/Bell_test "Bell test") will be constrained in a particular, quantifiable way. Many Bell tests have been performed and they have shown results incompatible with the constraints imposed by local hidden variables.[\[16\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-wiseman15-16)[\[17\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-wolchover17-17) It is not possible to present these concepts in more than a superficial way without introducing the mathematics involved; understanding quantum mechanics requires not only manipulating complex numbers, but also [linear algebra](https://en.wikipedia.org/wiki/Linear_algebra "Linear algebra"), [differential equations](https://en.wikipedia.org/wiki/Differential_equation "Differential equation"), [group theory](https://en.wikipedia.org/wiki/Group_theory "Group theory"), and other more advanced subjects.[\[18\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-18)[\[19\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-19) Accordingly, this article will present a mathematical formulation of quantum mechanics and survey its application to some useful and oft-studied examples. ## Mathematical formulation Main article: [Mathematical formulation of quantum mechanics](https://en.wikipedia.org/wiki/Mathematical_formulation_of_quantum_mechanics "Mathematical formulation of quantum mechanics") In the mathematically rigorous formulation of quantum mechanics, the state of a quantum mechanical system is a vector ψ {\\displaystyle \\psi } ![{\\displaystyle \\psi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a) belonging to a ([separable](https://en.wikipedia.org/wiki/Separable_space "Separable space")) complex [Hilbert space](https://en.wikipedia.org/wiki/Hilbert_space "Hilbert space") H {\\displaystyle {\\mathcal {H}}} ![{\\displaystyle {\\mathcal {H}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/19ef4c7b923a5125ac91aa491838a95ee15b804f). This vector is postulated to be normalized under the Hilbert space inner product, that is, it obeys ⟨ ψ , ψ ⟩ \= 1 {\\displaystyle \\langle \\psi ,\\psi \\rangle =1} ![{\\displaystyle \\langle \\psi ,\\psi \\rangle =1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d5758e7a60b4e54bc46e01b0618919c65b787a0), and it is well-defined up to a complex number of modulus 1 (the global phase), that is, ψ {\\displaystyle \\psi } ![{\\displaystyle \\psi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a) and e i α ψ {\\displaystyle e^{i\\alpha }\\psi } ![{\\displaystyle e^{i\\alpha }\\psi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/7193e9b0479da798b9cf2dcc492f49166d0d7103) represent the same physical system. In other words, the possible states are points in the [projective space](https://en.wikipedia.org/wiki/Projective_space "Projective space") of a Hilbert space, usually called the [complex projective space](https://en.wikipedia.org/wiki/Complex_projective_space "Complex projective space"). The exact nature of this Hilbert space is dependent on the system – for example, for describing position and momentum the Hilbert space is the space of complex [square-integrable](https://en.wikipedia.org/wiki/Square-integrable "Square-integrable") functions L 2 ( C ) {\\displaystyle L^{2}(\\mathbb {C} )} ![{\\displaystyle L^{2}(\\mathbb {C} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a3649087af50f42a95da2163656076529ca3139),[\[20\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Holevo2001-20): 13 while the Hilbert space for the [spin](https://en.wikipedia.org/wiki/Spin_\(physics\) "Spin (physics)") of a single proton is simply the space of two-dimensional complex vectors C 2 {\\displaystyle \\mathbb {C} ^{2}} ![{\\displaystyle \\mathbb {C} ^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f43d6ec8a1e1fe5a85aec0dd9bdcd45ae09b06b) with the usual inner product.[\[20\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Holevo2001-20): 20 Physical quantities of interest – position, momentum, energy, spin – are represented by observables, which are [Hermitian](https://en.wikipedia.org/wiki/Hermitian_adjoint#Hermitian_operators "Hermitian adjoint") (more precisely, [self-adjoint](https://en.wikipedia.org/wiki/Self-adjoint_operator "Self-adjoint operator")) linear [operators](https://en.wikipedia.org/wiki/Operator_\(physics\) "Operator (physics)") acting on the Hilbert space.[\[20\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Holevo2001-20): 17 A quantum state can be an [eigenvector](https://en.wikipedia.org/wiki/Eigenvector "Eigenvector") of an observable, in which case it is called an [eigenstate](https://en.wikipedia.org/wiki/Eigenstate "Eigenstate"), and the associated [eigenvalue](https://en.wikipedia.org/wiki/Eigenvalue "Eigenvalue") corresponds to the value of the observable in that eigenstate. More generally, a quantum state will be a linear combination of the eigenstates, known as a [quantum superposition](https://en.wikipedia.org/wiki/Quantum_superposition "Quantum superposition"). When an observable is measured, the result will be one of its eigenvalues with probability given by the [Born rule](https://en.wikipedia.org/wiki/Born_rule "Born rule"): in the simplest case the eigenvalue λ {\\displaystyle \\lambda } ![{\\displaystyle \\lambda }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a) is non-degenerate and the probability is given by \| ⟨ λ → , ψ ⟩ \| 2 {\\displaystyle \|\\langle {\\vec {\\lambda }},\\psi \\rangle \|^{2}} ![{\\displaystyle \|\\langle {\\vec {\\lambda }},\\psi \\rangle \|^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de4b465224a805b558697f44dcf3824fb2fbfe20), where λ → {\\displaystyle {\\vec {\\lambda }}} ![{\\displaystyle {\\vec {\\lambda }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d8c69573b9c55ec8ea726662102b8e693bf32f01) is its associated unit-length eigenvector. More generally, the eigenvalue is degenerate and the probability is given by ⟨ ψ , P λ ψ ⟩ {\\displaystyle \\langle \\psi ,P\_{\\lambda }\\psi \\rangle } ![{\\displaystyle \\langle \\psi ,P\_{\\lambda }\\psi \\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/6ba3648378b52e2e710937b36948cd908b2d6531), where P λ {\\displaystyle P\_{\\lambda }} ![{\\displaystyle P\_{\\lambda }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/330591f9b6fffc93ca78514576fd0d8cfac6f0c7) is the projector onto its associated eigenspace.[\[21\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-21) In the continuous case, these formulas give instead the [probability density](https://en.wikipedia.org/wiki/Probability_density "Probability density"). After the [measurement](https://en.wikipedia.org/wiki/Measurement_in_quantum_mechanics "Measurement in quantum mechanics"), if result λ {\\displaystyle \\lambda } ![{\\displaystyle \\lambda }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a) was obtained, the quantum state is postulated to [collapse](https://en.wikipedia.org/wiki/Collapse_of_the_wavefunction "Collapse of the wavefunction") to λ → {\\displaystyle {\\vec {\\lambda }}} ![{\\displaystyle {\\vec {\\lambda }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d8c69573b9c55ec8ea726662102b8e693bf32f01), in the non-degenerate case, or to P λ ψ / ⟨ ψ , P λ ψ ⟩ {\\textstyle P\_{\\lambda }\\psi {\\big /}\\!{\\sqrt {\\langle \\psi ,P\_{\\lambda }\\psi \\rangle }}} ![{\\textstyle P\_{\\lambda }\\psi {\\big /}\\!{\\sqrt {\\langle \\psi ,P\_{\\lambda }\\psi \\rangle }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e38d8eb9122b686c4d8f937fe59e2ec511be43ed), in the general case. The [probabilistic](https://en.wikipedia.org/wiki/Probabilistic "Probabilistic") nature of quantum mechanics thus stems from the act of measurement. This is one of the most debated aspects of quantum theory, with different [interpretations of quantum mechanics](https://en.wikipedia.org/wiki/Interpretations_of_quantum_mechanics "Interpretations of quantum mechanics") giving radically different answers to questions regarding quantum-state collapse, as discussed [below](https://en.wikipedia.org/wiki/Quantum_mechanics#Philosophical_implications). ### Time evolution of a quantum state The time evolution of a quantum state is described by the Schrödinger equation: i ℏ ∂ ∂ t ψ ( t ) \= H ψ ( t ) . {\\displaystyle i\\hbar {\\frac {\\partial }{\\partial t}}\\psi (t)=H\\psi (t).} ![{\\displaystyle i\\hbar {\\frac {\\partial }{\\partial t}}\\psi (t)=H\\psi (t).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5c41b5a62afa3aa83e6ea98e69d692ac0f77c2c6) Here H {\\displaystyle H} ![{\\displaystyle H}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b) denotes the [Hamiltonian](https://en.wikipedia.org/wiki/Hamiltonian_\(quantum_mechanics\) "Hamiltonian (quantum mechanics)"), the observable corresponding to the [total energy](https://en.wikipedia.org/wiki/Total_energy "Total energy") of the system, and ℏ {\\displaystyle \\hbar } ![{\\displaystyle \\hbar }](https://wikimedia.org/api/rest_v1/media/math/render/svg/de68de3a92517953436c93b5a76461d49160cc41) is the reduced [Planck constant](https://en.wikipedia.org/wiki/Planck_constant "Planck constant"). The constant i ℏ {\\displaystyle i\\hbar } ![{\\displaystyle i\\hbar }](https://wikimedia.org/api/rest_v1/media/math/render/svg/3d724d600903f701570a9f97c2a8378a6f3a80e5) is introduced so that the Hamiltonian is reduced to the [classical Hamiltonian](https://en.wikipedia.org/wiki/Hamiltonian_mechanics "Hamiltonian mechanics") in cases where the quantum system can be approximated by a classical system; the ability to make such an approximation in certain limits is called the [correspondence principle](https://en.wikipedia.org/wiki/Correspondence_principle "Correspondence principle"). The solution of this differential equation is given by ψ ( t ) \= e − i H t / ℏ ψ ( 0 ) . {\\displaystyle \\psi (t)=e^{-iHt/\\hbar }\\psi (0).} ![{\\displaystyle \\psi (t)=e^{-iHt/\\hbar }\\psi (0).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b0a822b842f8d952c4e5827f312597e72a45f4d) The operator U ( t ) \= e − i H t / ℏ {\\displaystyle U(t)=e^{-iHt/\\hbar }} ![{\\displaystyle U(t)=e^{-iHt/\\hbar }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/17a17e78953fa7d61788cc8bc41a25ef254fc8e0) is known as the time-evolution operator, and has the crucial property that it is [unitary](https://en.wikipedia.org/wiki/Unitarity_\(physics\) "Unitarity (physics)"). This time evolution is [deterministic](https://en.wikipedia.org/wiki/Deterministic "Deterministic") in the sense that – given an initial quantum state ψ ( 0 ) {\\displaystyle \\psi (0)} ![{\\displaystyle \\psi (0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fcfdf3fefa6feee8b2192815ac15a2650d8a8db2) – it makes a definite prediction of what the quantum state ψ ( t ) {\\displaystyle \\psi (t)} ![{\\displaystyle \\psi (t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6cf4a36b5f945be90a527b3dbe3d55d3f0439cdb) will be at any later time.[\[22\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-22) [![](https://upload.wikimedia.org/wikipedia/commons/thumb/3/35/Atomic-orbital-clouds_spd_m0.png/330px-Atomic-orbital-clouds_spd_m0.png)](https://en.wikipedia.org/wiki/File:Atomic-orbital-clouds_spd_m0.png) Fig. 1: [Probability densities](https://en.wikipedia.org/wiki/Probability_density_function "Probability density function") corresponding to the wave functions of an electron in a hydrogen atom possessing definite energy levels (increasing from the top of the image to the bottom: *n* = 1, 2, 3, ...) and angular momenta (increasing across from left to right: *s*, *p*, *d*, ...). Denser areas correspond to higher probability density in a position measurement. Such wave functions are directly comparable to [Chladni's figures](https://en.wikipedia.org/wiki/Chladni%27s_figures "Chladni's figures") of [acoustic](https://en.wikipedia.org/wiki/Acoustics "Acoustics") modes of vibration in classical physics and are modes of oscillation as well, possessing a sharp energy and thus, a definite frequency. The [angular momentum](https://en.wikipedia.org/wiki/Angular_momentum "Angular momentum") and energy are [quantized](https://en.wikipedia.org/wiki/Quantization_\(physics\) "Quantization (physics)") and take *only* discrete values like those shown – as is the case for [resonant frequencies](https://en.wikipedia.org/wiki/Resonant_frequencies "Resonant frequencies") in acoustics. Some wave functions produce probability distributions that are independent of time, such as [eigenstates](https://en.wikipedia.org/wiki/Eigenstate "Eigenstate") of the Hamiltonian.[\[7\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Zwiebach2022-7): 133–137 Many systems that are treated dynamically in classical mechanics are described by such "static" wave functions. For example, a single electron in an unexcited atom is pictured classically as a particle moving in a circular trajectory around the [atomic nucleus](https://en.wikipedia.org/wiki/Atomic_nucleus "Atomic nucleus"), whereas in quantum mechanics, it is described by a static wave function surrounding the nucleus. For example, the electron wave function for an unexcited hydrogen atom is a spherically symmetric function known as an [*s* orbital](https://en.wikipedia.org/wiki/Atomic_orbital "Atomic orbital") ([Fig. 1](https://en.wikipedia.org/wiki/Quantum_mechanics#fig1)). Analytic solutions of the Schrödinger equation are known for [very few relatively simple model Hamiltonians](https://en.wikipedia.org/wiki/List_of_quantum-mechanical_systems_with_analytical_solutions "List of quantum-mechanical systems with analytical solutions") including the [quantum harmonic oscillator](https://en.wikipedia.org/wiki/Quantum_harmonic_oscillator "Quantum harmonic oscillator"), the [particle in a box](https://en.wikipedia.org/wiki/Particle_in_a_box "Particle in a box"), the [dihydrogen cation](https://en.wikipedia.org/wiki/Dihydrogen_cation "Dihydrogen cation"), and the [hydrogen atom](https://en.wikipedia.org/wiki/Hydrogen_atom "Hydrogen atom"). Even the [helium](https://en.wikipedia.org/wiki/Helium "Helium") atom – which contains just two electrons – has defied all attempts at a fully analytic treatment, admitting no solution in [closed form](https://en.wikipedia.org/wiki/Closed-form_expression "Closed-form expression").[\[23\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-23)[\[24\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-24)[\[25\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-25) However, there are techniques for finding approximate solutions. One method, called [perturbation theory](https://en.wikipedia.org/wiki/Perturbation_theory_\(quantum_mechanics\) "Perturbation theory (quantum mechanics)"), uses the analytic result for a simple quantum mechanical model to create a result for a related but more complicated model by (for example) the addition of a weak [potential energy](https://en.wikipedia.org/wiki/Potential_energy "Potential energy").[\[7\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Zwiebach2022-7): 793 Another approximation method applies to systems for which quantum mechanics produces only small deviations from classical behavior. These deviations can then be computed based on the classical motion.[\[7\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Zwiebach2022-7): 849 ### Uncertainty principle One consequence of the basic quantum formalism is the uncertainty principle. In its most familiar form, this states that no preparation of a quantum particle can imply simultaneously precise predictions both for a measurement of its position and for a measurement of its momentum.[\[26\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Cohen-Tannoudji-26)[\[27\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-L&L-27) Both position and momentum are observables, meaning that they are represented by [Hermitian operators](https://en.wikipedia.org/wiki/Hermitian_operators "Hermitian operators"). The position operator X ^ {\\displaystyle {\\hat {X}}} ![{\\displaystyle {\\hat {X}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/acc59ad6d9a06d55b96b65beb0fdfc89acc1e40e) and momentum operator P ^ {\\displaystyle {\\hat {P}}} ![{\\displaystyle {\\hat {P}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a46a8cf7bc789e8c4f8de7ca0d9946a46fb8842) do not commute, but rather satisfy the [canonical commutation relation](https://en.wikipedia.org/wiki/Canonical_commutation_relation "Canonical commutation relation"): \[ X ^ , P ^ \] \= i ℏ . {\\displaystyle \[{\\hat {X}},{\\hat {P}}\]=i\\hbar .} ![{\\displaystyle \[{\\hat {X}},{\\hat {P}}\]=i\\hbar .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/803fe39b0eeaff8d1570df480e738cf5a968cc71) Given a quantum state, the Born rule lets us compute expectation values for both X {\\displaystyle X} ![{\\displaystyle X}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab) and P {\\displaystyle P} ![{\\displaystyle P}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a), and moreover for powers of them. Defining the uncertainty for an observable by a [standard deviation](https://en.wikipedia.org/wiki/Standard_deviation "Standard deviation"), we have σ X \= ⟨ X 2 ⟩ − ⟨ X ⟩ 2 , {\\displaystyle \\sigma \_{X}={\\textstyle {\\sqrt {\\left\\langle X^{2}\\right\\rangle -\\left\\langle X\\right\\rangle ^{2}}}},} ![{\\displaystyle \\sigma \_{X}={\\textstyle {\\sqrt {\\left\\langle X^{2}\\right\\rangle -\\left\\langle X\\right\\rangle ^{2}}}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/457ec20972d63dfb1ecc9087e18d1f949f908c8f) and likewise for the momentum: σ P \= ⟨ P 2 ⟩ − ⟨ P ⟩ 2 . {\\displaystyle \\sigma \_{P}={\\sqrt {\\left\\langle P^{2}\\right\\rangle -\\left\\langle P\\right\\rangle ^{2}}}.} ![{\\displaystyle \\sigma \_{P}={\\sqrt {\\left\\langle P^{2}\\right\\rangle -\\left\\langle P\\right\\rangle ^{2}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/63ec8f5f7b9e5957ea6d06c56068b06244acc184) The uncertainty principle states that σ X σ P ≥ ℏ 2 . {\\displaystyle \\sigma \_{X}\\sigma \_{P}\\geq {\\frac {\\hbar }{2}}.} ![{\\displaystyle \\sigma \_{X}\\sigma \_{P}\\geq {\\frac {\\hbar }{2}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/538cada7fa57155ece387e5d53d90ca366e323fe) Either standard deviation can in principle be made arbitrarily small, but not both simultaneously.[\[28\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-ballentine1970-28) This inequality generalizes to arbitrary pairs of self-adjoint operators A {\\displaystyle A} ![{\\displaystyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3) and B {\\displaystyle B} ![{\\displaystyle B}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a). The [commutator](https://en.wikipedia.org/wiki/Commutator "Commutator") of these two operators is \[ A , B \] \= A B − B A , {\\displaystyle \[A,B\]=AB-BA,} ![{\\displaystyle \[A,B\]=AB-BA,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2a47259b42e63c048c65f67d304404867841951) and this provides the lower bound on the product of standard deviations: σ A σ B ≥ 1 2 \| ⟨ \[ A , B \] ⟩ \| . {\\displaystyle \\sigma \_{A}\\sigma \_{B}\\geq {\\tfrac {1}{2}}\\left\|{\\bigl \\langle }\[A,B\]{\\bigr \\rangle }\\right\|.} ![{\\displaystyle \\sigma \_{A}\\sigma \_{B}\\geq {\\tfrac {1}{2}}\\left\|{\\bigl \\langle }\[A,B\]{\\bigr \\rangle }\\right\|.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d0fd768b447334e150b8b98181f74b475e41ee52) Another consequence of the canonical commutation relation is that the position and momentum operators are [Fourier transforms](https://en.wikipedia.org/wiki/Fourier_transform#Uncertainty_principle "Fourier transform") of each other, so that a description of an object according to its momentum is the Fourier transform of its description according to its position. The fact that dependence in momentum is the Fourier transform of the dependence in position means that the momentum operator is equivalent (up to an i / ℏ {\\displaystyle i/\\hbar } ![{\\displaystyle i/\\hbar }](https://wikimedia.org/api/rest_v1/media/math/render/svg/44a04e1f19b5e7bea2bfa8002a841bf8d1b4e66a) factor) to taking the derivative according to the position, since in Fourier analysis [differentiation corresponds to multiplication in the dual space](https://en.wikipedia.org/wiki/Fourier_transform#Differentiation "Fourier transform"). This is why in quantum equations in position space, the momentum p i {\\displaystyle p\_{i}} ![{\\displaystyle p\_{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5bab39399bf5424f25d957cdc57c84a0622626d2) is replaced by − i ℏ ∂ ∂ x {\\displaystyle -i\\hbar {\\frac {\\partial }{\\partial x}}} ![{\\displaystyle -i\\hbar {\\frac {\\partial }{\\partial x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab7fffcee704fc55eb36b137e0cc25132b5dc1bf), and in particular in the [non-relativistic Schrödinger equation in position space](https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation#Equation "Schrödinger equation") the momentum-squared term is replaced with a Laplacian times − ℏ 2 {\\displaystyle -\\hbar ^{2}} ![{\\displaystyle -\\hbar ^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/185934c9e69984a0905432abe63d3b41741342b4).[\[26\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Cohen-Tannoudji-26) ### Composite systems and entanglement When two different quantum systems are considered together, the Hilbert space of the combined system is the [tensor product](https://en.wikipedia.org/wiki/Tensor_product "Tensor product") of the Hilbert spaces of the two components. For example, let A and B be two quantum systems, with Hilbert spaces H A {\\displaystyle {\\mathcal {H}}\_{A}} ![{\\displaystyle {\\mathcal {H}}\_{A}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4611f8542d6c72789b7e82bfcf29014d44c13aa3) and H B {\\displaystyle {\\mathcal {H}}\_{B}} ![{\\displaystyle {\\mathcal {H}}\_{B}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bfea1102a35aff8c6435147dc9dffc21b4cfe4ef), respectively. The Hilbert space of the composite system is then H A B \= H A ⊗ H B . {\\displaystyle {\\mathcal {H}}\_{AB}={\\mathcal {H}}\_{A}\\otimes {\\mathcal {H}}\_{B}.} ![{\\displaystyle {\\mathcal {H}}\_{AB}={\\mathcal {H}}\_{A}\\otimes {\\mathcal {H}}\_{B}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/54e4f6aff2be5ec9f3eb4657a133d1be17f6e2b3) If the state for the first system is the vector ψ A {\\displaystyle \\psi \_{A}} ![{\\displaystyle \\psi \_{A}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a04e49109107e7765357dd92e975895840fa7700) and the state for the second system is ψ B {\\displaystyle \\psi \_{B}} ![{\\displaystyle \\psi \_{B}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ca3b90bba42c451129c4969322e175f5a1d3368), then the state of the composite system is ψ A ⊗ ψ B . {\\displaystyle \\psi \_{A}\\otimes \\psi \_{B}.} ![{\\displaystyle \\psi \_{A}\\otimes \\psi \_{B}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/852c1e09787e91af3af1a10d1eee4eadc232d48c) Not all states in the joint Hilbert space H A B {\\displaystyle {\\mathcal {H}}\_{AB}} ![{\\displaystyle {\\mathcal {H}}\_{AB}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/951b54ea87d5bd62464abaf1be530e6a37a86f1b) can be written in this form, however, because the superposition principle implies that linear combinations of these "separable" or "product states" are also valid. For example, if ψ A {\\displaystyle \\psi \_{A}} ![{\\displaystyle \\psi \_{A}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a04e49109107e7765357dd92e975895840fa7700) and ϕ A {\\displaystyle \\phi \_{A}} ![{\\displaystyle \\phi \_{A}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e38d84dcc21ea769a51ab502691f566c830b430) are both possible states for system A {\\displaystyle A} ![{\\displaystyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3), and likewise ψ B {\\displaystyle \\psi \_{B}} ![{\\displaystyle \\psi \_{B}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ca3b90bba42c451129c4969322e175f5a1d3368) and ϕ B {\\displaystyle \\phi \_{B}} ![{\\displaystyle \\phi \_{B}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a829a7a85bb29904dec5e93605d0b976baf45755) are both possible states for system B {\\displaystyle B} ![{\\displaystyle B}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a), then 1 2 ( ψ A ⊗ ψ B \+ ϕ A ⊗ ϕ B ) {\\displaystyle {\\tfrac {1}{\\sqrt {2}}}\\left(\\psi \_{A}\\otimes \\psi \_{B}+\\phi \_{A}\\otimes \\phi \_{B}\\right)} ![{\\displaystyle {\\tfrac {1}{\\sqrt {2}}}\\left(\\psi \_{A}\\otimes \\psi \_{B}+\\phi \_{A}\\otimes \\phi \_{B}\\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/53d226ba585a99942ec855197c23ed885b635258) is a valid joint state that is not separable. States that are not separable are called [entangled](https://en.wikipedia.org/wiki/Quantum_entanglement "Quantum entanglement").[\[29\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-:0-29)[\[30\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-:1-30) If the state for a composite system is entangled, it is impossible to describe either component system A or system B by a state vector. One can instead define [reduced density matrices](https://en.wikipedia.org/wiki/Reduced_density_matrix "Reduced density matrix") that describe the statistics that can be obtained by making measurements on either component system alone. This necessarily causes a loss of information, though: knowing the reduced density matrices of the individual systems is not enough to reconstruct the state of the composite system.[\[29\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-:0-29)[\[30\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-:1-30) Just as density matrices specify the state of a subsystem of a larger system, analogously, [positive operator-valued measures](https://en.wikipedia.org/wiki/POVM "POVM") (POVMs) describe the effect on a subsystem of a measurement performed on a larger system. POVMs are extensively used in quantum information theory.[\[29\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-:0-29)[\[31\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-wilde-31) As described above, entanglement is a key feature of models of measurement processes in which an apparatus becomes entangled with the system being measured. Systems interacting with the environment in which they reside generally become entangled with that environment, a phenomenon known as [quantum decoherence](https://en.wikipedia.org/wiki/Quantum_decoherence "Quantum decoherence"). This can explain why, in practice, quantum effects are difficult to observe in systems larger than microscopic.[\[32\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-32) ### Equivalence between formulations There are many mathematically equivalent formulations of quantum mechanics. One of the oldest and most common is the "[transformation theory](https://en.wikipedia.org/wiki/Transformation_theory_\(quantum_mechanics\) "Transformation theory (quantum mechanics)")" proposed by [Paul Dirac](https://en.wikipedia.org/wiki/Paul_Dirac "Paul Dirac"), which unifies and generalizes the two earliest formulations of quantum mechanics – [matrix mechanics](https://en.wikipedia.org/wiki/Matrix_mechanics "Matrix mechanics") (invented by [Werner Heisenberg](https://en.wikipedia.org/wiki/Werner_Heisenberg "Werner Heisenberg")) and wave mechanics (invented by [Erwin Schrödinger](https://en.wikipedia.org/wiki/Erwin_Schr%C3%B6dinger "Erwin Schrödinger")).[\[33\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-33) An alternative formulation of quantum mechanics is [Feynman](https://en.wikipedia.org/wiki/Feynman "Feynman")'s [path integral formulation](https://en.wikipedia.org/wiki/Path_integral_formulation "Path integral formulation"), in which a quantum-mechanical amplitude is considered as a sum over all possible classical and non-classical paths between the initial and final states. This is the quantum-mechanical counterpart of the [action principle](https://en.wikipedia.org/wiki/Action_principle "Action principle") in classical mechanics.[\[34\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-34) ### Symmetries and conservation laws Main article: [Noether's theorem](https://en.wikipedia.org/wiki/Noether%27s_theorem "Noether's theorem") The Hamiltonian H {\\displaystyle H} ![{\\displaystyle H}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b) is known as the *generator* of time evolution, since it defines a unitary time-evolution operator U ( t ) \= e − i H t / ℏ {\\displaystyle U(t)=e^{-iHt/\\hbar }} ![{\\displaystyle U(t)=e^{-iHt/\\hbar }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/17a17e78953fa7d61788cc8bc41a25ef254fc8e0) for each value of t {\\displaystyle t} ![{\\displaystyle t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560). From this relation between U ( t ) {\\displaystyle U(t)} ![{\\displaystyle U(t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/666c639df532e88616357c4991cabce9a57b5611) and H {\\displaystyle H} ![{\\displaystyle H}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b), it follows that any observable A {\\displaystyle A} ![{\\displaystyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3) that commutes with H {\\displaystyle H} ![{\\displaystyle H}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b) will be *conserved*: its expectation value will not change over time.[\[7\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Zwiebach2022-7): 471 This statement generalizes, as mathematically, any Hermitian operator A {\\displaystyle A} ![{\\displaystyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3) can generate a family of unitary operators parameterized by a variable t {\\displaystyle t} ![{\\displaystyle t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560). Under the evolution generated by A {\\displaystyle A} ![{\\displaystyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3), any observable B {\\displaystyle B} ![{\\displaystyle B}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a) that commutes with A {\\displaystyle A} ![{\\displaystyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3) will be conserved. Moreover, if B {\\displaystyle B} ![{\\displaystyle B}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a) is conserved by evolution under A {\\displaystyle A} ![{\\displaystyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3), then A {\\displaystyle A} ![{\\displaystyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3) is conserved under the evolution generated by B {\\displaystyle B} ![{\\displaystyle B}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a). This implies a quantum version of the result proven by [Emmy Noether](https://en.wikipedia.org/wiki/Emmy_Noether "Emmy Noether") in classical ([Lagrangian](https://en.wikipedia.org/wiki/Lagrangian_mechanics "Lagrangian mechanics")) mechanics: for every [differentiable](https://en.wikipedia.org/wiki/Differentiable "Differentiable") [symmetry](https://en.wikipedia.org/wiki/Symmetry_\(physics\) "Symmetry (physics)") of a Hamiltonian, there exists a corresponding [conservation law](https://en.wikipedia.org/wiki/Conservation_law "Conservation law"). ## Examples ### Free particle Main article: [Free particle](https://en.wikipedia.org/wiki/Free_particle "Free particle") [![](https://upload.wikimedia.org/wikipedia/commons/5/56/Guassian_Dispersion.gif)](https://en.wikipedia.org/wiki/File:Guassian_Dispersion.gif) Position space probability density of a Gaussian [wave packet](https://en.wikipedia.org/wiki/Wave_packet "Wave packet") moving in one dimension in free space The simplest example of a quantum system with a position degree of freedom is a free particle in a single spatial dimension. A free particle is one which is not subject to external influences, so that its Hamiltonian consists only of its kinetic energy: H \= 1 2 m P 2 \= − ℏ 2 2 m d 2 d x 2 . {\\displaystyle H={\\frac {1}{2m}}P^{2}=-{\\frac {\\hbar ^{2}}{2m}}{\\frac {d^{2}}{dx^{2}}}.} ![{\\displaystyle H={\\frac {1}{2m}}P^{2}=-{\\frac {\\hbar ^{2}}{2m}}{\\frac {d^{2}}{dx^{2}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/084f585ee5c6cd8a34e323a6de7943227128afc3) The general solution of the Schrödinger equation is given by ψ ( x , t ) \= 1 2 π ∫ − ∞ ∞ ψ ^ ( k , 0 ) e i ( k x − ℏ k 2 2 m t ) d k , {\\displaystyle \\psi (x,t)={\\frac {1}{\\sqrt {2\\pi }}}\\int \_{-\\infty }^{\\infty }{\\hat {\\psi }}(k,0)e^{i(kx-{\\frac {\\hbar k^{2}}{2m}}t)}\\mathrm {d} k,} ![{\\displaystyle \\psi (x,t)={\\frac {1}{\\sqrt {2\\pi }}}\\int \_{-\\infty }^{\\infty }{\\hat {\\psi }}(k,0)e^{i(kx-{\\frac {\\hbar k^{2}}{2m}}t)}\\mathrm {d} k,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef4f021ba945856e3676808b11724109a8a74dad) which is a superposition of all possible [plane waves](https://en.wikipedia.org/wiki/Plane_wave "Plane wave") e i ( k x − ℏ k 2 2 m t ) {\\displaystyle e^{i(kx-{\\frac {\\hbar k^{2}}{2m}}t)}} ![{\\displaystyle e^{i(kx-{\\frac {\\hbar k^{2}}{2m}}t)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fb4cd9a9984c84a493ce547babcf58e31b04f7e7), which are eigenstates of the momentum operator with momentum p \= ℏ k {\\displaystyle p=\\hbar k} ![{\\displaystyle p=\\hbar k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24fee69175538303b28ac54e907baf53d0a58dbf). The coefficients of the superposition are ψ ^ ( k , 0 ) {\\displaystyle {\\hat {\\psi }}(k,0)} ![{\\displaystyle {\\hat {\\psi }}(k,0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b8323c08418da8bc376c6d78b578d4729b927ea), which is the Fourier transform of the initial quantum state ψ ( x , 0 ) {\\displaystyle \\psi (x,0)} ![{\\displaystyle \\psi (x,0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/55ad442e07ca2d7986ef0787f9129fc325cde137). It is not possible for the solution to be a single momentum eigenstate, or a single position eigenstate, as these are not normalizable quantum states.[\[note 1\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-35) Instead, we can consider a Gaussian [wave packet](https://en.wikipedia.org/wiki/Wave_packet "Wave packet"): ψ ( x , 0 ) \= 1 π a 4 e − x 2 2 a {\\displaystyle \\psi (x,0)={\\frac {1}{\\sqrt\[{4}\]{\\pi a}}}e^{-{\\frac {x^{2}}{2a}}}} ![{\\displaystyle \\psi (x,0)={\\frac {1}{\\sqrt\[{4}\]{\\pi a}}}e^{-{\\frac {x^{2}}{2a}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a4c2dae82312897d5fd4c58986c426a6009e6840) which has Fourier transform, and therefore momentum distribution ψ ^ ( k , 0 ) \= a π 4 e − a k 2 2 . {\\displaystyle {\\hat {\\psi }}(k,0)={\\sqrt\[{4}\]{\\frac {a}{\\pi }}}e^{-{\\frac {ak^{2}}{2}}}.} ![{\\displaystyle {\\hat {\\psi }}(k,0)={\\sqrt\[{4}\]{\\frac {a}{\\pi }}}e^{-{\\frac {ak^{2}}{2}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c4991535bba434314af8c27c16fff74f49ce367e) We see that as we make a {\\displaystyle a} ![{\\displaystyle a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc) smaller the spread in position gets smaller, but the spread in momentum gets larger. Conversely, by making a {\\displaystyle a} ![{\\displaystyle a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc) larger we make the spread in momentum smaller, but the spread in position gets larger. This illustrates the uncertainty principle. As we let the Gaussian wave packet evolve in time, we see that its center moves through space at a constant velocity (like a classical particle with no forces acting on it). However, the wave packet will also spread out as time progresses, which means that the position becomes more and more uncertain. The uncertainty in momentum, however, stays constant.[\[35\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-36) ### Particle in a box [![](https://upload.wikimedia.org/wikipedia/commons/thumb/2/27/Infinite_potential_well.svg/250px-Infinite_potential_well.svg.png)](https://en.wikipedia.org/wiki/File:Infinite_potential_well.svg) 1-dimensional potential energy box (or infinite potential well) Main article: [Particle in a box](https://en.wikipedia.org/wiki/Particle_in_a_box "Particle in a box") The particle in a one-dimensional potential energy box is the most mathematically simple example where restraints lead to the quantization of energy levels. The box is defined as having zero potential energy everywhere *inside* a certain region, and therefore infinite potential energy everywhere *outside* that region.[\[26\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Cohen-Tannoudji-26): 77–78 For the one-dimensional case in the x {\\displaystyle x} ![{\\displaystyle x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4) direction, the time-independent Schrödinger equation may be written − ℏ 2 2 m d 2 ψ d x 2 \= E ψ . {\\displaystyle -{\\frac {\\hbar ^{2}}{2m}}{\\frac {d^{2}\\psi }{dx^{2}}}=E\\psi .} ![{\\displaystyle -{\\frac {\\hbar ^{2}}{2m}}{\\frac {d^{2}\\psi }{dx^{2}}}=E\\psi .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc73e9216faf1390c3ed550b72be21fc068ec747) With the differential operator defined by p ^ x \= − i ℏ d d x {\\displaystyle {\\hat {p}}\_{x}=-i\\hbar {\\frac {d}{dx}}} ![{\\displaystyle {\\hat {p}}\_{x}=-i\\hbar {\\frac {d}{dx}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2511885975007002c7582a9c8175689076df210a)the previous equation is evocative of the [classic kinetic energy analogue](https://en.wikipedia.org/wiki/Kinetic_energy#Kinetic_energy_of_rigid_bodies "Kinetic energy"), 1 2 m p ^ x 2 \= E , {\\displaystyle {\\frac {1}{2m}}{\\hat {p}}\_{x}^{2}=E,} ![{\\displaystyle {\\frac {1}{2m}}{\\hat {p}}\_{x}^{2}=E,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/422f57813ede91b8e348e786616c6297742b4fd7) with state ψ {\\displaystyle \\psi } ![{\\displaystyle \\psi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a) in this case having energy E {\\displaystyle E} ![{\\displaystyle E}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b) coincident with the kinetic energy of the particle. The general solutions of the Schrödinger equation for the particle in a box are ψ ( x ) \= A e i k x \+ B e − i k x E \= ℏ 2 k 2 2 m {\\displaystyle \\psi (x)=Ae^{ikx}+Be^{-ikx}\\qquad \\qquad E={\\frac {\\hbar ^{2}k^{2}}{2m}}} ![{\\displaystyle \\psi (x)=Ae^{ikx}+Be^{-ikx}\\qquad \\qquad E={\\frac {\\hbar ^{2}k^{2}}{2m}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5fb1b2f1d5afb42edb4eb98bf89791d283c1e53) or, from [Euler's formula](https://en.wikipedia.org/wiki/Euler%27s_formula "Euler's formula"), ψ ( x ) \= C sin ⁡ ( k x ) \+ D cos ⁡ ( k x ) . {\\displaystyle \\psi (x)=C\\sin(kx)+D\\cos(kx).\\!} ![{\\displaystyle \\psi (x)=C\\sin(kx)+D\\cos(kx).\\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/229ebbbe587e0c678318cb85089a74ed289c971a) The infinite potential walls of the box determine the values of C , D , {\\displaystyle C,D,} ![{\\displaystyle C,D,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e153d2c75e3492a32d4fafefec88846862c3b9b) and k {\\displaystyle k} ![{\\displaystyle k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40) at x \= 0 {\\displaystyle x=0} ![{\\displaystyle x=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/953917eaf52f2e1baad54c8c9e3d6f9bb3710cdc) and x \= L {\\displaystyle x=L} ![{\\displaystyle x=L}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5fe40c588800aaab69041986b49a59664cd767a) where ψ {\\displaystyle \\psi } ![{\\displaystyle \\psi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a) must be zero. Thus, at x \= 0 {\\displaystyle x=0} ![{\\displaystyle x=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/953917eaf52f2e1baad54c8c9e3d6f9bb3710cdc), ψ ( 0 ) \= 0 \= C sin ⁡ ( 0 ) \+ D cos ⁡ ( 0 ) \= D {\\displaystyle \\psi (0)=0=C\\sin(0)+D\\cos(0)=D} ![{\\displaystyle \\psi (0)=0=C\\sin(0)+D\\cos(0)=D}](https://wikimedia.org/api/rest_v1/media/math/render/svg/34188e64cbf80486b3ad311b53b512fa36de59c6) and D \= 0 {\\displaystyle D=0} ![{\\displaystyle D=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d375dfda80ee8df1d1d7aa8b962114044e464305). At x \= L {\\displaystyle x=L} ![{\\displaystyle x=L}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5fe40c588800aaab69041986b49a59664cd767a), ψ ( L ) \= 0 \= C sin ⁡ ( k L ) , {\\displaystyle \\psi (L)=0=C\\sin(kL),} ![{\\displaystyle \\psi (L)=0=C\\sin(kL),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c7a5bc9ef330efb5b6eba5d89c2ee026bbeea6a8) in which C {\\displaystyle C} ![{\\displaystyle C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029) cannot be zero as this would conflict with the postulate that ψ {\\displaystyle \\psi } ![{\\displaystyle \\psi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a) has norm 1. Therefore, since sin ⁡ ( k L ) \= 0 {\\displaystyle \\sin(kL)=0} ![{\\displaystyle \\sin(kL)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5177ccdb2057c5c1be728af20b8ef3d61f79999), k L {\\displaystyle kL} ![{\\displaystyle kL}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d8865c1184b2c1dff6226dae50d3be91f4f01cfe) must be an integer multiple of π {\\displaystyle \\pi } ![{\\displaystyle \\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a), k \= n π L n \= 1 , 2 , 3 , … . {\\displaystyle k={\\frac {n\\pi }{L}}\\qquad \\qquad n=1,2,3,\\ldots .} ![{\\displaystyle k={\\frac {n\\pi }{L}}\\qquad \\qquad n=1,2,3,\\ldots .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3fd91af3024e1f59cf57e04884494fd2c55664f8) This constraint on k {\\displaystyle k} ![{\\displaystyle k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40) implies a constraint on the energy levels, yielding E n \= ℏ 2 π 2 n 2 2 m L 2 \= n 2 h 2 8 m L 2 . {\\displaystyle E\_{n}={\\frac {\\hbar ^{2}\\pi ^{2}n^{2}}{2mL^{2}}}={\\frac {n^{2}h^{2}}{8mL^{2}}}.} ![{\\displaystyle E\_{n}={\\frac {\\hbar ^{2}\\pi ^{2}n^{2}}{2mL^{2}}}={\\frac {n^{2}h^{2}}{8mL^{2}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe507fd1f92ebfe133aeb4f1da46ea27f569f38b) A [finite potential well](https://en.wikipedia.org/wiki/Finite_potential_well "Finite potential well") is the generalization of the infinite potential well problem to potential wells having finite depth. The finite potential well problem is mathematically more complicated than the infinite particle-in-a-box problem as the wave function is not pinned to zero at the walls of the well. Instead, the wave function must satisfy more complicated mathematical boundary conditions as it is nonzero in regions outside the well. Another related problem is that of the [rectangular potential barrier](https://en.wikipedia.org/wiki/Rectangular_potential_barrier "Rectangular potential barrier"), which furnishes a model for the [quantum tunneling](https://en.wikipedia.org/wiki/Quantum_tunneling "Quantum tunneling") effect that plays an important role in the performance of modern technologies such as [flash memory](https://en.wikipedia.org/wiki/Flash_memory "Flash memory") and [scanning tunneling microscopy](https://en.wikipedia.org/wiki/Scanning_tunneling_microscopy "Scanning tunneling microscopy"). ### Harmonic oscillator Main article: [Quantum harmonic oscillator](https://en.wikipedia.org/wiki/Quantum_harmonic_oscillator "Quantum harmonic oscillator") [![](https://upload.wikimedia.org/wikipedia/commons/9/90/QuantumHarmonicOscillatorAnimation.gif)](https://en.wikipedia.org/wiki/File:QuantumHarmonicOscillatorAnimation.gif) Some trajectories of a [harmonic oscillator](https://en.wikipedia.org/wiki/Harmonic_oscillator "Harmonic oscillator") (i.e. a ball attached to a [spring](https://en.wikipedia.org/wiki/Hooke%27s_law "Hooke's law")) in [classical mechanics](https://en.wikipedia.org/wiki/Classical_mechanics "Classical mechanics") (A-B) and quantum mechanics (C-H). In quantum mechanics, the position of the ball is represented by a [wave](https://en.wikipedia.org/wiki/Wave "Wave") (called the wave function), with the [real part](https://en.wikipedia.org/wiki/Real_part "Real part") shown in blue and the [imaginary part](https://en.wikipedia.org/wiki/Imaginary_part "Imaginary part") shown in red. Some of the trajectories (such as C, D, E, and F) are [standing waves](https://en.wikipedia.org/wiki/Standing_wave "Standing wave") (or "[stationary states](https://en.wikipedia.org/wiki/Stationary_state "Stationary state")"). Each standing-wave frequency is proportional to a possible [energy level](https://en.wikipedia.org/wiki/Energy_level "Energy level") of the oscillator. This "energy quantization" does not occur in classical physics, where the oscillator can have *any* energy. As in the classical case, the potential for the quantum harmonic oscillator is given by[\[7\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Zwiebach2022-7): 234 V ( x ) \= 1 2 m ω 2 x 2 . {\\displaystyle V(x)={\\frac {1}{2}}m\\omega ^{2}x^{2}.} ![{\\displaystyle V(x)={\\frac {1}{2}}m\\omega ^{2}x^{2}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac9d8735e10198c090b576c765dc2778fe458b9c) This problem can either be treated by directly solving the Schrödinger equation, which is not trivial, or by using the more elegant "ladder method" first proposed by Paul Dirac. The [eigenstates](https://en.wikipedia.org/wiki/Eigenstate "Eigenstate") are given by ψ n ( x ) \= 1 2 n n \! ⋅ ( m ω π ℏ ) 1 / 4 ⋅ e − m ω x 2 2 ℏ ⋅ H n ( m ω ℏ x ) , {\\displaystyle \\psi \_{n}(x)={\\sqrt {\\frac {1}{2^{n}\\,n!}}}\\cdot \\left({\\frac {m\\omega }{\\pi \\hbar }}\\right)^{1/4}\\cdot e^{-{\\frac {m\\omega x^{2}}{2\\hbar }}}\\cdot H\_{n}\\left({\\sqrt {\\frac {m\\omega }{\\hbar }}}x\\right),\\qquad } ![{\\displaystyle \\psi \_{n}(x)={\\sqrt {\\frac {1}{2^{n}\\,n!}}}\\cdot \\left({\\frac {m\\omega }{\\pi \\hbar }}\\right)^{1/4}\\cdot e^{-{\\frac {m\\omega x^{2}}{2\\hbar }}}\\cdot H\_{n}\\left({\\sqrt {\\frac {m\\omega }{\\hbar }}}x\\right),\\qquad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a0031b1b0c9fa4a9e4e6957718d509846522c4d) n \= 0 , 1 , 2 , … . {\\displaystyle n=0,1,2,\\ldots .} ![{\\displaystyle n=0,1,2,\\ldots .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d652f9756f144ea772e4e632cb1398652d36372c) where *Hn* are the [Hermite polynomials](https://en.wikipedia.org/wiki/Hermite_polynomials "Hermite polynomials") H n ( x ) \= ( − 1 ) n e x 2 d n d x n ( e − x 2 ) , {\\displaystyle H\_{n}(x)=(-1)^{n}e^{x^{2}}{\\frac {d^{n}}{dx^{n}}}\\left(e^{-x^{2}}\\right),} ![{\\displaystyle H\_{n}(x)=(-1)^{n}e^{x^{2}}{\\frac {d^{n}}{dx^{n}}}\\left(e^{-x^{2}}\\right),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0da8b4a24291ce8b32bcc0665f490f3c76afc7d3) and the corresponding energy levels are E n \= ℏ ω ( n \+ 1 2 ) . {\\displaystyle E\_{n}=\\hbar \\omega \\left(n+{1 \\over 2}\\right).} ![{\\displaystyle E\_{n}=\\hbar \\omega \\left(n+{1 \\over 2}\\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/635fb242f4c34cd5ba947f4c8132ddd47f2872c0) This is another example illustrating the discretization of energy for [bound states](https://en.wikipedia.org/wiki/Bound_state "Bound state"). ### Mach–Zehnder interferometer [![](https://upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Mach-Zehnder_interferometer.svg/330px-Mach-Zehnder_interferometer.svg.png)](https://en.wikipedia.org/wiki/File:Mach-Zehnder_interferometer.svg) Schematic of a Mach–Zehnder interferometer The [Mach–Zehnder interferometer](https://en.wikipedia.org/wiki/Mach%E2%80%93Zehnder_interferometer "Mach–Zehnder interferometer") (MZI) illustrates the concepts of superposition and interference with linear algebra in dimension 2, rather than differential equations. It can be seen as a simplified version of the double-slit experiment, but it is of interest in its own right, for example in the [delayed choice quantum eraser](https://en.wikipedia.org/wiki/Delayed_choice_quantum_eraser "Delayed choice quantum eraser"), the [Elitzur–Vaidman bomb tester](https://en.wikipedia.org/wiki/Elitzur%E2%80%93Vaidman_bomb_tester "Elitzur–Vaidman bomb tester"), and in studies of quantum entanglement.[\[36\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Paris1999-37)[\[37\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Haack2010-38) We can model a photon going through the interferometer by considering that at each point it can be in a superposition of only two paths: the "lower" path which starts from the left, goes straight through both beam splitters, and ends at the top, and the "upper" path which starts from the bottom, goes straight through both beam splitters, and ends at the right. The quantum state of the photon is therefore a vector ψ ∈ C 2 {\\displaystyle \\psi \\in \\mathbb {C} ^{2}} ![{\\displaystyle \\psi \\in \\mathbb {C} ^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a7a51a8280039fc22cd88d90915a952f8e020f47) that is a superposition of the "lower" path ψ l \= ( 1 0 ) {\\displaystyle \\psi \_{l}={\\begin{pmatrix}1\\\\0\\end{pmatrix}}} ![{\\displaystyle \\psi \_{l}={\\begin{pmatrix}1\\\\0\\end{pmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/feca39f1f03b4dc63be6f7a7c2060430b1217e2f) and the "upper" path ψ u \= ( 0 1 ) {\\displaystyle \\psi \_{u}={\\begin{pmatrix}0\\\\1\\end{pmatrix}}} ![{\\displaystyle \\psi \_{u}={\\begin{pmatrix}0\\\\1\\end{pmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/400e9751fcd7816718398d3892bec2ad26bb4713), that is, ψ \= α ψ l \+ β ψ u {\\displaystyle \\psi =\\alpha \\psi \_{l}+\\beta \\psi \_{u}} ![{\\displaystyle \\psi =\\alpha \\psi \_{l}+\\beta \\psi \_{u}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fb88364b79f3c611ca2ca1edb5356139bf4e1085) for complex α , β {\\displaystyle \\alpha ,\\beta } ![{\\displaystyle \\alpha ,\\beta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4b46b57cfa0011b643037751809904d915c1b48). In order to respect the postulate that ⟨ ψ , ψ ⟩ \= 1 {\\displaystyle \\langle \\psi ,\\psi \\rangle =1} ![{\\displaystyle \\langle \\psi ,\\psi \\rangle =1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d5758e7a60b4e54bc46e01b0618919c65b787a0) we require that \| α \| 2 \+ \| β \| 2 \= 1 {\\displaystyle \|\\alpha \|^{2}+\|\\beta \|^{2}=1} ![{\\displaystyle \|\\alpha \|^{2}+\|\\beta \|^{2}=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/18cd7473cdb894839d10852890517b1fb687c73b). Both [beam splitters](https://en.wikipedia.org/wiki/Beam_splitter "Beam splitter") are modelled as the unitary matrix B \= 1 2 ( 1 i i 1 ) {\\displaystyle B={\\frac {1}{\\sqrt {2}}}{\\begin{pmatrix}1\&i\\\\i&1\\end{pmatrix}}} ![{\\displaystyle B={\\frac {1}{\\sqrt {2}}}{\\begin{pmatrix}1\&i\\\\i&1\\end{pmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ddf502efcb65d0cbac5bb8ef1a6f163ac9cf2145), which means that when a photon meets the beam splitter it will either stay on the same path with a probability amplitude of 1 / 2 {\\displaystyle 1/{\\sqrt {2}}} ![{\\displaystyle 1/{\\sqrt {2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75a0bbdb60fcb73ac67d9970a5eb0808b87fd37d), or be reflected to the other path with a probability amplitude of i / 2 {\\displaystyle i/{\\sqrt {2}}} ![{\\displaystyle i/{\\sqrt {2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e846b6a5731f3799960a4968399d85bc0b7fb9fd). The phase shifter on the upper arm is modelled as the unitary matrix P \= ( 1 0 0 e i Δ Φ ) {\\displaystyle P={\\begin{pmatrix}1&0\\\\0\&e^{i\\Delta \\Phi }\\end{pmatrix}}} ![{\\displaystyle P={\\begin{pmatrix}1&0\\\\0\&e^{i\\Delta \\Phi }\\end{pmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3df9457946dd8035c51a39e0926be9f07c7f0a3e), which means that if the photon is on the "upper" path it will gain a relative phase of Δ Φ {\\displaystyle \\Delta \\Phi } ![{\\displaystyle \\Delta \\Phi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/20cedb08e6edea3cad9b2829ef67311bbe518dd2), and it will stay unchanged if it is in the lower path. A photon that enters the interferometer from the left will then be acted upon with a beam splitter B {\\displaystyle B} ![{\\displaystyle B}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a), a phase shifter P {\\displaystyle P} ![{\\displaystyle P}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a), and another beam splitter B {\\displaystyle B} ![{\\displaystyle B}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a), and so end up in the state B P B ψ l \= i e i Δ Φ / 2 ( − sin ⁡ ( Δ Φ / 2 ) cos ⁡ ( Δ Φ / 2 ) ) , {\\displaystyle BPB\\psi \_{l}=ie^{i\\Delta \\Phi /2}{\\begin{pmatrix}-\\sin(\\Delta \\Phi /2)\\\\\\cos(\\Delta \\Phi /2)\\end{pmatrix}},} ![{\\displaystyle BPB\\psi \_{l}=ie^{i\\Delta \\Phi /2}{\\begin{pmatrix}-\\sin(\\Delta \\Phi /2)\\\\\\cos(\\Delta \\Phi /2)\\end{pmatrix}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7927a94da54f5d57b8accffdb9ad456a1e3b5033) and the probabilities that it will be detected at the right or at the top are given respectively by p ( u ) \= \| ⟨ ψ u , B P B ψ l ⟩ \| 2 \= cos 2 ⁡ Δ Φ 2 , {\\displaystyle p(u)=\|\\langle \\psi \_{u},BPB\\psi \_{l}\\rangle \|^{2}=\\cos ^{2}{\\frac {\\Delta \\Phi }{2}},} ![{\\displaystyle p(u)=\|\\langle \\psi \_{u},BPB\\psi \_{l}\\rangle \|^{2}=\\cos ^{2}{\\frac {\\Delta \\Phi }{2}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/110acb8ee7dc4e309de846470778f4767fe97f8f) p ( l ) \= \| ⟨ ψ l , B P B ψ l ⟩ \| 2 \= sin 2 ⁡ Δ Φ 2 . {\\displaystyle p(l)=\|\\langle \\psi \_{l},BPB\\psi \_{l}\\rangle \|^{2}=\\sin ^{2}{\\frac {\\Delta \\Phi }{2}}.} ![{\\displaystyle p(l)=\|\\langle \\psi \_{l},BPB\\psi \_{l}\\rangle \|^{2}=\\sin ^{2}{\\frac {\\Delta \\Phi }{2}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86880ce53051688a4f591ebfb5183fbae013deee) One can therefore use the Mach–Zehnder interferometer to estimate the [phase shift](https://en.wikipedia.org/wiki/Phase_\(waves\) "Phase (waves)") by estimating these probabilities. It is interesting to consider what would happen if the photon were definitely in either the "lower" or "upper" paths between the beam splitters. This can be accomplished by blocking one of the paths, or equivalently by removing the first beam splitter (and feeding the photon from the left or the bottom, as desired). In both cases, there will be no interference between the paths anymore, and the probabilities are given by p ( u ) \= p ( l ) \= 1 / 2 {\\displaystyle p(u)=p(l)=1/2} ![{\\displaystyle p(u)=p(l)=1/2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d2a9b3426984de1a64c07261c3e14b485320fee4), independently of the phase Δ Φ {\\displaystyle \\Delta \\Phi } ![{\\displaystyle \\Delta \\Phi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/20cedb08e6edea3cad9b2829ef67311bbe518dd2). From this we can conclude that the photon does not take one path or another after the first beam splitter, but rather that it is in a genuine quantum superposition of the two paths.[\[38\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-vedral-39) ## Applications Main article: [Applications of quantum mechanics](https://en.wikipedia.org/wiki/Applications_of_quantum_mechanics "Applications of quantum mechanics") Quantum mechanics has had enormous success in explaining many of the features of our universe, with regard to small-scale and discrete quantities and interactions which cannot be explained by [classical methods](https://en.wikipedia.org/wiki/Classical_physics "Classical physics").[\[note 2\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-feynmanIII-40) Quantum mechanics is often the only theory that can reveal the individual behaviors of the subatomic particles that make up all forms of matter (electrons, [protons](https://en.wikipedia.org/wiki/Proton "Proton"), [neutrons](https://en.wikipedia.org/wiki/Neutron "Neutron"), [photons](https://en.wikipedia.org/wiki/Photon "Photon"), and others). [Solid-state physics](https://en.wikipedia.org/wiki/Solid-state_physics "Solid-state physics") and [materials science](https://en.wikipedia.org/wiki/Materials_science "Materials science") are dependent upon quantum mechanics.[\[39\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-marvincohen2008-41) In many aspects, modern technology operates at a scale where quantum effects are significant. Important applications of quantum theory include [quantum chemistry](https://en.wikipedia.org/wiki/Quantum_chemistry "Quantum chemistry"), [quantum optics](https://en.wikipedia.org/wiki/Quantum_optics "Quantum optics"), [quantum computing](https://en.wikipedia.org/wiki/Quantum_computing "Quantum computing"), [superconducting magnets](https://en.wikipedia.org/wiki/Superconducting_magnet "Superconducting magnet"), [light-emitting diodes](https://en.wikipedia.org/wiki/Light-emitting_diode "Light-emitting diode"), the [optical amplifier](https://en.wikipedia.org/wiki/Optical_amplifier "Optical amplifier") and the laser, the [transistor](https://en.wikipedia.org/wiki/Transistor "Transistor") and [semiconductors](https://en.wikipedia.org/wiki/Semiconductor "Semiconductor") such as the [microprocessor](https://en.wikipedia.org/wiki/Microprocessor "Microprocessor"), [medical and research imaging](https://en.wikipedia.org/wiki/Medical_imaging "Medical imaging") such as [magnetic resonance imaging](https://en.wikipedia.org/wiki/Magnetic_resonance_imaging "Magnetic resonance imaging") and [electron microscopy](https://en.wikipedia.org/wiki/Electron_microscopy "Electron microscopy").[\[40\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-42) Explanations for many biological and physical phenomena are rooted in the nature of the chemical bond, most notably the macro-molecule [DNA](https://en.wikipedia.org/wiki/DNA "DNA"). ## Relation to other scientific theories | [Modern physics](https://en.wikipedia.org/wiki/Modern_physics "Modern physics") | |---| | H ^ \| ψ n ( t ) ⟩ \= i ℏ d d t \| ψ n ( t ) ⟩ {\\displaystyle {\\hat {H}}\|\\psi \_{n}(t)\\rangle =i\\hbar {\\frac {d}{dt}}\|\\psi \_{n}(t)\\rangle } ![{\\displaystyle {\\hat {H}}\|\\psi \_{n}(t)\\rangle =i\\hbar {\\frac {d}{dt}}\|\\psi \_{n}(t)\\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/6227214310d46ffac904ca257878eee1fb6ce726) G μ ν \+ Λ g μ ν \= κ T μ ν {\\displaystyle G\_{\\mu \\nu }+\\Lambda g\_{\\mu \\nu }={\\kappa }T\_{\\mu \\nu }} ![{\\displaystyle G\_{\\mu \\nu }+\\Lambda g\_{\\mu \\nu }={\\kappa }T\_{\\mu \\nu }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/124ab80fcb17e2733cc17ff6f93da5e52f355c77)[Schrödinger](https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation "Schrödinger equation") and [Einstein field equations](https://en.wikipedia.org/wiki/Einstein_field_equations "Einstein field equations") | ### Classical mechanics The rules of quantum mechanics assert that the state space of a system is a Hilbert space and that observables of the system are Hermitian operators acting on vectors in that space – although they do not tell us which Hilbert space or which operators. These can be chosen appropriately in order to obtain a quantitative description of a quantum system, a necessary step in making physical predictions. An important guide for making these choices is the [correspondence principle](https://en.wikipedia.org/wiki/Correspondence_principle "Correspondence principle"), a heuristic which states that the predictions of quantum mechanics reduce to those of [classical mechanics](https://en.wikipedia.org/wiki/Classical_mechanics "Classical mechanics") in the regime of large [quantum numbers](https://en.wikipedia.org/wiki/Quantum_number "Quantum number").[\[41\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Tipler-43) One can also start from an established classical model of a particular system, and then try to guess the underlying quantum model that would give rise to the classical model in the correspondence limit. This approach is known as [quantization](https://en.wikipedia.org/wiki/Canonical_quantization "Canonical quantization").[\[42\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Peres1993-44): 299 [\[43\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-45) When quantum mechanics was originally formulated, it was applied to models whose correspondence limit was [non-relativistic](https://en.wikipedia.org/wiki/Theory_of_relativity "Theory of relativity") classical mechanics. For instance, the well-known model of the [quantum harmonic oscillator](https://en.wikipedia.org/wiki/Quantum_harmonic_oscillator "Quantum harmonic oscillator") uses an explicitly non-relativistic expression for the [kinetic energy](https://en.wikipedia.org/wiki/Kinetic_energy "Kinetic energy") of the oscillator, and is thus a quantum version of the [classical harmonic oscillator](https://en.wikipedia.org/wiki/Harmonic_oscillator "Harmonic oscillator").[\[7\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Zwiebach2022-7): 234 Complications arise with [chaotic systems](https://en.wikipedia.org/wiki/Chaotic_systems "Chaotic systems"), which do not have good quantum numbers, and [quantum chaos](https://en.wikipedia.org/wiki/Quantum_chaos "Quantum chaos") studies the relationship between classical and quantum descriptions in these systems.[\[42\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Peres1993-44): 353 [Quantum decoherence](https://en.wikipedia.org/wiki/Quantum_decoherence "Quantum decoherence") is a mechanism through which quantum systems lose [coherence](https://en.wikipedia.org/wiki/Quantum_coherence "Quantum coherence"), and thus become incapable of displaying many typically quantum effects: [quantum superpositions](https://en.wikipedia.org/wiki/Quantum_superposition "Quantum superposition") become simply probabilistic mixtures, and quantum entanglement becomes simply classical correlations.[\[7\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Zwiebach2022-7): 687–730 Quantum coherence is not typically evident at macroscopic scales, though at temperatures approaching [absolute zero](https://en.wikipedia.org/wiki/Absolute_zero "Absolute zero") quantum behavior may manifest macroscopically.[\[note 3\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-46) Many macroscopic properties of a classical system are a direct consequence of the quantum behavior of its parts. For example, the stability of bulk matter (consisting of atoms and [molecules](https://en.wikipedia.org/wiki/Molecule "Molecule") which would quickly collapse under electric forces alone), the rigidity of solids, and the mechanical, thermal, chemical, optical and magnetic properties of matter are all results of the interaction of [electric charges](https://en.wikipedia.org/wiki/Electric_charge "Electric charge") under the rules of quantum mechanics.[\[44\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-47) ### Special relativity and electrodynamics Early attempts to merge quantum mechanics with [special relativity](https://en.wikipedia.org/wiki/Special_relativity "Special relativity") involved the replacement of the Schrödinger equation with a covariant equation such as the [Klein–Gordon equation](https://en.wikipedia.org/wiki/Klein%E2%80%93Gordon_equation "Klein–Gordon equation") or the [Dirac equation](https://en.wikipedia.org/wiki/Dirac_equation "Dirac equation"). While these theories were successful in explaining many experimental results, they had certain unsatisfactory qualities stemming from their neglect of the relativistic creation and annihilation of particles. A fully relativistic quantum theory required the development of quantum field theory, which applies quantization to a field (rather than a fixed set of particles). The first complete quantum field theory, [quantum electrodynamics](https://en.wikipedia.org/wiki/Quantum_electrodynamics "Quantum electrodynamics"), provides a fully quantum description of the [electromagnetic interaction](https://en.wikipedia.org/wiki/Electromagnetic_interaction "Electromagnetic interaction"). Quantum electrodynamics is, along with [general relativity](https://en.wikipedia.org/wiki/General_relativity "General relativity"), one of the most accurate physical theories ever devised.[\[45\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-48)[\[46\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-49) The full apparatus of quantum field theory is often unnecessary for describing electrodynamic systems. A simpler approach, one that has been used since the inception of quantum mechanics, is to treat [charged](https://en.wikipedia.org/wiki/Electric_charge "Electric charge") particles as quantum mechanical objects being acted on by a classical [electromagnetic field](https://en.wikipedia.org/wiki/Electromagnetic_field "Electromagnetic field"). For example, the elementary quantum model of the [hydrogen atom](https://en.wikipedia.org/wiki/Hydrogen_atom "Hydrogen atom") describes the [electric field](https://en.wikipedia.org/wiki/Electric_field "Electric field") of the hydrogen atom using a classical − e 2 / ( 4 π ϵ 0 r ) {\\displaystyle \\textstyle -e^{2}/(4\\pi \\epsilon \_{\_{0}}r)} ![{\\displaystyle \\textstyle -e^{2}/(4\\pi \\epsilon \_{\_{0}}r)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c4876353362da0a234af137590c6379d8f8c273) [Coulomb potential](https://en.wikipedia.org/wiki/Electric_potential "Electric potential").[\[7\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Zwiebach2022-7): 285 Likewise, in a [Stern–Gerlach experiment](https://en.wikipedia.org/wiki/Stern%E2%80%93Gerlach_experiment "Stern–Gerlach experiment"), a charged particle is modeled as a quantum system, while the background magnetic field is described classically.[\[42\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Peres1993-44): 26 This "semi-classical" approach fails if quantum fluctuations in the electromagnetic field play an important role, such as in the emission of photons by [charged particles](https://en.wikipedia.org/wiki/Charged_particle "Charged particle"). [Quantum field](https://en.wikipedia.org/wiki/Field_\(physics\) "Field (physics)") theories for the [strong nuclear force](https://en.wikipedia.org/wiki/Strong_nuclear_force "Strong nuclear force") and the [weak nuclear force](https://en.wikipedia.org/wiki/Weak_nuclear_force "Weak nuclear force") have also been developed. The quantum field theory of the strong nuclear force is called [quantum chromodynamics](https://en.wikipedia.org/wiki/Quantum_chromodynamics "Quantum chromodynamics"), and describes the interactions of subnuclear particles such as [quarks](https://en.wikipedia.org/wiki/Quark "Quark") and [gluons](https://en.wikipedia.org/wiki/Gluon "Gluon"). The weak nuclear force and the electromagnetic force were unified, in their quantized forms, into a single quantum field theory (known as [electroweak theory](https://en.wikipedia.org/wiki/Electroweak_theory "Electroweak theory")), by the physicists [Abdus Salam](https://en.wikipedia.org/wiki/Abdus_Salam "Abdus Salam"), [Sheldon Glashow](https://en.wikipedia.org/wiki/Sheldon_Glashow "Sheldon Glashow") and [Steven Weinberg](https://en.wikipedia.org/wiki/Steven_Weinberg "Steven Weinberg").[\[47\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-50) ### Relation to general relativity Even though the predictions of both quantum theory and general relativity have been supported by rigorous and repeated [empirical evidence](https://en.wikipedia.org/wiki/Empirical_evidence "Empirical evidence"), their abstract formalisms contradict each other and they have proven extremely difficult to incorporate into one consistent, cohesive model. Gravity is negligible in many areas of particle physics, so that unification between general relativity and quantum mechanics is not an urgent issue in those particular applications. However, the lack of a correct theory of [quantum gravity](https://en.wikipedia.org/wiki/Quantum_gravity "Quantum gravity") is an important issue in [physical cosmology](https://en.wikipedia.org/wiki/Physical_cosmology "Physical cosmology") and the search by physicists for an elegant "[Theory of Everything](https://en.wikipedia.org/wiki/Theory_of_Everything "Theory of Everything")" (TOE). Consequently, resolving the inconsistencies between both theories has been a major goal of 20th- and 21st-century physics. This TOE would combine not only the models of subatomic physics but also derive the four fundamental forces of nature from a single force or phenomenon.[\[48\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-NYT-20221010-51) One proposal for doing so is [string theory](https://en.wikipedia.org/wiki/String_theory "String theory"), which posits that the [point-like particles](https://en.wikipedia.org/wiki/Point_particle "Point particle") of [particle physics](https://en.wikipedia.org/wiki/Particle_physics "Particle physics") are replaced by [one-dimensional](https://en.wikipedia.org/wiki/Dimension_\(mathematics_and_physics\) "Dimension (mathematics and physics)") objects called [strings](https://en.wikipedia.org/wiki/String_\(physics\) "String (physics)"). String theory describes how these strings propagate through space and interact with each other. On distance scales larger than the string scale, a string looks just like an ordinary particle, with its [mass](https://en.wikipedia.org/wiki/Mass "Mass"), [charge](https://en.wikipedia.org/wiki/Charge_\(physics\) "Charge (physics)"), and other properties determined by the [vibrational](https://en.wikipedia.org/wiki/Vibration "Vibration") state of the string. In string theory, one of the many vibrational states of the string corresponds to the [graviton](https://en.wikipedia.org/wiki/Graviton "Graviton"), a quantum mechanical particle that carries gravitational force.[\[49\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-52)[\[50\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-53) Another popular theory is [loop quantum gravity](https://en.wikipedia.org/wiki/Loop_quantum_gravity "Loop quantum gravity") (LQG), which describes quantum properties of gravity and is thus a theory of [quantum spacetime](https://en.wikipedia.org/wiki/Quantum_spacetime "Quantum spacetime"). LQG is an attempt to merge and adapt standard quantum mechanics and standard general relativity. This theory describes space as an extremely fine fabric "woven" of finite loops called [spin networks](https://en.wikipedia.org/wiki/Spin_network "Spin network"). The evolution of a spin network over time is called a [spin foam](https://en.wikipedia.org/wiki/Spin_foam "Spin foam"). The characteristic length scale of a spin foam is the [Planck length](https://en.wikipedia.org/wiki/Planck_length "Planck length"), approximately 1.616×10−35 m, and so lengths shorter than the Planck length are not physically meaningful in LQG.[\[51\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-54) ## Philosophical implications Main article: [Interpretations of quantum mechanics](https://en.wikipedia.org/wiki/Interpretations_of_quantum_mechanics "Interpretations of quantum mechanics") Unsolved problem in physics Is there a preferred interpretation of quantum mechanics? How does the quantum description of reality, which includes elements such as the "[superposition](https://en.wikipedia.org/wiki/Superposition_principle "Superposition principle") of states" and "[wave function collapse](https://en.wikipedia.org/wiki/Wave_function_collapse "Wave function collapse")", give rise to the reality we perceive? [More unsolved problems in physics](https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_physics "List of unsolved problems in physics") Since its inception, the many counter-intuitive aspects and results of quantum mechanics have provoked strong [philosophical](https://en.wikipedia.org/wiki/Philosophical "Philosophical") debates and many [interpretations](https://en.wikipedia.org/wiki/Interpretations_of_quantum_mechanics "Interpretations of quantum mechanics"). The arguments centre on the probabilistic nature of quantum mechanics, the difficulties with [wavefunction collapse](https://en.wikipedia.org/wiki/Wavefunction_collapse "Wavefunction collapse") and the related [measurement problem](https://en.wikipedia.org/wiki/Measurement_problem "Measurement problem"), and [quantum nonlocality](https://en.wikipedia.org/wiki/Quantum_nonlocality "Quantum nonlocality"). Perhaps the only consensus that exists about these issues is that there is no consensus. [Richard Feynman](https://en.wikipedia.org/wiki/Richard_Feynman "Richard Feynman") once said, "I think I can safely say that nobody understands quantum mechanics."[\[52\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-55) According to [Steven Weinberg](https://en.wikipedia.org/wiki/Steven_Weinberg "Steven Weinberg"), "There is now in my opinion no entirely satisfactory interpretation of quantum mechanics."[\[53\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-56) The views of [Niels Bohr](https://en.wikipedia.org/wiki/Niels_Bohr "Niels Bohr"), Werner Heisenberg and other physicists are often grouped together as the "[Copenhagen interpretation](https://en.wikipedia.org/wiki/Copenhagen_interpretation "Copenhagen interpretation")".[\[54\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-57)[\[55\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-58) According to these views, the probabilistic nature of quantum mechanics is not a *temporary* feature which will eventually be replaced by a deterministic theory, but is instead a *final* renunciation of the classical idea of "causality". Bohr in particular emphasized that any well-defined application of the quantum mechanical formalism must always make reference to the experimental arrangement, due to the [complementary](https://en.wikipedia.org/wiki/Complementarity_\(physics\) "Complementarity (physics)") nature of evidence obtained under different experimental situations. Copenhagen-type interpretations were adopted by Nobel laureates in quantum physics, including Bohr,[\[56\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-BohrComo-59) Heisenberg,[\[57\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-60) Schrödinger,[\[58\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-61) Feynman,[\[2\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Feynman-2) and [Zeilinger](https://en.wikipedia.org/wiki/Anton_Zeilinger "Anton Zeilinger")[\[59\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-MaKoflerZeilinger-62) as well as 21st-century researchers in quantum foundations.[\[60\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-:25-63) [Albert Einstein](https://en.wikipedia.org/wiki/Albert_Einstein "Albert Einstein"), himself one of the founders of [quantum theory](https://en.wikipedia.org/wiki/Old_quantum_theory "Old quantum theory"), was troubled by its apparent failure to respect some cherished metaphysical principles, such as [determinism](https://en.wikipedia.org/wiki/Determinism "Determinism") and [locality](https://en.wikipedia.org/wiki/Principle_of_locality "Principle of locality"). Einstein's long-running exchanges with Bohr about the meaning and status of quantum mechanics are now known as the [Bohr–Einstein debates](https://en.wikipedia.org/wiki/Bohr%E2%80%93Einstein_debates "Bohr–Einstein debates"). Einstein believed that underlying quantum mechanics must be a theory that explicitly forbids [action at a distance](https://en.wikipedia.org/wiki/Action_at_a_distance "Action at a distance"). He argued that quantum mechanics was incomplete, a theory that was valid but not fundamental, analogous to how [thermodynamics](https://en.wikipedia.org/wiki/Thermodynamics "Thermodynamics") is valid, but the fundamental theory behind it is [statistical mechanics](https://en.wikipedia.org/wiki/Statistical_mechanics "Statistical mechanics"). In 1935, Einstein and his collaborators [Boris Podolsky](https://en.wikipedia.org/wiki/Boris_Podolsky "Boris Podolsky") and [Nathan Rosen](https://en.wikipedia.org/wiki/Nathan_Rosen "Nathan Rosen") published an argument that the principle of locality implies the incompleteness of quantum mechanics, a [thought experiment](https://en.wikipedia.org/wiki/Thought_experiment "Thought experiment") later termed the [Einstein–Podolsky–Rosen paradox](https://en.wikipedia.org/wiki/Einstein%E2%80%93Podolsky%E2%80%93Rosen_paradox "Einstein–Podolsky–Rosen paradox").[\[note 4\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-68) In 1964, [John Bell](https://en.wikipedia.org/wiki/John_Stewart_Bell "John Stewart Bell") showed that EPR's principle of locality, together with determinism, was actually incompatible with quantum mechanics: they implied constraints on the correlations produced by distance systems, now known as [Bell inequalities](https://en.wikipedia.org/wiki/Bell_inequalities "Bell inequalities"), that can be violated by entangled particles.[\[65\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-69) Since then [several experiments](https://en.wikipedia.org/wiki/Bell_test "Bell test") have been performed to obtain these correlations, with the result that they do in fact violate Bell inequalities, and thus falsify the conjunction of locality with determinism.[\[16\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-wiseman15-16)[\[17\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-wolchover17-17) [Bohmian mechanics](https://en.wikipedia.org/wiki/Bohmian_mechanics "Bohmian mechanics") shows that it is possible to reformulate quantum mechanics to make it deterministic, at the price of making it explicitly nonlocal. It attributes not only a wave function to a physical system, but in addition a real position, that evolves deterministically under a nonlocal guiding equation. The evolution of a physical system is given at all times by the Schrödinger equation together with the guiding equation; there is never a collapse of the wave function. This solves the measurement problem.[\[66\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-70) [![](https://upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Schroedingers_cat_film.svg/250px-Schroedingers_cat_film.svg.png)](https://en.wikipedia.org/wiki/File:Schroedingers_cat_film.svg) The [Schrödinger's cat](https://en.wikipedia.org/wiki/Schr%C3%B6dinger%27s_cat "Schrödinger's cat") thought experiment can be used to visualize the many-worlds interpretation of quantum mechanics, where a branching of the universe occurs through a superposition of two quantum mechanical states. Everett's [many-worlds interpretation](https://en.wikipedia.org/wiki/Many-worlds_interpretation "Many-worlds interpretation"), formulated in 1956, holds that *all* the possibilities described by quantum theory *simultaneously* occur in a multiverse composed of mostly independent parallel universes.[\[67\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-71) This is a consequence of removing the axiom of the collapse of the wave packet. All possible states of the measured system and the measuring apparatus, together with the observer, are present in a real physical quantum superposition. While the multiverse is deterministic, we perceive non-deterministic behavior governed by probabilities, because we do not observe the multiverse as a whole, but only one parallel universe at a time. Exactly how this is supposed to work has been the subject of much debate. Several attempts have been made to make sense of this and derive the Born rule,[\[68\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-dewitt73-72)[\[69\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-wallace2003-73) with no consensus on whether they have been successful.[\[70\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-ballentine1973-74)[\[71\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-75)[\[72\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-kent2009-76) [Relational quantum mechanics](https://en.wikipedia.org/wiki/Relational_quantum_mechanics "Relational quantum mechanics") appeared in the late 1990s as a modern derivative of Copenhagen-type ideas,[\[73\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-77)[\[74\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-78) and [QBism](https://en.wikipedia.org/wiki/QBism "QBism") was developed some years later.[\[75\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-:23-79)[\[76\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-80) ## History Main articles: [History of quantum mechanics](https://en.wikipedia.org/wiki/History_of_quantum_mechanics "History of quantum mechanics") and [Atomic theory](https://en.wikipedia.org/wiki/Atomic_theory "Atomic theory") Quantum mechanics was developed in the early decades of the 20th century, driven by the need to explain phenomena that, in some cases, had been observed in earlier times. Scientific inquiry into the wave nature of light began in the 17th and 18th centuries, when scientists such as [Robert Hooke](https://en.wikipedia.org/wiki/Robert_Hooke "Robert Hooke"), [Christiaan Huygens](https://en.wikipedia.org/wiki/Christiaan_Huygens "Christiaan Huygens") and [Leonhard Euler](https://en.wikipedia.org/wiki/Leonhard_Euler "Leonhard Euler") proposed a wave theory of light based on experimental observations.[\[77\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Born_&_Wolf-81) In 1803 English [polymath](https://en.wikipedia.org/wiki/Polymath "Polymath") [Thomas Young](https://en.wikipedia.org/wiki/Thomas_Young_\(scientist\) "Thomas Young (scientist)") described the famous [double-slit experiment](https://en.wikipedia.org/wiki/Young%27s_interference_experiment "Young's interference experiment").[\[78\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-82) This experiment played a major role in the general acceptance of the [wave theory of light](https://en.wikipedia.org/wiki/Wave_theory_of_light "Wave theory of light"). During the early 19th century, [chemical](https://en.wikipedia.org/wiki/Chemistry "Chemistry") research by [John Dalton](https://en.wikipedia.org/wiki/John_Dalton "John Dalton") and [Amedeo Avogadro](https://en.wikipedia.org/wiki/Amedeo_Avogadro "Amedeo Avogadro") lent weight to the [atomic theory](https://en.wikipedia.org/wiki/Atomic_theory "Atomic theory") of matter, an idea that [James Clerk Maxwell](https://en.wikipedia.org/wiki/James_Clerk_Maxwell "James Clerk Maxwell"), [Ludwig Boltzmann](https://en.wikipedia.org/wiki/Ludwig_Boltzmann "Ludwig Boltzmann") and others built upon to establish the [kinetic theory of gases](https://en.wikipedia.org/wiki/Kinetic_theory_of_gases "Kinetic theory of gases"). The successes of kinetic theory gave further credence to the idea that matter is composed of atoms, yet the theory also had shortcomings that would only be resolved by the development of quantum mechanics.[\[79\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Feynman-kinetic-theory-83) While the early conception of atoms from [Greek philosophy](https://en.wikipedia.org/wiki/Greek_philosophy "Greek philosophy") had been that they were indivisible units – the word "atom" deriving from the [Greek](https://en.wikipedia.org/wiki/Greek_language "Greek language") for 'uncuttable' – the 19th century saw the formulation of hypotheses about subatomic structure. One important discovery in that regard was [Michael Faraday](https://en.wikipedia.org/wiki/Michael_Faraday "Michael Faraday")'s 1838 observation of a glow caused by an electrical discharge inside a glass tube containing gas at low pressure. [Julius Plücker](https://en.wikipedia.org/wiki/Julius_Pl%C3%BCcker "Julius Plücker"), [Johann Wilhelm Hittorf](https://en.wikipedia.org/wiki/Johann_Wilhelm_Hittorf "Johann Wilhelm Hittorf") and [Eugen Goldstein](https://en.wikipedia.org/wiki/Eugen_Goldstein "Eugen Goldstein") carried on and improved upon Faraday's work, leading to the identification of [cathode rays](https://en.wikipedia.org/wiki/Cathode_rays "Cathode rays"), which [J. J. Thomson](https://en.wikipedia.org/wiki/J._J._Thomson "J. J. Thomson") found to consist of subatomic particles that would be called electrons.[\[80\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-84)[\[81\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-85) [![](https://upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Max_Planck_%281858-1947%29.jpg/250px-Max_Planck_%281858-1947%29.jpg)](https://en.wikipedia.org/wiki/File:Max_Planck_\(1858-1947\).jpg) [Max Planck](https://en.wikipedia.org/wiki/Max_Planck "Max Planck") is considered the father of the quantum theory. The [black-body radiation](https://en.wikipedia.org/wiki/Black-body_radiation "Black-body radiation") problem was discovered by [Gustav Kirchhoff](https://en.wikipedia.org/wiki/Gustav_Kirchhoff "Gustav Kirchhoff") in 1859. In 1900, Max Planck proposed the hypothesis that energy is radiated and absorbed in discrete "quanta" (or energy packets), yielding a calculation that precisely matched the observed patterns of black-body radiation.[\[82\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-86) The word *quantum* derives from the [Latin](https://en.wikipedia.org/wiki/Latin "Latin"), meaning "how great" or "how much".[\[83\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-87) According to Planck, quantities of energy could be thought of as divided into "elements" whose size (*E*) would be proportional to their [frequency](https://en.wikipedia.org/wiki/Frequency "Frequency") (*ν*): E \= h ν {\\displaystyle E=h\\nu \\ } ![{\\displaystyle E=h\\nu \\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/24767663bb931daecf1eb628be4fea46f32e3622), where *h* is the [Planck constant](https://en.wikipedia.org/wiki/Planck_constant "Planck constant"). Planck cautiously insisted that this was only an aspect of the processes of absorption and emission of radiation and was not the *physical reality* of the radiation.[\[84\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-88) In fact, he considered his quantum hypothesis a mathematical trick to get the right answer rather than a sizable discovery.[\[85\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Kragh-89) However, in 1905 Albert Einstein interpreted Planck's quantum hypothesis [realistically](https://en.wikipedia.org/wiki/Local_realism "Local realism") and used it to explain the [photoelectric effect](https://en.wikipedia.org/wiki/Photoelectric_effect "Photoelectric effect"), in which shining light on certain materials can eject electrons from the material. Niels Bohr then developed Planck's ideas about radiation into a [model of the hydrogen atom](https://en.wikipedia.org/wiki/Bohr_model "Bohr model") that successfully predicted the [spectral lines](https://en.wikipedia.org/wiki/Spectral_line "Spectral line") of hydrogen.[\[86\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-90) Einstein further developed this idea to show that an [electromagnetic wave](https://en.wikipedia.org/wiki/Electromagnetic_wave "Electromagnetic wave") such as light could also be described as a particle (later called the photon), with a discrete amount of energy that depends on its frequency.[\[87\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-91) In his paper "On the Quantum Theory of Radiation", Einstein expanded on the interaction between energy and matter to explain the absorption and emission of energy by atoms. Although overshadowed at the time by his general theory of relativity, this paper articulated the mechanism underlying the stimulated emission of radiation,[\[88\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-92) which became the basis of the laser.[\[89\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-93) [![](https://upload.wikimedia.org/wikipedia/commons/thumb/6/6e/Solvay_conference_1927.jpg/500px-Solvay_conference_1927.jpg)](https://en.wikipedia.org/wiki/File:Solvay_conference_1927.jpg) The 1927 [Solvay Conference](https://en.wikipedia.org/wiki/Solvay_Conference "Solvay Conference") in [Brussels](https://en.wikipedia.org/wiki/Brussels "Brussels") was the fifth world physics conference. This phase is known as the [old quantum theory](https://en.wikipedia.org/wiki/Old_quantum_theory "Old quantum theory"). Never complete or self-consistent, the old quantum theory was rather a set of [heuristic](https://en.wikipedia.org/wiki/Heuristic "Heuristic") corrections to classical mechanics.[\[90\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-terHaar-94)[\[91\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-95) The theory is now understood as a [semi-classical approximation](https://en.wikipedia.org/wiki/WKB_approximation#Application_to_the_Schr%C3%B6dinger_equation "WKB approximation") to modern quantum mechanics.[\[92\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-96)[\[93\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-97) Notable results from this period include, in addition to the work of Planck, Einstein and Bohr mentioned above, Einstein and [Peter Debye](https://en.wikipedia.org/wiki/Peter_Debye "Peter Debye")'s work on the [specific heat](https://en.wikipedia.org/wiki/Specific_heat "Specific heat") of solids, Bohr and [Hendrika Johanna van Leeuwen](https://en.wikipedia.org/wiki/Hendrika_Johanna_van_Leeuwen "Hendrika Johanna van Leeuwen")'s [proof](https://en.wikipedia.org/wiki/Bohr%E2%80%93Van_Leeuwen_theorem "Bohr–Van Leeuwen theorem") that classical physics cannot account for [diamagnetism](https://en.wikipedia.org/wiki/Diamagnetism "Diamagnetism"), and [Arnold Sommerfeld](https://en.wikipedia.org/wiki/Arnold_Sommerfeld "Arnold Sommerfeld")'s extension of the Bohr model to include special-relativistic effects.[\[90\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-terHaar-94)[\[94\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Aharoni-98) In the mid-1920s quantum mechanics was developed to become the standard formulation for atomic physics. In 1923, the French physicist [Louis de Broglie](https://en.wikipedia.org/wiki/Louis_de_Broglie "Louis de Broglie") put forward his theory of matter waves by stating that particles can exhibit wave characteristics and vice versa. Building on de Broglie's approach, modern quantum mechanics was born in 1925, when the German physicists Werner Heisenberg, Max Born, and [Pascual Jordan](https://en.wikipedia.org/wiki/Pascual_Jordan "Pascual Jordan")[\[95\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Edwards79-99)[\[96\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Edwards81-100) developed [matrix mechanics](https://en.wikipedia.org/wiki/Matrix_mechanics "Matrix mechanics") and the Austrian physicist Erwin Schrödinger invented [wave mechanics](https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation "Schrödinger equation"). Born introduced the probabilistic interpretation of Schrödinger's wave function in July 1926.[\[97\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-101) Thus, the entire field of quantum physics emerged, leading to its wider acceptance at the Fifth [Solvay Conference](https://en.wikipedia.org/wiki/Solvay_Conference "Solvay Conference") in 1927.[\[98\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-pais1997-102) By 1930, quantum mechanics had been further unified and formalized by [David Hilbert](https://en.wikipedia.org/wiki/David_Hilbert "David Hilbert"), Paul Dirac and [John von Neumann](https://en.wikipedia.org/wiki/John_von_Neumann "John von Neumann")[\[99\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-103) with greater emphasis on [measurement](https://en.wikipedia.org/wiki/Measurement_in_quantum_mechanics "Measurement in quantum mechanics"), the statistical nature of our knowledge of reality, and [philosophical speculation about the 'observer'](https://en.wikipedia.org/wiki/Interpretations_of_quantum_mechanics "Interpretations of quantum mechanics"). It has since permeated many disciplines, including quantum chemistry, [quantum electronics](https://en.wikipedia.org/wiki/Quantum_electronics "Quantum electronics"), [quantum optics](https://en.wikipedia.org/wiki/Quantum_optics "Quantum optics"), and [quantum information science](https://en.wikipedia.org/wiki/Quantum_information_science "Quantum information science"). It also provides a useful framework for many features of the modern [periodic table of elements](https://en.wikipedia.org/wiki/Periodic_table_of_elements "Periodic table of elements"), and describes the behaviors of [atoms](https://en.wikipedia.org/wiki/Atoms "Atoms") during [chemical bonding](https://en.wikipedia.org/wiki/Chemical_bond "Chemical bond") and the flow of electrons in computer [semiconductors](https://en.wikipedia.org/wiki/Semiconductor "Semiconductor"), and therefore plays a crucial role in many modern technologies. While quantum mechanics was constructed to describe the world of the very small, it is also needed to explain some [macroscopic](https://en.wikipedia.org/wiki/Macroscopic "Macroscopic") phenomena such as [superconductors](https://en.wikipedia.org/wiki/Superconductors "Superconductors")[\[100\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-feynman2015-104) and [superfluids](https://en.wikipedia.org/wiki/Superfluid "Superfluid").[\[101\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-105) ## See also - [Bra–ket notation](https://en.wikipedia.org/wiki/Bra%E2%80%93ket_notation "Bra–ket notation") - [Einstein's thought experiments](https://en.wikipedia.org/wiki/Einstein%27s_thought_experiments "Einstein's thought experiments") - [List of textbooks on classical mechanics and quantum mechanics](https://en.wikipedia.org/wiki/List_of_textbooks_on_classical_mechanics_and_quantum_mechanics "List of textbooks on classical mechanics and quantum mechanics") - [Macroscopic quantum phenomena](https://en.wikipedia.org/wiki/Macroscopic_quantum_phenomena "Macroscopic quantum phenomena") - [Phase-space formulation](https://en.wikipedia.org/wiki/Phase-space_formulation "Phase-space formulation") - [Regularization (physics)](https://en.wikipedia.org/wiki/Regularization_\(physics\) "Regularization (physics)") - [Two-state quantum system](https://en.wikipedia.org/wiki/Two-state_quantum_system "Two-state quantum system") ## Explanatory notes 1. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-35)** A momentum eigenstate would be a perfectly monochromatic wave of infinite extent, which is not square-integrable. Likewise, a position eigenstate would be a [Dirac delta distribution](https://en.wikipedia.org/wiki/Dirac_delta_distribution "Dirac delta distribution"), not square-integrable and technically not a function at all. Consequently, neither can belong to the particle's Hilbert space. Physicists sometimes introduce fictitious "bases" for a Hilbert space comprising elements outside that space. These are invented for calculational convenience and do not represent physical states.[\[26\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Cohen-Tannoudji-26): 100–105 2. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-feynmanIII_40-0)** See, for example, [the Feynman Lectures on Physics](https://en.wikipedia.org/wiki/The_Feynman_Lectures_on_Physics "The Feynman Lectures on Physics") for some of the technological applications which use quantum mechanics, e.g., [transistors](https://en.wikipedia.org/wiki/Transistor "Transistor") (vol **III**, pp. 14–11 ff), [integrated circuits](https://en.wikipedia.org/wiki/Integrated_circuit "Integrated circuit"), which are follow-on technology in solid-state physics (vol **II**, pp. 8–6), and [lasers](https://en.wikipedia.org/wiki/Laser "Laser") (vol **III**, pp. 9–13). 3. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-46)** See *[Macroscopic quantum phenomena](https://en.wikipedia.org/wiki/Macroscopic_quantum_phenomena "Macroscopic quantum phenomena")*, *[Bose–Einstein condensate](https://en.wikipedia.org/wiki/Bose%E2%80%93Einstein_condensate "Bose–Einstein condensate")*, and *[Quantum machine](https://en.wikipedia.org/wiki/Quantum_machine "Quantum machine")* 4. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-68)** The published form of the EPR argument was due to Podolsky, and Einstein himself was not satisfied with it. In his own publications and correspondence, Einstein used a different argument to insist that quantum mechanics is an incomplete theory.[\[61\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-spekkens-64)[\[62\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-howard-65)[\[63\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-66)[\[64\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-67) ## References 1. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Born1926_1-0)** [Born, M.](https://en.wikipedia.org/wiki/Max_Born "Max Born") (1926). "Zur Quantenmechanik der Stoßvorgänge" \[On the Quantum Mechanics of Collision Processes\]. *Zeitschrift für Physik* (in German). **37** (12): 863–867\. [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[1926ZPhy...37..863B](https://ui.adsabs.harvard.edu/abs/1926ZPhy...37..863B). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1007/BF01397477](https://doi.org/10.1007%2FBF01397477). [ISSN](https://en.wikipedia.org/wiki/ISSN_\(identifier\) "ISSN (identifier)") [1434-6001](https://search.worldcat.org/issn/1434-6001). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [119896026](https://api.semanticscholar.org/CorpusID:119896026). 2. ^ [***a***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Feynman_2-0) [***b***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Feynman_2-1) [***c***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Feynman_2-2) [***d***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Feynman_2-3) Feynman, Richard; Leighton, Robert; Sands, Matthew (1964). [*The Feynman Lectures on Physics*](https://feynmanlectures.caltech.edu/III_01.html). Vol. 3. California Institute of Technology. Retrieved 19 December 2020. Reprinted, Addison-Wesley, 1989, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-201-50064-6](https://en.wikipedia.org/wiki/Special:BookSources/978-0-201-50064-6 "Special:BookSources/978-0-201-50064-6") 3. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-3)** Jaeger, Gregg (September 2014). "What in the (quantum) world is macroscopic?". *American Journal of Physics*. **82** (9): 896–905\. [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[2014AmJPh..82..896J](https://ui.adsabs.harvard.edu/abs/2014AmJPh..82..896J). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1119/1.4878358](https://doi.org/10.1119%2F1.4878358). 4. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-4)** Fein, Yaakov Y.; Geyer, Philipp; Zwick, Patrick; Kiałka, Filip; Pedalino, Sebastian; Mayor, Marcel; Gerlich, Stefan; Arndt, Markus (September 2019). "Quantum superposition of molecules beyond 25 kDa". *Nature Physics*. **15** (12): 1242–1245\. [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[2019NatPh..15.1242F](https://ui.adsabs.harvard.edu/abs/2019NatPh..15.1242F). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1038/s41567-019-0663-9](https://doi.org/10.1038%2Fs41567-019-0663-9). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [203638258](https://api.semanticscholar.org/CorpusID:203638258). 5. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-5)** Bojowald, Martin (2015). "Quantum cosmology: a review". *Reports on Progress in Physics*. **78** (2) 023901. [arXiv](https://en.wikipedia.org/wiki/ArXiv_\(identifier\) "ArXiv (identifier)"):[1501\.04899](https://arxiv.org/abs/1501.04899). 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[PMID](https://en.wikipedia.org/wiki/PMID_\(identifier\) "PMID (identifier)") [36867820](https://pubmed.ncbi.nlm.nih.gov/36867820). 7. ^ [***a***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Zwiebach2022_7-0) [***b***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Zwiebach2022_7-1) [***c***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Zwiebach2022_7-2) [***d***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Zwiebach2022_7-3) [***e***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Zwiebach2022_7-4) [***f***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Zwiebach2022_7-5) [***g***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Zwiebach2022_7-6) [***h***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Zwiebach2022_7-7) [***i***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Zwiebach2022_7-8) [***j***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Zwiebach2022_7-9) [Zwiebach, Barton](https://en.wikipedia.org/wiki/Barton_Zwiebach "Barton Zwiebach") (2022). *Mastering Quantum Mechanics: Essentials, Theory, and Applications*. MIT Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-262-04613-8](https://en.wikipedia.org/wiki/Special:BookSources/978-0-262-04613-8 "Special:BookSources/978-0-262-04613-8") . 8. ^ [***a***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Lederman_8-0) [***b***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Lederman_8-1) [***c***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Lederman_8-2) Lederman, Leon M.; Hill, Christopher T. (2011). [*Quantum Physics for Poets*](https://books.google.com/books?id=qY_yOwHg_WYC&pg=PA102). US: Prometheus Books. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-1-61614-281-0](https://en.wikipedia.org/wiki/Special:BookSources/978-1-61614-281-0 "Special:BookSources/978-1-61614-281-0") . 9. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-M%C3%BCller-Kirsten_9-0)** Müller-Kirsten, H. J. W. (2006). [*Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral*](https://books.google.com/books?id=p1_Z81Le58MC&pg=PA14). US: World Scientific. p. 14. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-981-256-691-1](https://en.wikipedia.org/wiki/Special:BookSources/978-981-256-691-1 "Special:BookSources/978-981-256-691-1") . 10. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Plotnitsky_10-0)** Plotnitsky, Arkady (2012). [*Niels Bohr and Complementarity: An Introduction*](https://books.google.com/books?id=dmdUp97S4AYC&pg=PA75). US: Springer. pp. 75–76\. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-1-4614-4517-3](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4614-4517-3 "Special:BookSources/978-1-4614-4517-3") . 11. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-11)** [Griffiths, David J.](https://en.wikipedia.org/wiki/David_J._Griffiths "David J. Griffiths") (1995). [*Introduction to Quantum Mechanics*](https://en.wikipedia.org/wiki/Introduction_to_Quantum_Mechanics_\(book\) "Introduction to Quantum Mechanics (book)"). Prentice Hall. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [0-13-124405-1](https://en.wikipedia.org/wiki/Special:BookSources/0-13-124405-1 "Special:BookSources/0-13-124405-1") . 12. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Trixler2013_12-0)** Trixler, F. (2013). ["Quantum tunnelling to the origin and evolution of life"](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3768233). *Current Organic Chemistry*. **17** (16): 1758–1770\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.2174/13852728113179990083](https://doi.org/10.2174%2F13852728113179990083). [PMC](https://en.wikipedia.org/wiki/PMC_\(identifier\) "PMC (identifier)") [3768233](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3768233). [PMID](https://en.wikipedia.org/wiki/PMID_\(identifier\) "PMID (identifier)") [24039543](https://pubmed.ncbi.nlm.nih.gov/24039543). 13. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-13)** Phifer, Arnold (2012-03-27). ["Developing more energy-efficient transistors through quantum tunneling"](https://news.nd.edu/news/developing-more-energy-efficient-transistors-through-quantum-tunneling/). *Notre Dame News*. Retrieved 2024-06-07. 14. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-14)** [Bub, Jeffrey](https://en.wikipedia.org/wiki/Jeffrey_Bub "Jeffrey Bub") (2019). ["Quantum entanglement"](https://plato.stanford.edu/entries/qt-entangle/). In Zalta, Edward N. (ed.). *Stanford Encyclopedia of Philosophy*. Metaphysics Research Lab, Stanford University. 15. ^ [***a***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Caves_15-0) [***b***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Caves_15-1) [Caves, Carlton M.](https://en.wikipedia.org/wiki/Carlton_M._Caves "Carlton M. Caves") (2015). "Quantum Information Science: Emerging No More". In Kelley, Paul; Agrawal, Govind; Bass, Mike; Hecht, Jeff; Stroud, Carlos (eds.). *OSA Century of Optics*. [The Optical Society](https://en.wikipedia.org/wiki/The_Optical_Society "The Optical Society"). pp. 320–323\. [arXiv](https://en.wikipedia.org/wiki/ArXiv_\(identifier\) "ArXiv (identifier)"):[1302\.1864](https://arxiv.org/abs/1302.1864). [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[2013arXiv1302.1864C](https://ui.adsabs.harvard.edu/abs/2013arXiv1302.1864C). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-1-943580-04-0](https://en.wikipedia.org/wiki/Special:BookSources/978-1-943580-04-0 "Special:BookSources/978-1-943580-04-0") . 16. ^ [***a***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-wiseman15_16-0) [***b***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-wiseman15_16-1) [Wiseman, Howard](https://en.wikipedia.org/wiki/Howard_M._Wiseman "Howard M. Wiseman") (October 2015). ["Death by experiment for local realism"](https://doi.org/10.1038%2Fnature15631). *[Nature](https://en.wikipedia.org/wiki/Nature_\(journal\) "Nature (journal)")*. **526** (7575): 649–650\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1038/nature15631](https://doi.org/10.1038%2Fnature15631). [ISSN](https://en.wikipedia.org/wiki/ISSN_\(identifier\) "ISSN (identifier)") [0028-0836](https://search.worldcat.org/issn/0028-0836). [PMID](https://en.wikipedia.org/wiki/PMID_\(identifier\) "PMID (identifier)") [26503054](https://pubmed.ncbi.nlm.nih.gov/26503054). 17. ^ [***a***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-wolchover17_17-0) [***b***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-wolchover17_17-1) [Wolchover, Natalie](https://en.wikipedia.org/wiki/Natalie_Wolchover "Natalie Wolchover") (7 February 2017). ["Experiment Reaffirms Quantum Weirdness"](https://www.quantamagazine.org/20170207-bell-test-quantum-loophole/). *[Quanta Magazine](https://en.wikipedia.org/wiki/Quanta_Magazine "Quanta Magazine")*. Retrieved 8 February 2020. 18. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-18)** [Baez, John C.](https://en.wikipedia.org/wiki/John_C._Baez "John C. Baez") (20 March 2020). ["How to Learn Math and Physics"](https://math.ucr.edu/home/baez/books.html). *University of California, Riverside*. Retrieved 19 December 2020. "there's no way to understand the interpretation of quantum mechanics without also being able to *solve quantum mechanics problems* – to understand the theory, you need to be able to use it (and vice versa)" 19. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-19)** [Sagan, Carl](https://en.wikipedia.org/wiki/Carl_Sagan "Carl Sagan") (1996). [*The Demon-Haunted World: Science as a Candle in the Dark*](https://en.wikipedia.org/wiki/The_Demon-Haunted_World "The Demon-Haunted World"). Ballantine Books. p. 249. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [0-345-40946-9](https://en.wikipedia.org/wiki/Special:BookSources/0-345-40946-9 "Special:BookSources/0-345-40946-9") . ""For most physics students, (the "mathematical underpinning" of quantum mechanics) might occupy them from, say, third grade to early graduate school – roughly 15 years. ... The job of the popularizer of science, trying to get across some idea of quantum mechanics to a general audience that has not gone through these initiation rites, is daunting. Indeed, there are no successful popularizations of quantum mechanics in my opinion – partly for this reason." 20. ^ [***a***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Holevo2001_20-0) [***b***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Holevo2001_20-1) [***c***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Holevo2001_20-2) [Holevo, Alexander S.](https://en.wikipedia.org/wiki/Alexander_Holevo "Alexander Holevo") (2001). *Statistical Structure of Quantum Theory*. Lecture Notes in Physics. Springer. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [3-540-42082-7](https://en.wikipedia.org/wiki/Special:BookSources/3-540-42082-7 "Special:BookSources/3-540-42082-7") . [OCLC](https://en.wikipedia.org/wiki/OCLC_\(identifier\) "OCLC (identifier)") [318268606](https://search.worldcat.org/oclc/318268606). 21. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-21)** Hall, Brian C. (2013). *Quantum Theory for Mathematicians*. [Graduate Texts in Mathematics](https://en.wikipedia.org/wiki/Graduate_Texts_in_Mathematics "Graduate Texts in Mathematics"). Vol. 267. Springer. p. 125. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-1-4614-7115-8](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4614-7115-8 "Special:BookSources/978-1-4614-7115-8") . 22. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-22)** Weinberg, Steven (2010). [*Dreams Of A Final Theory: The Search for The Fundamental Laws of Nature*](https://books.google.com/books?id=OLrZkgPsZR0C). Random House. p. [82](https://books.google.com/books?id=OLrZkgPsZR0C&pg=PT82). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-1-4070-6396-6](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4070-6396-6 "Special:BookSources/978-1-4070-6396-6") . 23. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-23)** Zhang, Ruiqin; Deng, Conghao (1993). "Exact solutions of the Schrödinger equation for some quantum-mechanical many-body systems". *Physical Review A*. **47** (1): 71–77\. [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[1993PhRvA..47...71Z](https://ui.adsabs.harvard.edu/abs/1993PhRvA..47...71Z). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1103/PhysRevA.47.71](https://doi.org/10.1103%2FPhysRevA.47.71). [ISSN](https://en.wikipedia.org/wiki/ISSN_\(identifier\) "ISSN (identifier)") [1050-2947](https://search.worldcat.org/issn/1050-2947). [PMID](https://en.wikipedia.org/wiki/PMID_\(identifier\) "PMID (identifier)") [9908895](https://pubmed.ncbi.nlm.nih.gov/9908895). 24. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-24)** Li, Jing; Drummond, N. D.; Schuck, Peter; Olevano, Valerio (2019-04-01). ["Comparing many-body approaches against the helium atom exact solution"](https://doi.org/10.21468%2FSciPostPhys.6.4.040). *SciPost Physics*. **6** (4): 40. [arXiv](https://en.wikipedia.org/wiki/ArXiv_\(identifier\) "ArXiv (identifier)"):[1801\.09977](https://arxiv.org/abs/1801.09977). [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[2019ScPP....6...40L](https://ui.adsabs.harvard.edu/abs/2019ScPP....6...40L). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.21468/SciPostPhys.6.4.040](https://doi.org/10.21468%2FSciPostPhys.6.4.040). [ISSN](https://en.wikipedia.org/wiki/ISSN_\(identifier\) "ISSN (identifier)") [2542-4653](https://search.worldcat.org/issn/2542-4653). 25. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-25)** Drake, Gordon W. F. (2023). "High Precision Calculations for Helium". In Drake, Gordon W. F. (ed.). *Springer Handbook of Atomic, Molecular, and Optical Physics*. Springer Handbooks. Cham: Springer International Publishing. pp. 199–216\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1007/978-3-030-73893-8\_12](https://doi.org/10.1007%2F978-3-030-73893-8_12). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-3-030-73892-1](https://en.wikipedia.org/wiki/Special:BookSources/978-3-030-73892-1 "Special:BookSources/978-3-030-73892-1") . 26. ^ [***a***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Cohen-Tannoudji_26-0) [***b***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Cohen-Tannoudji_26-1) [***c***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Cohen-Tannoudji_26-2) [***d***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Cohen-Tannoudji_26-3) [Cohen-Tannoudji, Claude](https://en.wikipedia.org/wiki/Claude_Cohen-Tannoudji "Claude Cohen-Tannoudji"); Diu, Bernard; Laloë, Franck (2005). *Quantum Mechanics*. Translated by Hemley, Susan Reid; Ostrowsky, Nicole; Ostrowsky, Dan. John Wiley & Sons. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [0-471-16433-X](https://en.wikipedia.org/wiki/Special:BookSources/0-471-16433-X "Special:BookSources/0-471-16433-X") . 27. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-L&L_27-0)** [Landau, Lev D.](https://en.wikipedia.org/wiki/Lev_Landau "Lev Landau"); [Lifschitz, Evgeny M.](https://en.wikipedia.org/wiki/Evgeny_Lifshitz "Evgeny Lifshitz") (1977). [*Quantum Mechanics: Non-Relativistic Theory*](https://archive.org/details/QuantumMechanics_104). Vol. 3 (3rd ed.). [Pergamon Press](https://en.wikipedia.org/wiki/Pergamon_Press "Pergamon Press"). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-08-020940-1](https://en.wikipedia.org/wiki/Special:BookSources/978-0-08-020940-1 "Special:BookSources/978-0-08-020940-1") . [OCLC](https://en.wikipedia.org/wiki/OCLC_\(identifier\) "OCLC (identifier)") [2284121](https://search.worldcat.org/oclc/2284121). 28. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-ballentine1970_28-0)** Section 3.2 of Ballentine, Leslie E. (1970), "The Statistical Interpretation of Quantum Mechanics", *Reviews of Modern Physics*, **42** (4): 358–381, [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[1970RvMP...42..358B](https://ui.adsabs.harvard.edu/abs/1970RvMP...42..358B), [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1103/RevModPhys.42.358](https://doi.org/10.1103%2FRevModPhys.42.358), [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [120024263](https://api.semanticscholar.org/CorpusID:120024263) . This fact is experimentally well-known for example in quantum optics; see e.g. chap. 2 and Fig. 2.1 Leonhardt, Ulf (1997), [*Measuring the Quantum State of Light*](https://archive.org/details/measuringquantum0000leon), Cambridge: Cambridge University Press, [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[1997mqsl.book.....L](https://ui.adsabs.harvard.edu/abs/1997mqsl.book.....L), [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [0-521-49730-2](https://en.wikipedia.org/wiki/Special:BookSources/0-521-49730-2 "Special:BookSources/0-521-49730-2") . 29. ^ [***a***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-:0_29-0) [***b***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-:0_29-1) [***c***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-:0_29-2) [Nielsen, Michael A.](https://en.wikipedia.org/wiki/Michael_Nielsen "Michael Nielsen"); [Chuang, Isaac L.](https://en.wikipedia.org/wiki/Isaac_Chuang "Isaac Chuang") (2010). *Quantum Computation and Quantum Information* (2nd ed.). Cambridge University Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-1-107-00217-3](https://en.wikipedia.org/wiki/Special:BookSources/978-1-107-00217-3 "Special:BookSources/978-1-107-00217-3") . [OCLC](https://en.wikipedia.org/wiki/OCLC_\(identifier\) "OCLC (identifier)") [844974180](https://search.worldcat.org/oclc/844974180). 30. ^ [***a***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-:1_30-0) [***b***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-:1_30-1) [Rieffel, Eleanor G.](https://en.wikipedia.org/wiki/Eleanor_Rieffel "Eleanor Rieffel"); Polak, Wolfgang H. (2011). [*Quantum Computing: A Gentle Introduction*](https://en.wikipedia.org/wiki/Quantum_Computing:_A_Gentle_Introduction "Quantum Computing: A Gentle Introduction"). MIT Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-262-01506-6](https://en.wikipedia.org/wiki/Special:BookSources/978-0-262-01506-6 "Special:BookSources/978-0-262-01506-6") . 31. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-wilde_31-0)** Wilde, Mark M. (2017). *Quantum Information Theory* (2nd ed.). Cambridge University Press. [arXiv](https://en.wikipedia.org/wiki/ArXiv_\(identifier\) "ArXiv (identifier)"):[1106\.1445](https://arxiv.org/abs/1106.1445). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1017/9781316809976.001](https://doi.org/10.1017%2F9781316809976.001). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-1-107-17616-4](https://en.wikipedia.org/wiki/Special:BookSources/978-1-107-17616-4 "Special:BookSources/978-1-107-17616-4") . [OCLC](https://en.wikipedia.org/wiki/OCLC_\(identifier\) "OCLC (identifier)") [973404322](https://search.worldcat.org/oclc/973404322). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [2515538](https://api.semanticscholar.org/CorpusID:2515538). 32. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-32)** Schlosshauer, Maximilian (October 2019). "Quantum decoherence". *Physics Reports*. **831**: 1–57\. [arXiv](https://en.wikipedia.org/wiki/ArXiv_\(identifier\) "ArXiv (identifier)"):[1911\.06282](https://arxiv.org/abs/1911.06282). [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[2019PhR...831....1S](https://ui.adsabs.harvard.edu/abs/2019PhR...831....1S). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1016/j.physrep.2019.10.001](https://doi.org/10.1016%2Fj.physrep.2019.10.001). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [208006050](https://api.semanticscholar.org/CorpusID:208006050). 33. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-33)** [Rechenberg, Helmut](https://en.wikipedia.org/wiki/Helmut_Rechenberg "Helmut Rechenberg") (1987). ["Erwin Schrödinger and the creation of wave mechanics"](http://www.actaphys.uj.edu.pl/fulltext?series=Reg&vol=19&page=683) (PDF). *[Acta Physica Polonica B](https://en.wikipedia.org/wiki/Acta_Physica_Polonica_B "Acta Physica Polonica B")*. **19** (8): 683–695. Retrieved 13 June 2016. 34. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-34)** Feynman, Richard P.; Hibbs, Albert R. (2005). Steyer, Daniel F. (ed.). *Quantum Mechanics and Path Integrals* (Emended ed.). McGraw-Hill. pp. v–vii. 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"Entanglement and visibility at the output of a Mach–Zehnder interferometer". *[Physical Review A](https://en.wikipedia.org/wiki/Physical_Review_A "Physical Review A")*. **59** (2): 1615–1621\. [arXiv](https://en.wikipedia.org/wiki/ArXiv_\(identifier\) "ArXiv (identifier)"):[quant-ph/9811078](https://arxiv.org/abs/quant-ph/9811078). [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[1999PhRvA..59.1615P](https://ui.adsabs.harvard.edu/abs/1999PhRvA..59.1615P). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1103/PhysRevA.59.1615](https://doi.org/10.1103%2FPhysRevA.59.1615). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [13963928](https://api.semanticscholar.org/CorpusID:13963928). 37. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Haack2010_38-0)** Haack, G. R.; Förster, H.; Büttiker, M. (2010). "Parity detection and entanglement with a Mach-Zehnder interferometer". *[Physical Review B](https://en.wikipedia.org/wiki/Physical_Review_B "Physical Review B")*. **82** (15) 155303. [arXiv](https://en.wikipedia.org/wiki/ArXiv_\(identifier\) "ArXiv (identifier)"):[1005\.3976](https://arxiv.org/abs/1005.3976). [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[2010PhRvB..82o5303H](https://ui.adsabs.harvard.edu/abs/2010PhRvB..82o5303H). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1103/PhysRevB.82.155303](https://doi.org/10.1103%2FPhysRevB.82.155303). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [119261326](https://api.semanticscholar.org/CorpusID:119261326). 38. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-vedral_39-0)** [Vedral, Vlatko](https://en.wikipedia.org/wiki/Vlatko_Vedral "Vlatko Vedral") (2006). *Introduction to Quantum Information Science*. Oxford University Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-19-921570-6](https://en.wikipedia.org/wiki/Special:BookSources/978-0-19-921570-6 "Special:BookSources/978-0-19-921570-6") . [OCLC](https://en.wikipedia.org/wiki/OCLC_\(identifier\) "OCLC (identifier)") [442351498](https://search.worldcat.org/oclc/442351498). 39. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-marvincohen2008_41-0)** Cohen, Marvin L. (2008). ["Essay: Fifty Years of Condensed Matter Physics"](http://prl.aps.org/edannounce/PhysRevLett.101.250001). *Physical Review Letters*. **101** (25) 250001. [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[2008PhRvL.101y0001C](https://ui.adsabs.harvard.edu/abs/2008PhRvL.101y0001C). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1103/PhysRevLett.101.250001](https://doi.org/10.1103%2FPhysRevLett.101.250001). [PMID](https://en.wikipedia.org/wiki/PMID_\(identifier\) "PMID (identifier)") [19113681](https://pubmed.ncbi.nlm.nih.gov/19113681). Retrieved 31 March 2012. 40. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-42)** Matson, John. ["What Is Quantum Mechanics Good for?"](http://www.scientificamerican.com/article/everyday-quantum-physics/). *Scientific American*. Retrieved 18 May 2016. 41. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Tipler_43-0)** Tipler, Paul; Llewellyn, Ralph (2008). *Modern Physics* (5th ed.). W. H. Freeman and Company. pp. 160–161\. 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[OCLC](https://en.wikipedia.org/wiki/OCLC_\(identifier\) "OCLC (identifier)") [1151058062](https://search.worldcat.org/oclc/1151058062). 78. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-82)** Scheider, Walter (April 1986). ["Bringing one of the great moments of science to the classroom"](http://www.cavendishscience.org/phys/tyoung/tyoung.htm). *[The Physics Teacher](https://en.wikipedia.org/wiki/The_Physics_Teacher "The Physics Teacher")*. **24** (4): 217–219\. [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[1986PhTea..24..217S](https://ui.adsabs.harvard.edu/abs/1986PhTea..24..217S). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1119/1.2341987](https://doi.org/10.1119%2F1.2341987). [ISSN](https://en.wikipedia.org/wiki/ISSN_\(identifier\) "ISSN (identifier)") [0031-921X](https://search.worldcat.org/issn/0031-921X). 79. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Feynman-kinetic-theory_83-0)** Feynman, Richard; Leighton, Robert; Sands, Matthew (1964). [*The Feynman Lectures on Physics*](https://feynmanlectures.caltech.edu/I_40.html). Vol. 1. California Institute of Technology. Retrieved 30 September 2021. Reprinted, Addison-Wesley, 1989, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-201-50064-6](https://en.wikipedia.org/wiki/Special:BookSources/978-0-201-50064-6 "Special:BookSources/978-0-201-50064-6") 80. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-84)** Martin, Andre (1986), "Cathode Ray Tubes for Industrial and Military Applications", in Hawkes, Peter (ed.), *Advances in Electronics and Electron Physics, Volume 67*, Academic Press, p. 183, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-08-057733-3](https://en.wikipedia.org/wiki/Special:BookSources/978-0-08-057733-3 "Special:BookSources/978-0-08-057733-3") , "Evidence for the existence of "cathode-rays" was first found by Plücker and Hittorf ..." 81. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-85)** Dahl, Per F. (1997). [*Flash of the Cathode Rays: A History of J. J. Thomson's Electron*](https://books.google.com/books?id=xUzaWGocMdMC). CRC Press. pp. 47–57\. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-7503-0453-5](https://en.wikipedia.org/wiki/Special:BookSources/978-0-7503-0453-5 "Special:BookSources/978-0-7503-0453-5") . 82. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-86)** [Mehra, J.](https://en.wikipedia.org/wiki/Jagdish_Mehra "Jagdish Mehra"); Rechenberg, H. (1982). *The Historical Development of Quantum Theory, Vol. 1: The Quantum Theory of Planck, Einstein, Bohr and Sommerfeld. Its Foundation and the Rise of Its Difficulties (1900–1925)*. New York: Springer-Verlag. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-387-90642-3](https://en.wikipedia.org/wiki/Special:BookSources/978-0-387-90642-3 "Special:BookSources/978-0-387-90642-3") . 83. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-87)** ["Quantum"](http://www.merriam-webster.com/dictionary/quantum). *Merriam-Webster Dictionary*. [Archived](https://web.archive.org/web/20121026104456/http://www.merriam-webster.com/dictionary/quantum) from the original on Oct 26, 2012. Retrieved 18 August 2012. 84. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-88)** [Kuhn, T. S.](https://en.wikipedia.org/wiki/Thomas_Samuel_Kuhn "Thomas Samuel Kuhn") (1978). *Black-body theory and the quantum discontinuity 1894–1912*. Oxford: Clarendon Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-19-502383-1](https://en.wikipedia.org/wiki/Special:BookSources/978-0-19-502383-1 "Special:BookSources/978-0-19-502383-1") . 85. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Kragh_89-0)** [Kragh, Helge](https://en.wikipedia.org/wiki/Helge_Kragh "Helge Kragh") (1 December 2000). ["Max Planck: the reluctant revolutionary"](https://physicsworld.com/a/max-planck-the-reluctant-revolutionary/). *[Physics World](https://en.wikipedia.org/wiki/Physics_World "Physics World")*. Retrieved 12 December 2020. 86. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-90)** [Stachel, John](https://en.wikipedia.org/wiki/John_Stachel "John Stachel") (2009). "Bohr and the Photon". *Quantum Reality, Relativistic Causality, and Closing the Epistemic Circle*. The Western Ontario Series in Philosophy of Science. Vol. 73. Dordrecht: Springer. pp. 69–83\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1007/978-1-4020-9107-0\_5](https://doi.org/10.1007%2F978-1-4020-9107-0_5). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-1-4020-9106-3](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4020-9106-3 "Special:BookSources/978-1-4020-9106-3") . 87. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-91)** Einstein, Albert (1905). ["Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt"](https://doi.org/10.1002%2Fandp.19053220607) \[On a heuristic point of view concerning the production and transformation of light\]. *[Annalen der Physik](https://en.wikipedia.org/wiki/Annalen_der_Physik "Annalen der Physik")* (in German). **17** (6): 132–148\. [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[1905AnP...322..132E](https://ui.adsabs.harvard.edu/abs/1905AnP...322..132E). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1002/andp.19053220607](https://doi.org/10.1002%2Fandp.19053220607). Reprinted in [Stachel, John](https://en.wikipedia.org/wiki/John_Stachel "John Stachel"), ed. (1989). *The Collected Papers of Albert Einstein* (in German). Vol. 2. Princeton University Press. pp. 149–166\. See also "Einstein's early work on the quantum hypothesis", ibid. pp. 134–148. 88. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-92)** [Einstein, Albert](https://en.wikipedia.org/wiki/Albert_Einstein "Albert Einstein") (1917). "Zur Quantentheorie der Strahlung" \[On the Quantum Theory of Radiation\]. *[Physikalische Zeitschrift](https://en.wikipedia.org/wiki/Physikalische_Zeitschrift "Physikalische Zeitschrift")* (in German). **18**: 121–128\. [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[1917PhyZ...18..121E](https://ui.adsabs.harvard.edu/abs/1917PhyZ...18..121E). Translated in Einstein, A. (1967). "On the Quantum Theory of Radiation". *The Old Quantum Theory*. Elsevier. pp. 167–183\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1016/b978-0-08-012102-4.50018-8](https://doi.org/10.1016%2Fb978-0-08-012102-4.50018-8). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-08-012102-4](https://en.wikipedia.org/wiki/Special:BookSources/978-0-08-012102-4 "Special:BookSources/978-0-08-012102-4") . 89. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-93)** [Ball, Philip](https://en.wikipedia.org/wiki/Philip_Ball "Philip Ball") (2017-08-31). ["A century ago Einstein sparked the notion of the laser"](https://physicsworld.com/a/a-century-ago-einstein-sparked-the-notion-of-the-laser/). *[Physics World](https://en.wikipedia.org/wiki/Physics_World "Physics World")*. Retrieved 2024-03-23. 90. ^ [***a***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-terHaar_94-0) [***b***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-terHaar_94-1) ter Haar, D. (1967). [*The Old Quantum Theory*](https://archive.org/details/oldquantumtheory0000haar). Pergamon Press. pp. 3–75\. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-08-012101-7](https://en.wikipedia.org/wiki/Special:BookSources/978-0-08-012101-7 "Special:BookSources/978-0-08-012101-7") . [LCCN](https://en.wikipedia.org/wiki/LCCN_\(identifier\) "LCCN (identifier)") [66-29628](https://lccn.loc.gov/66-29628). 91. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-95)** Bokulich, Alisa; Bokulich, Peter (2020-08-13). ["Bohr's Correspondence Principle"](https://plato.stanford.edu/entries/bohr-correspondence/). In [Zalta, Edward N.](https://en.wikipedia.org/wiki/Edward_N._Zalta "Edward N. Zalta") (ed.). *[Stanford Encyclopedia of Philosophy](https://en.wikipedia.org/wiki/Stanford_Encyclopedia_of_Philosophy "Stanford Encyclopedia of Philosophy")*. [ISSN](https://en.wikipedia.org/wiki/ISSN_\(identifier\) "ISSN (identifier)") [1095-5054](https://search.worldcat.org/issn/1095-5054). [OCLC](https://en.wikipedia.org/wiki/OCLC_\(identifier\) "OCLC (identifier)") [429049174](https://search.worldcat.org/oclc/429049174). 92. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-96)** ["Semi-classical approximation"](https://web.archive.org/web/20221007190530/https://encyclopediaofmath.org/index.php?title=Semi-classical_approximation). *Encyclopedia of Mathematics*. Archived from [the original](https://www.encyclopediaofmath.org/index.php?title=Semi-classical_approximation) on 7 October 2022. Retrieved 1 February 2020. 93. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-97)** [Sakurai, J. J.](https://en.wikipedia.org/wiki/J._J._Sakurai "J. J. Sakurai"); Napolitano, J. (2014). "Quantum Dynamics". [*Modern Quantum Mechanics*](https://en.wikipedia.org/wiki/Modern_Quantum_Mechanics "Modern Quantum Mechanics"). Pearson. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-1-292-02410-3](https://en.wikipedia.org/wiki/Special:BookSources/978-1-292-02410-3 "Special:BookSources/978-1-292-02410-3") . [OCLC](https://en.wikipedia.org/wiki/OCLC_\(identifier\) "OCLC (identifier)") [929609283](https://search.worldcat.org/oclc/929609283). 94. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Aharoni_98-0)** [Aharoni, Amikam](https://en.wikipedia.org/wiki/Amikam_Aharoni "Amikam Aharoni") (1996). [*Introduction to the Theory of Ferromagnetism*](https://archive.org/details/introductiontoth00ahar/page/6). [Clarendon Press](https://en.wikipedia.org/wiki/Clarendon_Press "Clarendon Press"). pp. [6–7](https://archive.org/details/introductiontoth00ahar/page/6). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [0-19-851791-2](https://en.wikipedia.org/wiki/Special:BookSources/0-19-851791-2 "Special:BookSources/0-19-851791-2") . 95. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Edwards79_99-0)** David Edwards, "The Mathematical Foundations of Quantum Mechanics", *Synthese*, Volume 42, Number 1/September, 1979, pp. 1–70. 96. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Edwards81_100-0)** David Edwards, "The Mathematical Foundations of Quantum Field Theory: Fermions, Gauge Fields, and Super-symmetry, Part I: Lattice Field Theories", *International Journal of Theoretical Physics*, Vol. 20, No. 7 (1981). 97. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-101)** [Bernstein, Jeremy](https://en.wikipedia.org/wiki/Jeremy_Bernstein "Jeremy Bernstein") (November 2005). ["Max Born and the quantum theory"](https://doi.org/10.1119%2F1.2060717). *[American Journal of Physics](https://en.wikipedia.org/wiki/American_Journal_of_Physics "American Journal of Physics")*. **73** (11): 999–1008\. [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[2005AmJPh..73..999B](https://ui.adsabs.harvard.edu/abs/2005AmJPh..73..999B). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1119/1.2060717](https://doi.org/10.1119%2F1.2060717). [ISSN](https://en.wikipedia.org/wiki/ISSN_\(identifier\) "ISSN (identifier)") [0002-9505](https://search.worldcat.org/issn/0002-9505). 98. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-pais1997_102-0)** [Pais, Abraham](https://en.wikipedia.org/wiki/Abraham_Pais "Abraham Pais") (1997). [*A Tale of Two Continents: A Physicist's Life in a Turbulent World*](https://archive.org/details/taleoftwocontine00pais). Princeton, New Jersey: Princeton University Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [0-691-01243-1](https://en.wikipedia.org/wiki/Special:BookSources/0-691-01243-1 "Special:BookSources/0-691-01243-1") . 99. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-103)** Van Hove, Leon (1958). ["Von Neumann's contributions to quantum mechanics"](https://www.ams.org/journals/bull/1958-64-03/S0002-9904-1958-10206-2/S0002-9904-1958-10206-2.pdf) (PDF). *[Bulletin of the American Mathematical Society](https://en.wikipedia.org/wiki/Bulletin_of_the_American_Mathematical_Society "Bulletin of the American Mathematical Society")*. **64** (3): Part 2:95–99. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1090/s0002-9904-1958-10206-2](https://doi.org/10.1090%2Fs0002-9904-1958-10206-2). [Archived](https://web.archive.org/web/20240120073106/https://www.ams.org/journals/bull/1958-64-03/S0002-9904-1958-10206-2/S0002-9904-1958-10206-2.pdf) (PDF) from the original on Jan 20, 2024. 100. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-feynman2015_104-0)** [Feynman, Richard](https://en.wikipedia.org/wiki/Richard_Feynman "Richard Feynman"). ["The Feynman Lectures on Physics Vol. III Ch. 21: The Schrödinger Equation in a Classical Context: A Seminar on Superconductivity, 21-4"](https://feynmanlectures.caltech.edu/III_21.html#Ch21-S5). [California Institute of Technology](https://en.wikipedia.org/wiki/California_Institute_of_Technology "California Institute of Technology"). Retrieved 24 November 2015. "...it was long believed that the wave function of the Schrödinger equation would never have a macroscopic representation analogous to the macroscopic representation of the amplitude for photons. On the other hand, it is now realized that the phenomena of superconductivity presents us with just this situation." `{{cite web}}`: CS1 maint: url-status ([link](https://en.wikipedia.org/wiki/Category:CS1_maint:_url-status "Category:CS1 maint: url-status")) 101. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-105)** Packard, Richard (2006). ["Berkeley Experiments on Superfluid Macroscopic Quantum Effects"](https://web.archive.org/web/20151125112132/http://research.physics.berkeley.edu/packard/publications/Articles/LT24_Berk_expts_on_macro_sup_effects.pdf) (PDF). Physics Department, University of California, Berkeley. Archived from [the original](http://physics.berkeley.edu/sites/default/files/_/lt24_berk_expts_on_macro_sup_effects.pdf) (PDF) on 25 November 2015. Retrieved 24 November 2015. ## Further reading The following titles, all by working physicists, attempt to communicate quantum theory to lay people, using a minimum of technical apparatus: - [Chester, Marvin](https://en.wikipedia.org/wiki/Marvin_Chester "Marvin Chester") (1987). *Primer of Quantum Mechanics*. John Wiley. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [0-486-42878-8](https://en.wikipedia.org/wiki/Special:BookSources/0-486-42878-8 "Special:BookSources/0-486-42878-8") - [Cox, Brian](https://en.wikipedia.org/wiki/Brian_Cox_\(physicist\) "Brian Cox (physicist)"); [Forshaw, Jeff](https://en.wikipedia.org/wiki/Jeff_Forshaw "Jeff Forshaw") (2011). [*The Quantum Universe: Everything That Can Happen Does Happen*](https://en.wikipedia.org/wiki/The_Quantum_Universe "The Quantum Universe"). Allen Lane. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-1-84614-432-5](https://en.wikipedia.org/wiki/Special:BookSources/978-1-84614-432-5 "Special:BookSources/978-1-84614-432-5") . - [Richard Feynman](https://en.wikipedia.org/wiki/Richard_Feynman "Richard Feynman"), 1985. *[QED: The Strange Theory of Light and Matter](https://en.wikipedia.org/wiki/QED:_The_Strange_Theory_of_Light_and_Matter "QED: The Strange Theory of Light and Matter")*, Princeton University Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [0-691-08388-6](https://en.wikipedia.org/wiki/Special:BookSources/0-691-08388-6 "Special:BookSources/0-691-08388-6") . Four elementary lectures on quantum electrodynamics and [quantum field theory](https://en.wikipedia.org/wiki/Quantum_field_theory "Quantum field theory"), yet containing many insights for the expert. - [Ghirardi, GianCarlo](https://en.wikipedia.org/wiki/Giancarlo_Ghirardi "Giancarlo Ghirardi"), 2004. *Sneaking a Look at God's Cards*, Gerald Malsbary, trans. Princeton Univ. Press. The most technical of the works cited here. Passages using [algebra](https://en.wikipedia.org/wiki/Algebra "Algebra"), [trigonometry](https://en.wikipedia.org/wiki/Trigonometry "Trigonometry"), and [bra–ket notation](https://en.wikipedia.org/wiki/Bra%E2%80%93ket_notation "Bra–ket notation") can be passed over on a first reading. - [N. David Mermin](https://en.wikipedia.org/wiki/N._David_Mermin "N. David Mermin"), 1990, "Spooky actions at a distance: mysteries of the QT" in his *Boojums All the Way Through*. Cambridge University Press: 110–76. - [Victor Stenger](https://en.wikipedia.org/wiki/Victor_Stenger "Victor Stenger"), 2000. *Timeless Reality: Symmetry, Simplicity, and Multiple Universes*. Buffalo, New York: Prometheus Books. Chpts. 5–8. Includes cosmological and philosophical considerations. More technical: - [Bernstein, Jeremy](https://en.wikipedia.org/wiki/Jeremy_Bernstein "Jeremy Bernstein") (2009). [*Quantum Leaps*](https://books.google.com/books?id=j0Me3brYOL0C). Cambridge, Massachusetts: Belknap Press of Harvard University Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-674-03541-6](https://en.wikipedia.org/wiki/Special:BookSources/978-0-674-03541-6 "Special:BookSources/978-0-674-03541-6") . - [Bohm, David](https://en.wikipedia.org/wiki/David_Bohm "David Bohm") (1989). [*Quantum Theory*](https://archive.org/details/quantumtheory0000bohm). Dover Publications. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-486-65969-5](https://en.wikipedia.org/wiki/Special:BookSources/978-0-486-65969-5 "Special:BookSources/978-0-486-65969-5") . - [Binney, James](https://en.wikipedia.org/wiki/James_Binney "James Binney"); Skinner, David (2008). *The Physics of Quantum Mechanics*. Oxford University Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-19-968857-9](https://en.wikipedia.org/wiki/Special:BookSources/978-0-19-968857-9 "Special:BookSources/978-0-19-968857-9") . - Eisberg, Robert; [Resnick, Robert](https://en.wikipedia.org/wiki/Robert_Resnick "Robert Resnick") (1985). [*Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles*](https://archive.org/details/quantumphysicsof00eisb) (2nd ed.). Wiley. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-471-87373-0](https://en.wikipedia.org/wiki/Special:BookSources/978-0-471-87373-0 "Special:BookSources/978-0-471-87373-0") . - [Bryce DeWitt](https://en.wikipedia.org/wiki/Bryce_DeWitt "Bryce DeWitt"), R. Neill Graham, eds., 1973. *The Many-Worlds Interpretation of Quantum Mechanics*, Princeton Series in Physics, Princeton University Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [0-691-08131-X](https://en.wikipedia.org/wiki/Special:BookSources/0-691-08131-X "Special:BookSources/0-691-08131-X") - [Everett, Hugh](https://en.wikipedia.org/wiki/Hugh_Everett "Hugh Everett") (1957). "Relative State Formulation of Quantum Mechanics". *Reviews of Modern Physics*. **29** (3): 454–462\. [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[1957RvMP...29..454E](https://ui.adsabs.harvard.edu/abs/1957RvMP...29..454E). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1103/RevModPhys.29.454](https://doi.org/10.1103%2FRevModPhys.29.454). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [17178479](https://api.semanticscholar.org/CorpusID:17178479). - [Feynman, Richard P.](https://en.wikipedia.org/wiki/Richard_Feynman "Richard Feynman"); [Leighton, Robert B.](https://en.wikipedia.org/wiki/Robert_B._Leighton "Robert B. Leighton"); Sands, Matthew (1965). [*The Feynman Lectures on Physics*](https://en.wikipedia.org/wiki/The_Feynman_Lectures_on_Physics "The Feynman Lectures on Physics"). Vol. 1–3\. Addison-Wesley. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-7382-0008-8](https://en.wikipedia.org/wiki/Special:BookSources/978-0-7382-0008-8 "Special:BookSources/978-0-7382-0008-8") . - [D. Greenberger](https://en.wikipedia.org/wiki/Daniel_Greenberger "Daniel Greenberger"), [K. Hentschel](https://en.wikipedia.org/wiki/Klaus_Hentschel "Klaus Hentschel"), F. Weinert, eds., 2009. *Compendium of quantum physics, Concepts, experiments, history and philosophy*, Springer-Verlag, Berlin, Heidelberg. Short articles on many QM topics. - [Griffiths, David J.](https://en.wikipedia.org/wiki/David_J._Griffiths "David J. Griffiths") (2004). *[Introduction to Quantum Mechanics](https://en.wikipedia.org/wiki/Introduction_to_Quantum_Mechanics_\(book\) "Introduction to Quantum Mechanics (book)")* (2nd ed.). Prentice Hall. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-13-111892-8](https://en.wikipedia.org/wiki/Special:BookSources/978-0-13-111892-8 "Special:BookSources/978-0-13-111892-8") . [OCLC](https://en.wikipedia.org/wiki/OCLC_\(identifier\) "OCLC (identifier)") [40251748](https://search.worldcat.org/oclc/40251748). A standard undergraduate text. - [Max Jammer](https://en.wikipedia.org/wiki/Max_Jammer "Max Jammer"), 1966. *The Conceptual Development of Quantum Mechanics*. McGraw Hill. - [Hagen Kleinert](https://en.wikipedia.org/wiki/Hagen_Kleinert "Hagen Kleinert"), 2004. *Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets*, 3rd ed. Singapore: World Scientific. [Draft of 4th edition.](http://www.physik.fu-berlin.de/~kleinert/b5) [Archived](https://web.archive.org/web/20080615134934/http://www.physik.fu-berlin.de/~kleinert/b5) 2008-06-15 at the [Wayback Machine](https://en.wikipedia.org/wiki/Wayback_Machine "Wayback Machine") - Landau, L. D.; Lifshitz, E. M. (1977). *Quantum Mechanics: Non-Relativistic Theory*. Vol. 3 (3rd ed.). [Pergamon Press](https://en.wikipedia.org/wiki/Pergamon_Press "Pergamon Press"). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-08-020940-1](https://en.wikipedia.org/wiki/Special:BookSources/978-0-08-020940-1 "Special:BookSources/978-0-08-020940-1") . [Online copy](https://archive.org/details/QuantumMechanics_104) - [Liboff, Richard L.](https://en.wikipedia.org/wiki/Liboff,_Richard "Liboff, Richard") (2002). *Introductory Quantum Mechanics*. Addison-Wesley. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-8053-8714-8](https://en.wikipedia.org/wiki/Special:BookSources/978-0-8053-8714-8 "Special:BookSources/978-0-8053-8714-8") . - Gunther Ludwig, 1968. *Wave Mechanics*. London: Pergamon Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [0-08-203204-1](https://en.wikipedia.org/wiki/Special:BookSources/0-08-203204-1 "Special:BookSources/0-08-203204-1") - [George Mackey](https://en.wikipedia.org/wiki/George_Mackey "George Mackey") (2004). *The mathematical foundations of quantum mechanics*. Dover Publications. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [0-486-43517-2](https://en.wikipedia.org/wiki/Special:BookSources/0-486-43517-2 "Special:BookSources/0-486-43517-2") . - [Merzbacher, Eugen](https://en.wikipedia.org/wiki/Eugen_Merzbacher "Eugen Merzbacher") (1998). *Quantum Mechanics*. Wiley, John & Sons, Inc. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-471-88702-7](https://en.wikipedia.org/wiki/Special:BookSources/978-0-471-88702-7 "Special:BookSources/978-0-471-88702-7") . - [Albert Messiah](https://en.wikipedia.org/wiki/Albert_Messiah "Albert Messiah"), 1966. *Quantum Mechanics* (Vol. I), English translation from French by G. M. Temmer. North Holland, John Wiley & Sons. Cf. chpt. IV, section III. - [Omnès, Roland](https://en.wikipedia.org/wiki/Roland_Omn%C3%A8s "Roland Omnès") (1999). [*Understanding Quantum Mechanics*](https://archive.org/details/understandingqua00omne). Princeton University Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-691-00435-8](https://en.wikipedia.org/wiki/Special:BookSources/978-0-691-00435-8 "Special:BookSources/978-0-691-00435-8") . [OCLC](https://en.wikipedia.org/wiki/OCLC_\(identifier\) "OCLC (identifier)") [39849482](https://search.worldcat.org/oclc/39849482). - [Scerri, Eric. R.](https://en.wikipedia.org/wiki/Eric_R._Scerri "Eric R. Scerri") (2006). *The Periodic Table: Its Story and Its Significance*. Oxford University Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [0-19-530573-6](https://en.wikipedia.org/wiki/Special:BookSources/0-19-530573-6 "Special:BookSources/0-19-530573-6") . Considers the extent to which chemistry and the periodic system have been reduced to quantum mechanics. - [Schiff, Leonard I.](https://en.wikipedia.org/wiki/Leonard_I._Schiff "Leonard I. Schiff") (1955). *Quantum Mechanics*. McGraw Hill. - [Shankar, R.](https://en.wikipedia.org/wiki/Ramamurti_Shankar "Ramamurti Shankar") (1994). *Principles of Quantum Mechanics*. Springer. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-306-44790-7](https://en.wikipedia.org/wiki/Special:BookSources/978-0-306-44790-7 "Special:BookSources/978-0-306-44790-7") . - [Stone, A. Douglas](https://en.wikipedia.org/wiki/A._Douglas_Stone "A. Douglas Stone") (2013). [*Einstein and the Quantum*](https://archive.org/details/einsteinquantumq0000ston). Princeton University Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-691-13968-5](https://en.wikipedia.org/wiki/Special:BookSources/978-0-691-13968-5 "Special:BookSources/978-0-691-13968-5") . - *What is Quantum Mechanics? A Physics Adventure*. Boston: [Transnational College](https://en.wikipedia.org/w/index.php?title=Transnational_College&action=edit&redlink=1 "Transnational College (page does not exist)"), Language Research Foundation. 1996. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-9643504-1-0](https://en.wikipedia.org/wiki/Special:BookSources/978-0-9643504-1-0 "Special:BookSources/978-0-9643504-1-0") . [OCLC](https://en.wikipedia.org/wiki/OCLC_\(identifier\) "OCLC (identifier)") [34661512](https://search.worldcat.org/oclc/34661512). - [Veltman, Martinus J. G.](https://en.wikipedia.org/wiki/Martinus_J._G._Veltman "Martinus J. G. Veltman") (2003), *Facts and Mysteries in Elementary Particle Physics*. ## External links **Quantum mechanics** at Wikipedia's [sister projects](https://en.wikipedia.org/wiki/Wikipedia:Wikimedia_sister_projects "Wikipedia:Wikimedia sister projects") - [![Wiktionary logo](https://upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/40px-Wiktionary-logo-en-v2.svg.png)](https://en.wikipedia.org/wiki/File:Wiktionary-logo-en-v2.svg)[Definitions](https://en.wiktionary.org/wiki/Special:Search/Quantum_mechanics "wikt:Special:Search/Quantum mechanics") from Wiktionary - [![Wikimedia Commons logo](https://upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/20px-Commons-logo.svg.png)](https://en.wikipedia.org/wiki/File:Commons-logo.svg)[Media](https://commons.wikimedia.org/wiki/Category:Quantum_mechanics "c:Category:Quantum mechanics") from Commons - [![Wikinews logo](https://upload.wikimedia.org/wikipedia/commons/thumb/2/24/Wikinews-logo.svg/40px-Wikinews-logo.svg.png)](https://en.wikipedia.org/wiki/File:Wikinews-logo.svg)[News](https://en.wikinews.org/wiki/Special:Search/Quantum_mechanics "n:Special:Search/Quantum mechanics") from Wikinews - ![](https://upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/40px-Wikiquote-logo.svg.png)[Quotations](https://en.wikiquote.org/wiki/Quantum_mechanics "q:Quantum mechanics") from Wikiquote - [![Wikisource logo](https://upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/40px-Wikisource-logo.svg.png)](https://en.wikipedia.org/wiki/File:Wikisource-logo.svg)[Texts](https://en.wikisource.org/wiki/Quantum_mechanics "s:Quantum mechanics") from Wikisource - [![Wikibooks logo](https://upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/40px-Wikibooks-logo.svg.png)](https://en.wikipedia.org/wiki/File:Wikibooks-logo.svg)[Textbooks](https://en.wikibooks.org/wiki/Quantum_Mechanics "b:Quantum Mechanics") from Wikibooks - [![Wikiversity logo](https://upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/40px-Wikiversity_logo_2017.svg.png)](https://en.wikipedia.org/wiki/File:Wikiversity_logo_2017.svg)[Resources](https://en.wikiversity.org/wiki/Quantum_mechanics "v:Quantum mechanics") from Wikiversity - [Introduction to quantum mechanics by Timon Idema](https://interactivetextbooks.tudelft.nl/introduction-to-quantum-mechanics/) - [Quantum Physics Made Relatively Simple](https://bethe.cornell.edu/): three video lectures by [Hans Bethe](https://en.wikipedia.org/wiki/Hans_Bethe "Hans Bethe"). **Course material** - [Quantum Cook Book](http://oyc.yale.edu/sites/default/files/notes_quantum_cookbook.pdf) and [PHYS 201: Fundamentals of Physics II](http://oyc.yale.edu/physics/phys-201#sessions) by [Ramamurti Shankar](https://en.wikipedia.org/wiki/Ramamurti_Shankar "Ramamurti Shankar"), Yale OpenCourseware. - *[Modern Physics: With waves, thermodynamics, and optics](https://lightandmatter.com/mod/)* – an online textbook. - [MIT OpenCourseWare](https://en.wikipedia.org/wiki/MIT_OpenCourseWare "MIT OpenCourseWare"): [Chemistry](https://ocw.mit.edu/courses/chemistry/) and [Physics](https://ocw.mit.edu/courses/physics/). See [8\.04](https://ocw.mit.edu/courses/physics/8-04-quantum-physics-i-spring-2016/), [8\.05](https://ocw.mit.edu/courses/physics/8-05-quantum-physics-ii-fall-2013/index.htm) and [8\.06](https://ocw.mit.edu/courses/physics/8-06-quantum-physics-iii-spring-2018/index.htm). - [⁠5\+1/2⁠ Examples in Quantum Mechanics](http://physics.csbsju.edu/QM/). **Philosophy** - Ismael, Jenann. ["Quantum Mechanics"](https://plato.stanford.edu/entries/qm/). In [Zalta, Edward N.](https://en.wikipedia.org/wiki/Edward_N._Zalta "Edward N. Zalta") (ed.). *[Stanford Encyclopedia of Philosophy](https://en.wikipedia.org/wiki/Stanford_Encyclopedia_of_Philosophy "Stanford Encyclopedia of Philosophy")*. [ISSN](https://en.wikipedia.org/wiki/ISSN_\(identifier\) "ISSN (identifier)") [1095-5054](https://search.worldcat.org/issn/1095-5054). [OCLC](https://en.wikipedia.org/wiki/OCLC_\(identifier\) "OCLC (identifier)") [429049174](https://search.worldcat.org/oclc/429049174). - [Zalta, Edward N.](https://en.wikipedia.org/wiki/Edward_N._Zalta "Edward N. Zalta") (ed.). ["Philosophical Issues in Quantum Theory"](https://plato.stanford.edu/entries/qt-issues/). *[Stanford Encyclopedia of Philosophy](https://en.wikipedia.org/wiki/Stanford_Encyclopedia_of_Philosophy "Stanford Encyclopedia of Philosophy")*. [ISSN](https://en.wikipedia.org/wiki/ISSN_\(identifier\) "ISSN (identifier)") [1095-5054](https://search.worldcat.org/issn/1095-5054). [OCLC](https://en.wikipedia.org/wiki/OCLC_\(identifier\) "OCLC (identifier)") [429049174](https://search.worldcat.org/oclc/429049174). | [v](https://en.wikipedia.org/wiki/Template:Quantum_mechanics_topics "Template:Quantum mechanics topics") [t](https://en.wikipedia.org/wiki/Template_talk:Quantum_mechanics_topics "Template talk:Quantum mechanics topics") [e](https://en.wikipedia.org/wiki/Special:EditPage/Template:Quantum_mechanics_topics "Special:EditPage/Template:Quantum mechanics topics")[Quantum mechanics]() | | |---|---| | Background | [Introduction](https://en.wikipedia.org/wiki/Introduction_to_quantum_mechanics "Introduction to quantum mechanics") [History](https://en.wikipedia.org/wiki/History_of_quantum_mechanics "History of quantum mechanics") [Timeline](https://en.wikipedia.org/wiki/Timeline_of_quantum_mechanics "Timeline of quantum mechanics") [Classical mechanics](https://en.wikipedia.org/wiki/Classical_mechanics "Classical mechanics") [Old quantum theory](https://en.wikipedia.org/wiki/Old_quantum_theory "Old quantum theory") [Glossary](https://en.wikipedia.org/wiki/Glossary_of_elementary_quantum_mechanics "Glossary of elementary quantum mechanics") | | Fundamentals | [Born rule](https://en.wikipedia.org/wiki/Born_rule "Born rule") [Bra–ket notation](https://en.wikipedia.org/wiki/Bra%E2%80%93ket_notation "Bra–ket notation") [Complementarity](https://en.wikipedia.org/wiki/Complementarity_\(physics\) "Complementarity (physics)") [Density matrix](https://en.wikipedia.org/wiki/Density_matrix "Density matrix") [Energy level](https://en.wikipedia.org/wiki/Energy_level "Energy level") [Ground state](https://en.wikipedia.org/wiki/Ground_state "Ground state") [Excited state](https://en.wikipedia.org/wiki/Excited_state "Excited state") [Degenerate levels](https://en.wikipedia.org/wiki/Degenerate_energy_levels "Degenerate energy levels") [Zero-point energy](https://en.wikipedia.org/wiki/Zero-point_energy "Zero-point energy") [Entanglement](https://en.wikipedia.org/wiki/Quantum_entanglement "Quantum entanglement") [Hamiltonian](https://en.wikipedia.org/wiki/Hamiltonian_\(quantum_mechanics\) "Hamiltonian (quantum mechanics)") [Interference](https://en.wikipedia.org/wiki/Wave_interference "Wave interference") [Decoherence](https://en.wikipedia.org/wiki/Quantum_decoherence "Quantum decoherence") [Measurement](https://en.wikipedia.org/wiki/Measurement_in_quantum_mechanics "Measurement in quantum mechanics") [Nonlocality](https://en.wikipedia.org/wiki/Quantum_nonlocality "Quantum nonlocality") [Quantum state](https://en.wikipedia.org/wiki/Quantum_state "Quantum state") [quantum jump](https://en.wikipedia.org/wiki/Quantum_jump "Quantum jump") [Superposition](https://en.wikipedia.org/wiki/Quantum_superposition "Quantum superposition") [Tunnelling](https://en.wikipedia.org/wiki/Quantum_tunnelling "Quantum tunnelling") [Scattering theory](https://en.wikipedia.org/wiki/Scattering#Theory "Scattering") [Symmetry in quantum mechanics](https://en.wikipedia.org/wiki/Symmetry_in_quantum_mechanics "Symmetry in quantum mechanics") [Uncertainty](https://en.wikipedia.org/wiki/Uncertainty_principle "Uncertainty principle") [Wave function](https://en.wikipedia.org/wiki/Wave_function "Wave function") [Collapse](https://en.wikipedia.org/wiki/Wave_function_collapse "Wave function collapse") [Wave–particle duality](https://en.wikipedia.org/wiki/Wave%E2%80%93particle_duality "Wave–particle duality") | | Formulations | [Formulations](https://en.wikipedia.org/wiki/Mathematical_formulation_of_quantum_mechanics "Mathematical formulation of quantum mechanics") [Heisenberg](https://en.wikipedia.org/wiki/Heisenberg_picture "Heisenberg picture") [Interaction](https://en.wikipedia.org/wiki/Interaction_picture "Interaction picture") [Matrix mechanics](https://en.wikipedia.org/wiki/Matrix_mechanics "Matrix mechanics") [Schrödinger](https://en.wikipedia.org/wiki/Schr%C3%B6dinger_picture "Schrödinger picture") [Path integral formulation](https://en.wikipedia.org/wiki/Path_integral_formulation "Path integral formulation") [Phase space](https://en.wikipedia.org/wiki/Phase-space_formulation "Phase-space formulation") | | Equations | [Klein–Gordon](https://en.wikipedia.org/wiki/Klein%E2%80%93Gordon_equation "Klein–Gordon equation") [Dirac](https://en.wikipedia.org/wiki/Dirac_equation "Dirac equation") [Weyl](https://en.wikipedia.org/wiki/Weyl_equation "Weyl equation") [Majorana](https://en.wikipedia.org/wiki/Majorana_equation "Majorana equation") [Rarita–Schwinger](https://en.wikipedia.org/wiki/Rarita%E2%80%93Schwinger_equation "Rarita–Schwinger equation") [Pauli](https://en.wikipedia.org/wiki/Pauli_equation "Pauli equation") [Rydberg](https://en.wikipedia.org/wiki/Rydberg_formula "Rydberg formula") [Schrödinger](https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation "Schrödinger equation") | | [Interpretations](https://en.wikipedia.org/wiki/Interpretations_of_quantum_mechanics "Interpretations of quantum mechanics") | [Bayesian](https://en.wikipedia.org/wiki/Quantum_Bayesianism "Quantum Bayesianism") [Consciousness causes collapse](https://en.wikipedia.org/wiki/Consciousness_causes_collapse "Consciousness causes collapse") [Consistent histories](https://en.wikipedia.org/wiki/Consistent_histories "Consistent histories") [Copenhagen](https://en.wikipedia.org/wiki/Copenhagen_interpretation "Copenhagen interpretation") [de Broglie–Bohm](https://en.wikipedia.org/wiki/De_Broglie%E2%80%93Bohm_theory "De Broglie–Bohm theory") [Ensemble](https://en.wikipedia.org/wiki/Ensemble_interpretation "Ensemble interpretation") [Hidden-variable](https://en.wikipedia.org/wiki/Hidden-variable_theory "Hidden-variable theory") [Local](https://en.wikipedia.org/wiki/Local_hidden-variable_theory "Local hidden-variable theory") [Superdeterminism](https://en.wikipedia.org/wiki/Superdeterminism "Superdeterminism") [Many-worlds](https://en.wikipedia.org/wiki/Many-worlds_interpretation "Many-worlds interpretation") [Objective collapse](https://en.wikipedia.org/wiki/Objective-collapse_theory "Objective-collapse theory") [Quantum logic](https://en.wikipedia.org/wiki/Quantum_logic "Quantum logic") [Relational](https://en.wikipedia.org/wiki/Relational_quantum_mechanics "Relational quantum mechanics") [Transactional](https://en.wikipedia.org/wiki/Transactional_interpretation "Transactional interpretation") | | Experiments | [Bell test](https://en.wikipedia.org/wiki/Bell_test "Bell test") [Davisson–Germer](https://en.wikipedia.org/wiki/Davisson%E2%80%93Germer_experiment "Davisson–Germer experiment") [Delayed-choice quantum eraser](https://en.wikipedia.org/wiki/Delayed-choice_quantum_eraser "Delayed-choice quantum eraser") [Double-slit](https://en.wikipedia.org/wiki/Double-slit_experiment "Double-slit experiment") [Franck–Hertz](https://en.wikipedia.org/wiki/Franck%E2%80%93Hertz_experiment "Franck–Hertz experiment") [Mach–Zehnder interferometer](https://en.wikipedia.org/wiki/Mach%E2%80%93Zehnder_interferometer "Mach–Zehnder interferometer") [Elitzur–Vaidman](https://en.wikipedia.org/wiki/Elitzur%E2%80%93Vaidman_bomb_tester "Elitzur–Vaidman bomb tester") [Popper](https://en.wikipedia.org/wiki/Popper%27s_experiment "Popper's experiment") [Quantum eraser](https://en.wikipedia.org/wiki/Quantum_eraser_experiment "Quantum eraser experiment") [Stern–Gerlach](https://en.wikipedia.org/wiki/Stern%E2%80%93Gerlach_experiment "Stern–Gerlach experiment") [Wheeler's delayed choice](https://en.wikipedia.org/wiki/Wheeler%27s_delayed-choice_experiment "Wheeler's delayed-choice experiment") | | [Science](https://en.wikipedia.org/wiki/Nanotechnology "Nanotechnology") | [Quantum biology](https://en.wikipedia.org/wiki/Quantum_biology "Quantum biology") [Quantum chemistry](https://en.wikipedia.org/wiki/Quantum_chemistry "Quantum chemistry") [Quantum chaos](https://en.wikipedia.org/wiki/Quantum_chaos "Quantum chaos") [Quantum cosmology](https://en.wikipedia.org/wiki/Quantum_cosmology "Quantum cosmology") [Quantum differential calculus](https://en.wikipedia.org/wiki/Quantum_differential_calculus "Quantum differential calculus") [Quantum dynamics](https://en.wikipedia.org/wiki/Quantum_dynamics "Quantum dynamics") [Quantum geometry](https://en.wikipedia.org/wiki/Quantum_geometry "Quantum geometry") [Quantum measurement problem](https://en.wikipedia.org/wiki/Measurement_problem "Measurement problem") [Quantum mind](https://en.wikipedia.org/wiki/Quantum_mind "Quantum mind") [Quantum stochastic calculus](https://en.wikipedia.org/wiki/Quantum_stochastic_calculus "Quantum stochastic calculus") [Quantum spacetime](https://en.wikipedia.org/wiki/Quantum_spacetime "Quantum spacetime") | | [Technology](https://en.wikipedia.org/wiki/Quantum_engineering "Quantum engineering") | [Quantum algorithms](https://en.wikipedia.org/wiki/Quantum_algorithm "Quantum algorithm") [Quantum amplifier](https://en.wikipedia.org/wiki/Quantum_amplifier "Quantum amplifier") [Quantum bus](https://en.wikipedia.org/wiki/Quantum_bus "Quantum bus") [Quantum cellular automata](https://en.wikipedia.org/wiki/Quantum_cellular_automaton "Quantum cellular automaton") [Quantum finite automata](https://en.wikipedia.org/wiki/Quantum_finite_automaton "Quantum finite automaton") [Quantum channel](https://en.wikipedia.org/wiki/Quantum_channel "Quantum channel") [Quantum circuit](https://en.wikipedia.org/wiki/Quantum_circuit "Quantum circuit") [Quantum complexity theory](https://en.wikipedia.org/wiki/Quantum_complexity_theory "Quantum complexity theory") [Quantum computing](https://en.wikipedia.org/wiki/Quantum_computing "Quantum computing") [Timeline](https://en.wikipedia.org/wiki/Timeline_of_quantum_computing_and_communication "Timeline of quantum computing and communication") [Quantum cryptography](https://en.wikipedia.org/wiki/Quantum_cryptography "Quantum cryptography") [Quantum electronics](https://en.wikipedia.org/wiki/Quantum_optics#Quantum_electronics "Quantum optics") [Quantum error correction](https://en.wikipedia.org/wiki/Quantum_error_correction "Quantum error correction") [Quantum imaging](https://en.wikipedia.org/wiki/Quantum_imaging "Quantum imaging") [Quantum image processing](https://en.wikipedia.org/wiki/Quantum_image_processing "Quantum image processing") [Quantum information](https://en.wikipedia.org/wiki/Quantum_information "Quantum information") [Quantum key distribution](https://en.wikipedia.org/wiki/Quantum_key_distribution "Quantum key distribution") [Quantum logic](https://en.wikipedia.org/wiki/Quantum_logic "Quantum logic") [Quantum logic gates](https://en.wikipedia.org/wiki/Quantum_logic_gate "Quantum logic gate") [Quantum machine](https://en.wikipedia.org/wiki/Quantum_machine "Quantum machine") [Quantum machine learning](https://en.wikipedia.org/wiki/Quantum_machine_learning "Quantum machine learning") [Quantum metamaterial](https://en.wikipedia.org/wiki/Quantum_metamaterial "Quantum metamaterial") [Quantum metrology](https://en.wikipedia.org/wiki/Quantum_metrology "Quantum metrology") [Quantum network](https://en.wikipedia.org/wiki/Quantum_network "Quantum network") [Quantum neural network](https://en.wikipedia.org/wiki/Quantum_neural_network "Quantum neural network") [Quantum optics](https://en.wikipedia.org/wiki/Quantum_optics "Quantum optics") [Quantum programming](https://en.wikipedia.org/wiki/Quantum_programming "Quantum programming") [Quantum sensing](https://en.wikipedia.org/wiki/Quantum_sensor "Quantum sensor") [Quantum simulator](https://en.wikipedia.org/wiki/Quantum_simulator "Quantum simulator") [Quantum teleportation](https://en.wikipedia.org/wiki/Quantum_teleportation "Quantum teleportation") | | Extensions | [Quantum fluctuation](https://en.wikipedia.org/wiki/Quantum_fluctuation "Quantum fluctuation") [Casimir effect](https://en.wikipedia.org/wiki/Casimir_effect "Casimir effect") [Quantum statistical mechanics](https://en.wikipedia.org/wiki/Quantum_statistical_mechanics "Quantum statistical mechanics") [Quantum field theory](https://en.wikipedia.org/wiki/Quantum_field_theory "Quantum field theory") [History](https://en.wikipedia.org/wiki/History_of_quantum_field_theory "History of quantum field theory") [Quantum gravity](https://en.wikipedia.org/wiki/Quantum_gravity "Quantum gravity") [Relativistic quantum mechanics](https://en.wikipedia.org/wiki/Relativistic_quantum_mechanics "Relativistic quantum mechanics") | | Related | [Schrödinger's cat](https://en.wikipedia.org/wiki/Schr%C3%B6dinger%27s_cat "Schrödinger's cat") [in popular culture](https://en.wikipedia.org/wiki/Schr%C3%B6dinger%27s_cat_in_popular_culture "Schrödinger's cat in popular culture") [Wigner's friend](https://en.wikipedia.org/wiki/Wigner%27s_friend "Wigner's friend") [EPR paradox](https://en.wikipedia.org/wiki/Einstein%E2%80%93Podolsky%E2%80%93Rosen_paradox "Einstein–Podolsky–Rosen paradox") [Quantum mysticism](https://en.wikipedia.org/wiki/Quantum_mysticism "Quantum mysticism") | | ![](https://upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/20px-Symbol_category_class.svg.png) [Category](https://en.wikipedia.org/wiki/Category:Quantum_mechanics "Category:Quantum mechanics") | | | [v](https://en.wikipedia.org/wiki/Template:Branches_of_physics "Template:Branches of physics") [t](https://en.wikipedia.org/wiki/Template_talk:Branches_of_physics "Template talk:Branches of physics") [e](https://en.wikipedia.org/wiki/Special:EditPage/Template:Branches_of_physics "Special:EditPage/Template:Branches of physics")Major [branches of physics](https://en.wikipedia.org/wiki/Branches_of_physics "Branches of physics") | | |---|---| | Divisions | [Pure](https://en.wikipedia.org/wiki/Basic_research "Basic research") [Applied](https://en.wikipedia.org/wiki/Applied_physics "Applied physics") [Engineering](https://en.wikipedia.org/wiki/Engineering_physics "Engineering physics") | | Approaches | [Experimental](https://en.wikipedia.org/wiki/Experimental_physics "Experimental physics") [Theoretical](https://en.wikipedia.org/wiki/Theoretical_physics "Theoretical physics") [Computational](https://en.wikipedia.org/wiki/Computational_physics "Computational physics") | | [Classical](https://en.wikipedia.org/wiki/Classical_physics "Classical physics") | [Classical mechanics](https://en.wikipedia.org/wiki/Classical_mechanics "Classical mechanics") [Newtonian](https://en.wikipedia.org/wiki/Newton%27s_laws_of_motion "Newton's laws of motion") [Analytical](https://en.wikipedia.org/wiki/Analytical_mechanics "Analytical mechanics") [Celestial](https://en.wikipedia.org/wiki/Celestial_mechanics "Celestial mechanics") [Continuum](https://en.wikipedia.org/wiki/Continuum_mechanics "Continuum mechanics") [Acoustics](https://en.wikipedia.org/wiki/Acoustics "Acoustics") [Classical electromagnetism](https://en.wikipedia.org/wiki/Classical_electromagnetism 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"Quantum systems" redirects here. For the company, see [Quantum-Systems](https://en.wikipedia.org/wiki/Quantum-Systems "Quantum-Systems"). [![](https://upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Hydrogen_Density_Plots.png/500px-Hydrogen_Density_Plots.png)](https://en.wikipedia.org/wiki/File:Hydrogen_Density_Plots.png) [Wave functions](https://en.wikipedia.org/wiki/Wave_function "Wave function") of the [electron](https://en.wikipedia.org/wiki/Electron "Electron") in a hydrogen atom at different energy levels. Quantum mechanics cannot predict the exact location of a particle in space, only the probability of finding it at different locations.[\[1\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Born1926-1) The brighter areas represent a higher probability of finding the electron. **Quantum mechanics** is the fundamental physical [theory](https://en.wikipedia.org/wiki/Scientific_theory "Scientific theory") that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of [atoms](https://en.wikipedia.org/wiki/Atom "Atom").[\[2\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Feynman-2): 1.1 It is the foundation of all **quantum physics**, which includes [quantum chemistry](https://en.wikipedia.org/wiki/Quantum_chemistry "Quantum chemistry"), [quantum biology](https://en.wikipedia.org/wiki/Quantum_biology "Quantum biology"), [quantum field theory](https://en.wikipedia.org/wiki/Quantum_field_theory "Quantum field theory"), [quantum technology](https://en.wikipedia.org/wiki/Quantum_technology "Quantum technology"), and [quantum information science](https://en.wikipedia.org/wiki/Quantum_information_science "Quantum information science"). Quantum mechanics can describe many systems that [classical physics](https://en.wikipedia.org/wiki/Classical_physics "Classical physics") cannot. Classical physics can describe many aspects of nature at an ordinary ([macroscopic](https://en.wikipedia.org/wiki/Macroscopic "Macroscopic") and [(optical) microscopic](https://en.wikipedia.org/wiki/Microscopic_scale "Microscopic scale")) scale, however is insufficient for describing them at very small [submicroscopic](https://en.wikipedia.org/wiki/Submicroscopic "Submicroscopic") (atomic and [subatomic](https://en.wikipedia.org/wiki/Subatomic "Subatomic")) scales. Classical mechanics can be derived from quantum mechanics as an approximation that is valid at ordinary scales.[\[3\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-3) **Quantum systems** have [bound](https://en.wikipedia.org/wiki/Bound_state "Bound state") states that are [quantized](https://en.wikipedia.org/wiki/Quantization_\(physics\) "Quantization (physics)") to [discrete values](https://en.wikipedia.org/wiki/Discrete_mathematics "Discrete mathematics") of [energy](https://en.wikipedia.org/wiki/Energy "Energy"), [momentum](https://en.wikipedia.org/wiki/Momentum "Momentum"), [angular momentum](https://en.wikipedia.org/wiki/Angular_momentum "Angular momentum"), and other quantities, in contrast to classical systems where these quantities can be measured continuously. Measurements of quantum systems show characteristics of both [particles](https://en.wikipedia.org/wiki/Particle "Particle") and [waves](https://en.wikipedia.org/wiki/Wave "Wave") ([wave–particle duality](https://en.wikipedia.org/wiki/Wave%E2%80%93particle_duality "Wave–particle duality")), and there are limits to how accurately the value of a physical quantity can be predicted prior to its measurement, given a complete set of initial conditions (the [uncertainty principle](https://en.wikipedia.org/wiki/Uncertainty_principle "Uncertainty principle")). Quantum mechanics arose gradually from theories to explain observations that could not be reconciled with [classical physics](https://en.wikipedia.org/wiki/Classical_physics "Classical physics"), such as [Max Planck](https://en.wikipedia.org/wiki/Max_Planck "Max Planck")'s solution in 1900 to the [black-body radiation](https://en.wikipedia.org/wiki/Black-body_radiation "Black-body radiation") problem, and the correspondence between energy and frequency in [Albert Einstein](https://en.wikipedia.org/wiki/Albert_Einstein "Albert Einstein")'s [1905 paper](https://en.wikipedia.org/wiki/Annus_Mirabilis_papers#Photoelectric_effect "Annus Mirabilis papers"), which explained the [photoelectric effect](https://en.wikipedia.org/wiki/Photoelectric_effect "Photoelectric effect"). These early attempts to understand microscopic phenomena, now known as the "[old quantum theory](https://en.wikipedia.org/wiki/Old_quantum_theory "Old quantum theory")", led to the full development of quantum mechanics in the mid-1920s by [Niels Bohr](https://en.wikipedia.org/wiki/Niels_Bohr "Niels Bohr"), [Erwin Schrödinger](https://en.wikipedia.org/wiki/Erwin_Schr%C3%B6dinger "Erwin Schrödinger"), [Werner Heisenberg](https://en.wikipedia.org/wiki/Werner_Heisenberg "Werner Heisenberg"), [Max Born](https://en.wikipedia.org/wiki/Max_Born "Max Born"), [Paul Dirac](https://en.wikipedia.org/wiki/Paul_Dirac "Paul Dirac") and others. The modern theory is formulated in various [specially developed mathematical formalisms](https://en.wikipedia.org/wiki/Mathematical_formulations_of_quantum_mechanics "Mathematical formulations of quantum mechanics"). In one of them, a mathematical entity called the [wave function](https://en.wikipedia.org/wiki/Wave_function "Wave function") provides information, in the form of [probability amplitudes](https://en.wikipedia.org/wiki/Probability_amplitude "Probability amplitude"), about what measurements of a particle's energy, momentum, and other physical properties may yield. Overview and fundamental concepts Quantum mechanics allows the calculation of properties and behaviour of [physical systems](https://en.wikipedia.org/wiki/Physical_systems "Physical systems"). It is typically applied to microscopic systems: [molecules](https://en.wikipedia.org/wiki/Molecules "Molecules"), [atoms](https://en.wikipedia.org/wiki/Atoms "Atoms") and [subatomic particles](https://en.wikipedia.org/wiki/Subatomic_particle "Subatomic particle"). It has been demonstrated to hold for complex molecules with thousands of atoms,[\[4\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-4) but its application to human beings raises philosophical problems, such as [Wigner's friend](https://en.wikipedia.org/wiki/Wigner%27s_friend "Wigner's friend"), and its application to the universe as a whole remains speculative.[\[5\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-5) Predictions of quantum mechanics have been verified experimentally to an extremely high degree of [accuracy](https://en.wikipedia.org/wiki/Accuracy "Accuracy"). For example, the refinement of quantum mechanics for the interaction of light and matter, known as [quantum electrodynamics](https://en.wikipedia.org/wiki/Quantum_electrodynamics "Quantum electrodynamics") (QED), has been [shown to agree with experiment](https://en.wikipedia.org/wiki/Precision_tests_of_QED "Precision tests of QED") to within 1 part in 1012 when predicting the magnetic properties of an electron.[\[6\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-6) A fundamental feature of the theory is that it usually cannot predict with certainty what will happen, but only gives probabilities. Mathematically, a probability is found by taking the square of the absolute value of a [complex number](https://en.wikipedia.org/wiki/Complex_number "Complex number"), known as a probability amplitude. This is known as the [Born rule](https://en.wikipedia.org/wiki/Born_rule "Born rule"), named after physicist [Max Born](https://en.wikipedia.org/wiki/Max_Born "Max Born"). For example, a quantum particle like an [electron](https://en.wikipedia.org/wiki/Electron "Electron") can be described by a wave function, which associates to each point in space a probability amplitude. Applying the Born rule to these amplitudes gives a [probability density function](https://en.wikipedia.org/wiki/Probability_density_function "Probability density function") for the position that the electron will be found to have when an experiment is performed to measure it. This is the best the theory can do; it cannot say for certain where the electron will be found. The [Schrödinger equation](https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation "Schrödinger equation") relates the collection of probability amplitudes that pertain to one moment of time to the collection of probability amplitudes that pertain to another.[\[7\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Zwiebach2022-7): 67–87 One consequence of the mathematical rules of quantum mechanics is a tradeoff in predictability between measurable quantities. The most famous form of this [uncertainty principle](https://en.wikipedia.org/wiki/Uncertainty_principle "Uncertainty principle") says that no matter how a quantum particle is prepared or how carefully experiments upon it are arranged, it is impossible to have a precise prediction for a measurement of its position and also at the same time for a measurement of its [momentum](https://en.wikipedia.org/wiki/Momentum "Momentum").[\[7\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Zwiebach2022-7): 427–435 [![](https://upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Double-slit.svg/330px-Double-slit.svg.png)](https://en.wikipedia.org/wiki/File:Double-slit.svg) An illustration of the [double-slit experiment](https://en.wikipedia.org/wiki/Double-slit_experiment "Double-slit experiment") Another consequence of the mathematical rules of quantum mechanics is the phenomenon of [quantum interference](https://en.wikipedia.org/wiki/Quantum_interference "Quantum interference"), which is often illustrated with the [double-slit experiment](https://en.wikipedia.org/wiki/Double-slit_experiment "Double-slit experiment"). In the basic version of this experiment, a [coherent light source](https://en.wikipedia.org/wiki/Coherence_\(physics\) "Coherence (physics)"), such as a [laser](https://en.wikipedia.org/wiki/Laser "Laser") beam, illuminates a plate pierced by two parallel slits, and the light passing through the slits is observed on a screen behind the plate.[\[8\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Lederman-8): 102–111 [\[2\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Feynman-2): 1.1–1.8 The wave nature of light causes the light waves passing through the two slits to [interfere](https://en.wikipedia.org/wiki/Interference_\(wave_propagation\) "Interference (wave propagation)"), producing bright and dark bands on the screen – a result that would not be expected if light consisted of classical particles.[\[8\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Lederman-8) However, the light is always found to be absorbed at the screen at discrete points, as individual particles rather than waves; the interference pattern appears via the varying density of these particle hits on the screen. Furthermore, versions of the experiment that include detectors at the slits find that each detected [photon](https://en.wikipedia.org/wiki/Photon "Photon") passes through one slit (as would a classical particle), and not through both slits (as would a wave).[\[8\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Lederman-8): 109 [\[9\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-M%C3%BCller-Kirsten-9)[\[10\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Plotnitsky-10) However, [such experiments](https://en.wikipedia.org/wiki/Double-slit_experiment#Which_way "Double-slit experiment") demonstrate that particles do not form the interference pattern if one detects which slit they pass through. This behavior is known as [wave–particle duality](https://en.wikipedia.org/wiki/Wave%E2%80%93particle_duality "Wave–particle duality"). In addition to light, [electrons](https://en.wikipedia.org/wiki/Electrons "Electrons"), [atoms](https://en.wikipedia.org/wiki/Atoms "Atoms"), and [molecules](https://en.wikipedia.org/wiki/Molecules "Molecules") are all found to exhibit the same dual behavior when fired towards a double slit.[\[2\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Feynman-2) [![](https://upload.wikimedia.org/wikipedia/commons/thumb/5/5f/QuantumTunnel.jpg/330px-QuantumTunnel.jpg)](https://en.wikipedia.org/wiki/File:QuantumTunnel.jpg) A simplified diagram of [quantum tunneling](https://en.wikipedia.org/wiki/Quantum_tunneling "Quantum tunneling"), a phenomenon by which a particle may move through a barrier which would be impossible under classical mechanics Another non-classical phenomenon predicted by quantum mechanics is [quantum tunnelling](https://en.wikipedia.org/wiki/Quantum_tunnelling "Quantum tunnelling"): a particle that goes up against a [potential barrier](https://en.wikipedia.org/wiki/Potential_barrier "Potential barrier") can cross it, even if its kinetic energy is smaller than the maximum of the potential.[\[11\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-11) In classical mechanics this particle would be trapped. Quantum tunnelling has several important consequences, enabling [radioactive decay](https://en.wikipedia.org/wiki/Radioactive_decay "Radioactive decay"), [nuclear fusion](https://en.wikipedia.org/wiki/Nuclear_fusion "Nuclear fusion") in stars, and applications such as [scanning tunnelling microscopy](https://en.wikipedia.org/wiki/Scanning_tunnelling_microscopy "Scanning tunnelling microscopy"), [tunnel diode](https://en.wikipedia.org/wiki/Tunnel_diode "Tunnel diode") and [tunnel field-effect transistor](https://en.wikipedia.org/wiki/Tunnel_field-effect_transistor "Tunnel field-effect transistor").[\[12\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Trixler2013-12)[\[13\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-13) When quantum systems interact, the result can be the creation of [quantum entanglement](https://en.wikipedia.org/wiki/Quantum_entanglement "Quantum entanglement"): their properties become so intertwined that a description of the whole solely in terms of the individual parts is no longer possible. Erwin Schrödinger called entanglement "...*the* characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought".[\[14\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-14) Quantum entanglement enables [quantum computing](https://en.wikipedia.org/wiki/Quantum_computing "Quantum computing") and is part of quantum communication protocols, such as [quantum key distribution](https://en.wikipedia.org/wiki/Quantum_key_distribution "Quantum key distribution") and [superdense coding](https://en.wikipedia.org/wiki/Superdense_coding "Superdense coding").[\[15\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Caves-15) Contrary to popular misconception, entanglement does not allow sending signals [faster than light](https://en.wikipedia.org/wiki/Faster_than_light "Faster than light"), as demonstrated by the [no-communication theorem](https://en.wikipedia.org/wiki/No-communication_theorem "No-communication theorem").[\[15\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Caves-15) Another possibility opened by entanglement is testing for "[hidden variables](https://en.wikipedia.org/wiki/Hidden_variable_theory "Hidden variable theory")", hypothetical properties more fundamental than the quantities addressed in quantum theory itself, knowledge of which would allow more exact predictions than quantum theory provides. A collection of results, most significantly [Bell's theorem](https://en.wikipedia.org/wiki/Bell%27s_theorem "Bell's theorem"), have demonstrated that broad classes of such hidden-variable theories are in fact incompatible with quantum physics. According to Bell's theorem, if nature actually operates in accord with any theory of *local* hidden variables, then the results of a [Bell test](https://en.wikipedia.org/wiki/Bell_test "Bell test") will be constrained in a particular, quantifiable way. Many Bell tests have been performed and they have shown results incompatible with the constraints imposed by local hidden variables.[\[16\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-wiseman15-16)[\[17\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-wolchover17-17) It is not possible to present these concepts in more than a superficial way without introducing the mathematics involved; understanding quantum mechanics requires not only manipulating complex numbers, but also [linear algebra](https://en.wikipedia.org/wiki/Linear_algebra "Linear algebra"), [differential equations](https://en.wikipedia.org/wiki/Differential_equation "Differential equation"), [group theory](https://en.wikipedia.org/wiki/Group_theory "Group theory"), and other more advanced subjects.[\[18\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-18)[\[19\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-19) Accordingly, this article will present a mathematical formulation of quantum mechanics and survey its application to some useful and oft-studied examples. Mathematical formulation In the mathematically rigorous formulation of quantum mechanics, the state of a quantum mechanical system is a vector ![{\\displaystyle \\psi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a) belonging to a ([separable](https://en.wikipedia.org/wiki/Separable_space "Separable space")) complex [Hilbert space](https://en.wikipedia.org/wiki/Hilbert_space "Hilbert space") ![{\\displaystyle {\\mathcal {H}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/19ef4c7b923a5125ac91aa491838a95ee15b804f). This vector is postulated to be normalized under the Hilbert space inner product, that is, it obeys ![{\\displaystyle \\langle \\psi ,\\psi \\rangle =1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d5758e7a60b4e54bc46e01b0618919c65b787a0), and it is well-defined up to a complex number of modulus 1 (the global phase), that is, ![{\\displaystyle \\psi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a) and ![{\\displaystyle e^{i\\alpha }\\psi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/7193e9b0479da798b9cf2dcc492f49166d0d7103) represent the same physical system. In other words, the possible states are points in the [projective space](https://en.wikipedia.org/wiki/Projective_space "Projective space") of a Hilbert space, usually called the [complex projective space](https://en.wikipedia.org/wiki/Complex_projective_space "Complex projective space"). The exact nature of this Hilbert space is dependent on the system – for example, for describing position and momentum the Hilbert space is the space of complex [square-integrable](https://en.wikipedia.org/wiki/Square-integrable "Square-integrable") functions ![{\\displaystyle L^{2}(\\mathbb {C} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a3649087af50f42a95da2163656076529ca3139),[\[20\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Holevo2001-20): 13 while the Hilbert space for the [spin](https://en.wikipedia.org/wiki/Spin_\(physics\) "Spin (physics)") of a single proton is simply the space of two-dimensional complex vectors ![{\\displaystyle \\mathbb {C} ^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f43d6ec8a1e1fe5a85aec0dd9bdcd45ae09b06b) with the usual inner product.[\[20\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Holevo2001-20): 20 Physical quantities of interest – position, momentum, energy, spin – are represented by observables, which are [Hermitian](https://en.wikipedia.org/wiki/Hermitian_adjoint#Hermitian_operators "Hermitian adjoint") (more precisely, [self-adjoint](https://en.wikipedia.org/wiki/Self-adjoint_operator "Self-adjoint operator")) linear [operators](https://en.wikipedia.org/wiki/Operator_\(physics\) "Operator (physics)") acting on the Hilbert space.[\[20\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Holevo2001-20): 17 A quantum state can be an [eigenvector](https://en.wikipedia.org/wiki/Eigenvector "Eigenvector") of an observable, in which case it is called an [eigenstate](https://en.wikipedia.org/wiki/Eigenstate "Eigenstate"), and the associated [eigenvalue](https://en.wikipedia.org/wiki/Eigenvalue "Eigenvalue") corresponds to the value of the observable in that eigenstate. More generally, a quantum state will be a linear combination of the eigenstates, known as a [quantum superposition](https://en.wikipedia.org/wiki/Quantum_superposition "Quantum superposition"). When an observable is measured, the result will be one of its eigenvalues with probability given by the [Born rule](https://en.wikipedia.org/wiki/Born_rule "Born rule"): in the simplest case the eigenvalue ![{\\displaystyle \\lambda }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a) is non-degenerate and the probability is given by ![{\\displaystyle \|\\langle {\\vec {\\lambda }},\\psi \\rangle \|^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de4b465224a805b558697f44dcf3824fb2fbfe20), where ![{\\displaystyle {\\vec {\\lambda }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d8c69573b9c55ec8ea726662102b8e693bf32f01) is its associated unit-length eigenvector. More generally, the eigenvalue is degenerate and the probability is given by ![{\\displaystyle \\langle \\psi ,P\_{\\lambda }\\psi \\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/6ba3648378b52e2e710937b36948cd908b2d6531), where ![{\\displaystyle P\_{\\lambda }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/330591f9b6fffc93ca78514576fd0d8cfac6f0c7) is the projector onto its associated eigenspace.[\[21\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-21) In the continuous case, these formulas give instead the [probability density](https://en.wikipedia.org/wiki/Probability_density "Probability density"). After the [measurement](https://en.wikipedia.org/wiki/Measurement_in_quantum_mechanics "Measurement in quantum mechanics"), if result ![{\\displaystyle \\lambda }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a) was obtained, the quantum state is postulated to [collapse](https://en.wikipedia.org/wiki/Collapse_of_the_wavefunction "Collapse of the wavefunction") to ![{\\displaystyle {\\vec {\\lambda }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d8c69573b9c55ec8ea726662102b8e693bf32f01), in the non-degenerate case, or to ![{\\textstyle P\_{\\lambda }\\psi {\\big /}\\!{\\sqrt {\\langle \\psi ,P\_{\\lambda }\\psi \\rangle }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e38d8eb9122b686c4d8f937fe59e2ec511be43ed), in the general case. The [probabilistic](https://en.wikipedia.org/wiki/Probabilistic "Probabilistic") nature of quantum mechanics thus stems from the act of measurement. This is one of the most debated aspects of quantum theory, with different [interpretations of quantum mechanics](https://en.wikipedia.org/wiki/Interpretations_of_quantum_mechanics "Interpretations of quantum mechanics") giving radically different answers to questions regarding quantum-state collapse, as discussed [below](https://en.wikipedia.org/wiki/Quantum_mechanics#Philosophical_implications). Time evolution of a quantum state The time evolution of a quantum state is described by the Schrödinger equation: ![{\\displaystyle i\\hbar {\\frac {\\partial }{\\partial t}}\\psi (t)=H\\psi (t).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5c41b5a62afa3aa83e6ea98e69d692ac0f77c2c6) Here ![{\\displaystyle H}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b) denotes the [Hamiltonian](https://en.wikipedia.org/wiki/Hamiltonian_\(quantum_mechanics\) "Hamiltonian (quantum mechanics)"), the observable corresponding to the [total energy](https://en.wikipedia.org/wiki/Total_energy "Total energy") of the system, and ![{\\displaystyle \\hbar }](https://wikimedia.org/api/rest_v1/media/math/render/svg/de68de3a92517953436c93b5a76461d49160cc41) is the reduced [Planck constant](https://en.wikipedia.org/wiki/Planck_constant "Planck constant"). The constant ![{\\displaystyle i\\hbar }](https://wikimedia.org/api/rest_v1/media/math/render/svg/3d724d600903f701570a9f97c2a8378a6f3a80e5) is introduced so that the Hamiltonian is reduced to the [classical Hamiltonian](https://en.wikipedia.org/wiki/Hamiltonian_mechanics "Hamiltonian mechanics") in cases where the quantum system can be approximated by a classical system; the ability to make such an approximation in certain limits is called the [correspondence principle](https://en.wikipedia.org/wiki/Correspondence_principle "Correspondence principle"). The solution of this differential equation is given by ![{\\displaystyle \\psi (t)=e^{-iHt/\\hbar }\\psi (0).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b0a822b842f8d952c4e5827f312597e72a45f4d) The operator ![{\\displaystyle U(t)=e^{-iHt/\\hbar }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/17a17e78953fa7d61788cc8bc41a25ef254fc8e0) is known as the time-evolution operator, and has the crucial property that it is [unitary](https://en.wikipedia.org/wiki/Unitarity_\(physics\) "Unitarity (physics)"). This time evolution is [deterministic](https://en.wikipedia.org/wiki/Deterministic "Deterministic") in the sense that – given an initial quantum state ![{\\displaystyle \\psi (0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fcfdf3fefa6feee8b2192815ac15a2650d8a8db2) – it makes a definite prediction of what the quantum state ![{\\displaystyle \\psi (t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6cf4a36b5f945be90a527b3dbe3d55d3f0439cdb) will be at any later time.[\[22\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-22) [![](https://upload.wikimedia.org/wikipedia/commons/thumb/3/35/Atomic-orbital-clouds_spd_m0.png/330px-Atomic-orbital-clouds_spd_m0.png)](https://en.wikipedia.org/wiki/File:Atomic-orbital-clouds_spd_m0.png) Fig. 1: [Probability densities](https://en.wikipedia.org/wiki/Probability_density_function "Probability density function") corresponding to the wave functions of an electron in a hydrogen atom possessing definite energy levels (increasing from the top of the image to the bottom: *n* = 1, 2, 3, ...) and angular momenta (increasing across from left to right: *s*, *p*, *d*, ...). Denser areas correspond to higher probability density in a position measurement.Such wave functions are directly comparable to [Chladni's figures](https://en.wikipedia.org/wiki/Chladni%27s_figures "Chladni's figures") of [acoustic](https://en.wikipedia.org/wiki/Acoustics "Acoustics") modes of vibration in classical physics and are modes of oscillation as well, possessing a sharp energy and thus, a definite frequency. The [angular momentum](https://en.wikipedia.org/wiki/Angular_momentum "Angular momentum") and energy are [quantized](https://en.wikipedia.org/wiki/Quantization_\(physics\) "Quantization (physics)") and take *only* discrete values like those shown – as is the case for [resonant frequencies](https://en.wikipedia.org/wiki/Resonant_frequencies "Resonant frequencies") in acoustics. Some wave functions produce probability distributions that are independent of time, such as [eigenstates](https://en.wikipedia.org/wiki/Eigenstate "Eigenstate") of the Hamiltonian.[\[7\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Zwiebach2022-7): 133–137 Many systems that are treated dynamically in classical mechanics are described by such "static" wave functions. For example, a single electron in an unexcited atom is pictured classically as a particle moving in a circular trajectory around the [atomic nucleus](https://en.wikipedia.org/wiki/Atomic_nucleus "Atomic nucleus"), whereas in quantum mechanics, it is described by a static wave function surrounding the nucleus. For example, the electron wave function for an unexcited hydrogen atom is a spherically symmetric function known as an [*s* orbital](https://en.wikipedia.org/wiki/Atomic_orbital "Atomic orbital") ([Fig. 1](https://en.wikipedia.org/wiki/Quantum_mechanics#fig1)). Analytic solutions of the Schrödinger equation are known for [very few relatively simple model Hamiltonians](https://en.wikipedia.org/wiki/List_of_quantum-mechanical_systems_with_analytical_solutions "List of quantum-mechanical systems with analytical solutions") including the [quantum harmonic oscillator](https://en.wikipedia.org/wiki/Quantum_harmonic_oscillator "Quantum harmonic oscillator"), the [particle in a box](https://en.wikipedia.org/wiki/Particle_in_a_box "Particle in a box"), the [dihydrogen cation](https://en.wikipedia.org/wiki/Dihydrogen_cation "Dihydrogen cation"), and the [hydrogen atom](https://en.wikipedia.org/wiki/Hydrogen_atom "Hydrogen atom"). Even the [helium](https://en.wikipedia.org/wiki/Helium "Helium") atom – which contains just two electrons – has defied all attempts at a fully analytic treatment, admitting no solution in [closed form](https://en.wikipedia.org/wiki/Closed-form_expression "Closed-form expression").[\[23\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-23)[\[24\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-24)[\[25\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-25) However, there are techniques for finding approximate solutions. One method, called [perturbation theory](https://en.wikipedia.org/wiki/Perturbation_theory_\(quantum_mechanics\) "Perturbation theory (quantum mechanics)"), uses the analytic result for a simple quantum mechanical model to create a result for a related but more complicated model by (for example) the addition of a weak [potential energy](https://en.wikipedia.org/wiki/Potential_energy "Potential energy").[\[7\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Zwiebach2022-7): 793 Another approximation method applies to systems for which quantum mechanics produces only small deviations from classical behavior. These deviations can then be computed based on the classical motion.[\[7\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Zwiebach2022-7): 849 Uncertainty principle One consequence of the basic quantum formalism is the uncertainty principle. In its most familiar form, this states that no preparation of a quantum particle can imply simultaneously precise predictions both for a measurement of its position and for a measurement of its momentum.[\[26\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Cohen-Tannoudji-26)[\[27\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-L&L-27) Both position and momentum are observables, meaning that they are represented by [Hermitian operators](https://en.wikipedia.org/wiki/Hermitian_operators "Hermitian operators"). The position operator ![{\\displaystyle {\\hat {X}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/acc59ad6d9a06d55b96b65beb0fdfc89acc1e40e) and momentum operator ![{\\displaystyle {\\hat {P}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a46a8cf7bc789e8c4f8de7ca0d9946a46fb8842) do not commute, but rather satisfy the [canonical commutation relation](https://en.wikipedia.org/wiki/Canonical_commutation_relation "Canonical commutation relation"): ![{\\displaystyle \[{\\hat {X}},{\\hat {P}}\]=i\\hbar .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/803fe39b0eeaff8d1570df480e738cf5a968cc71) Given a quantum state, the Born rule lets us compute expectation values for both ![{\\displaystyle X}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab) and ![{\\displaystyle P}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a), and moreover for powers of them. Defining the uncertainty for an observable by a [standard deviation](https://en.wikipedia.org/wiki/Standard_deviation "Standard deviation"), we have ![{\\displaystyle \\sigma \_{X}={\\textstyle {\\sqrt {\\left\\langle X^{2}\\right\\rangle -\\left\\langle X\\right\\rangle ^{2}}}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/457ec20972d63dfb1ecc9087e18d1f949f908c8f) and likewise for the momentum: ![{\\displaystyle \\sigma \_{P}={\\sqrt {\\left\\langle P^{2}\\right\\rangle -\\left\\langle P\\right\\rangle ^{2}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/63ec8f5f7b9e5957ea6d06c56068b06244acc184) The uncertainty principle states that ![{\\displaystyle \\sigma \_{X}\\sigma \_{P}\\geq {\\frac {\\hbar }{2}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/538cada7fa57155ece387e5d53d90ca366e323fe) Either standard deviation can in principle be made arbitrarily small, but not both simultaneously.[\[28\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-ballentine1970-28) This inequality generalizes to arbitrary pairs of self-adjoint operators ![{\\displaystyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3) and ![{\\displaystyle B}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a). The [commutator](https://en.wikipedia.org/wiki/Commutator "Commutator") of these two operators is ![{\\displaystyle \[A,B\]=AB-BA,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2a47259b42e63c048c65f67d304404867841951) and this provides the lower bound on the product of standard deviations: ![{\\displaystyle \\sigma \_{A}\\sigma \_{B}\\geq {\\tfrac {1}{2}}\\left\|{\\bigl \\langle }\[A,B\]{\\bigr \\rangle }\\right\|.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d0fd768b447334e150b8b98181f74b475e41ee52) Another consequence of the canonical commutation relation is that the position and momentum operators are [Fourier transforms](https://en.wikipedia.org/wiki/Fourier_transform#Uncertainty_principle "Fourier transform") of each other, so that a description of an object according to its momentum is the Fourier transform of its description according to its position. The fact that dependence in momentum is the Fourier transform of the dependence in position means that the momentum operator is equivalent (up to an ![{\\displaystyle i/\\hbar }](https://wikimedia.org/api/rest_v1/media/math/render/svg/44a04e1f19b5e7bea2bfa8002a841bf8d1b4e66a) factor) to taking the derivative according to the position, since in Fourier analysis [differentiation corresponds to multiplication in the dual space](https://en.wikipedia.org/wiki/Fourier_transform#Differentiation "Fourier transform"). This is why in quantum equations in position space, the momentum ![{\\displaystyle p\_{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5bab39399bf5424f25d957cdc57c84a0622626d2) is replaced by ![{\\displaystyle -i\\hbar {\\frac {\\partial }{\\partial x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab7fffcee704fc55eb36b137e0cc25132b5dc1bf), and in particular in the [non-relativistic Schrödinger equation in position space](https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation#Equation "Schrödinger equation") the momentum-squared term is replaced with a Laplacian times ![{\\displaystyle -\\hbar ^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/185934c9e69984a0905432abe63d3b41741342b4).[\[26\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Cohen-Tannoudji-26) Composite systems and entanglement When two different quantum systems are considered together, the Hilbert space of the combined system is the [tensor product](https://en.wikipedia.org/wiki/Tensor_product "Tensor product") of the Hilbert spaces of the two components. For example, let A and B be two quantum systems, with Hilbert spaces ![{\\displaystyle {\\mathcal {H}}\_{A}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4611f8542d6c72789b7e82bfcf29014d44c13aa3) and ![{\\displaystyle {\\mathcal {H}}\_{B}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bfea1102a35aff8c6435147dc9dffc21b4cfe4ef), respectively. The Hilbert space of the composite system is then ![{\\displaystyle {\\mathcal {H}}\_{AB}={\\mathcal {H}}\_{A}\\otimes {\\mathcal {H}}\_{B}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/54e4f6aff2be5ec9f3eb4657a133d1be17f6e2b3) If the state for the first system is the vector ![{\\displaystyle \\psi \_{A}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a04e49109107e7765357dd92e975895840fa7700) and the state for the second system is ![{\\displaystyle \\psi \_{B}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ca3b90bba42c451129c4969322e175f5a1d3368), then the state of the composite system is ![{\\displaystyle \\psi \_{A}\\otimes \\psi \_{B}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/852c1e09787e91af3af1a10d1eee4eadc232d48c) Not all states in the joint Hilbert space ![{\\displaystyle {\\mathcal {H}}\_{AB}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/951b54ea87d5bd62464abaf1be530e6a37a86f1b) can be written in this form, however, because the superposition principle implies that linear combinations of these "separable" or "product states" are also valid. For example, if ![{\\displaystyle \\psi \_{A}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a04e49109107e7765357dd92e975895840fa7700) and ![{\\displaystyle \\phi \_{A}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e38d84dcc21ea769a51ab502691f566c830b430) are both possible states for system ![{\\displaystyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3), and likewise ![{\\displaystyle \\psi \_{B}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ca3b90bba42c451129c4969322e175f5a1d3368) and ![{\\displaystyle \\phi \_{B}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a829a7a85bb29904dec5e93605d0b976baf45755) are both possible states for system ![{\\displaystyle B}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a), then ![{\\displaystyle {\\tfrac {1}{\\sqrt {2}}}\\left(\\psi \_{A}\\otimes \\psi \_{B}+\\phi \_{A}\\otimes \\phi \_{B}\\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/53d226ba585a99942ec855197c23ed885b635258) is a valid joint state that is not separable. States that are not separable are called [entangled](https://en.wikipedia.org/wiki/Quantum_entanglement "Quantum entanglement").[\[29\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-:0-29)[\[30\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-:1-30) If the state for a composite system is entangled, it is impossible to describe either component system A or system B by a state vector. One can instead define [reduced density matrices](https://en.wikipedia.org/wiki/Reduced_density_matrix "Reduced density matrix") that describe the statistics that can be obtained by making measurements on either component system alone. This necessarily causes a loss of information, though: knowing the reduced density matrices of the individual systems is not enough to reconstruct the state of the composite system.[\[29\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-:0-29)[\[30\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-:1-30) Just as density matrices specify the state of a subsystem of a larger system, analogously, [positive operator-valued measures](https://en.wikipedia.org/wiki/POVM "POVM") (POVMs) describe the effect on a subsystem of a measurement performed on a larger system. POVMs are extensively used in quantum information theory.[\[29\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-:0-29)[\[31\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-wilde-31) As described above, entanglement is a key feature of models of measurement processes in which an apparatus becomes entangled with the system being measured. Systems interacting with the environment in which they reside generally become entangled with that environment, a phenomenon known as [quantum decoherence](https://en.wikipedia.org/wiki/Quantum_decoherence "Quantum decoherence"). This can explain why, in practice, quantum effects are difficult to observe in systems larger than microscopic.[\[32\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-32) Equivalence between formulations There are many mathematically equivalent formulations of quantum mechanics. One of the oldest and most common is the "[transformation theory](https://en.wikipedia.org/wiki/Transformation_theory_\(quantum_mechanics\) "Transformation theory (quantum mechanics)")" proposed by [Paul Dirac](https://en.wikipedia.org/wiki/Paul_Dirac "Paul Dirac"), which unifies and generalizes the two earliest formulations of quantum mechanics – [matrix mechanics](https://en.wikipedia.org/wiki/Matrix_mechanics "Matrix mechanics") (invented by [Werner Heisenberg](https://en.wikipedia.org/wiki/Werner_Heisenberg "Werner Heisenberg")) and wave mechanics (invented by [Erwin Schrödinger](https://en.wikipedia.org/wiki/Erwin_Schr%C3%B6dinger "Erwin Schrödinger")).[\[33\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-33) An alternative formulation of quantum mechanics is [Feynman](https://en.wikipedia.org/wiki/Feynman "Feynman")'s [path integral formulation](https://en.wikipedia.org/wiki/Path_integral_formulation "Path integral formulation"), in which a quantum-mechanical amplitude is considered as a sum over all possible classical and non-classical paths between the initial and final states. This is the quantum-mechanical counterpart of the [action principle](https://en.wikipedia.org/wiki/Action_principle "Action principle") in classical mechanics.[\[34\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-34) Symmetries and conservation laws The Hamiltonian ![{\\displaystyle H}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b) is known as the *generator* of time evolution, since it defines a unitary time-evolution operator ![{\\displaystyle U(t)=e^{-iHt/\\hbar }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/17a17e78953fa7d61788cc8bc41a25ef254fc8e0) for each value of ![{\\displaystyle t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560). From this relation between ![{\\displaystyle U(t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/666c639df532e88616357c4991cabce9a57b5611) and ![{\\displaystyle H}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b), it follows that any observable ![{\\displaystyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3) that commutes with ![{\\displaystyle H}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b) will be *conserved*: its expectation value will not change over time.[\[7\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Zwiebach2022-7): 471 This statement generalizes, as mathematically, any Hermitian operator ![{\\displaystyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3) can generate a family of unitary operators parameterized by a variable ![{\\displaystyle t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560). Under the evolution generated by ![{\\displaystyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3), any observable ![{\\displaystyle B}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a) that commutes with ![{\\displaystyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3) will be conserved. Moreover, if ![{\\displaystyle B}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a) is conserved by evolution under ![{\\displaystyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3), then ![{\\displaystyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3) is conserved under the evolution generated by ![{\\displaystyle B}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a). This implies a quantum version of the result proven by [Emmy Noether](https://en.wikipedia.org/wiki/Emmy_Noether "Emmy Noether") in classical ([Lagrangian](https://en.wikipedia.org/wiki/Lagrangian_mechanics "Lagrangian mechanics")) mechanics: for every [differentiable](https://en.wikipedia.org/wiki/Differentiable "Differentiable") [symmetry](https://en.wikipedia.org/wiki/Symmetry_\(physics\) "Symmetry (physics)") of a Hamiltonian, there exists a corresponding [conservation law](https://en.wikipedia.org/wiki/Conservation_law "Conservation law"). Examples Free particle [![](https://upload.wikimedia.org/wikipedia/commons/5/56/Guassian_Dispersion.gif)](https://en.wikipedia.org/wiki/File:Guassian_Dispersion.gif) Position space probability density of a Gaussian [wave packet](https://en.wikipedia.org/wiki/Wave_packet "Wave packet") moving in one dimension in free space The simplest example of a quantum system with a position degree of freedom is a free particle in a single spatial dimension. A free particle is one which is not subject to external influences, so that its Hamiltonian consists only of its kinetic energy: ![{\\displaystyle H={\\frac {1}{2m}}P^{2}=-{\\frac {\\hbar ^{2}}{2m}}{\\frac {d^{2}}{dx^{2}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/084f585ee5c6cd8a34e323a6de7943227128afc3) The general solution of the Schrödinger equation is given by ![{\\displaystyle \\psi (x,t)={\\frac {1}{\\sqrt {2\\pi }}}\\int \_{-\\infty }^{\\infty }{\\hat {\\psi }}(k,0)e^{i(kx-{\\frac {\\hbar k^{2}}{2m}}t)}\\mathrm {d} k,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef4f021ba945856e3676808b11724109a8a74dad) which is a superposition of all possible [plane waves](https://en.wikipedia.org/wiki/Plane_wave "Plane wave") ![{\\displaystyle e^{i(kx-{\\frac {\\hbar k^{2}}{2m}}t)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fb4cd9a9984c84a493ce547babcf58e31b04f7e7), which are eigenstates of the momentum operator with momentum ![{\\displaystyle p=\\hbar k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24fee69175538303b28ac54e907baf53d0a58dbf). The coefficients of the superposition are ![{\\displaystyle {\\hat {\\psi }}(k,0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b8323c08418da8bc376c6d78b578d4729b927ea), which is the Fourier transform of the initial quantum state ![{\\displaystyle \\psi (x,0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/55ad442e07ca2d7986ef0787f9129fc325cde137). It is not possible for the solution to be a single momentum eigenstate, or a single position eigenstate, as these are not normalizable quantum states.[\[note 1\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-35) Instead, we can consider a Gaussian [wave packet](https://en.wikipedia.org/wiki/Wave_packet "Wave packet"): ![{\\displaystyle \\psi (x,0)={\\frac {1}{\\sqrt\[{4}\]{\\pi a}}}e^{-{\\frac {x^{2}}{2a}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a4c2dae82312897d5fd4c58986c426a6009e6840) which has Fourier transform, and therefore momentum distribution ![{\\displaystyle {\\hat {\\psi }}(k,0)={\\sqrt\[{4}\]{\\frac {a}{\\pi }}}e^{-{\\frac {ak^{2}}{2}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c4991535bba434314af8c27c16fff74f49ce367e) We see that as we make ![{\\displaystyle a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc) smaller the spread in position gets smaller, but the spread in momentum gets larger. Conversely, by making ![{\\displaystyle a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc) larger we make the spread in momentum smaller, but the spread in position gets larger. This illustrates the uncertainty principle. As we let the Gaussian wave packet evolve in time, we see that its center moves through space at a constant velocity (like a classical particle with no forces acting on it). However, the wave packet will also spread out as time progresses, which means that the position becomes more and more uncertain. The uncertainty in momentum, however, stays constant.[\[35\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-36) Particle in a box [![](https://upload.wikimedia.org/wikipedia/commons/thumb/2/27/Infinite_potential_well.svg/250px-Infinite_potential_well.svg.png)](https://en.wikipedia.org/wiki/File:Infinite_potential_well.svg) 1-dimensional potential energy box (or infinite potential well) The particle in a one-dimensional potential energy box is the most mathematically simple example where restraints lead to the quantization of energy levels. The box is defined as having zero potential energy everywhere *inside* a certain region, and therefore infinite potential energy everywhere *outside* that region.[\[26\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Cohen-Tannoudji-26): 77–78 For the one-dimensional case in the ![{\\displaystyle x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4) direction, the time-independent Schrödinger equation may be written ![{\\displaystyle -{\\frac {\\hbar ^{2}}{2m}}{\\frac {d^{2}\\psi }{dx^{2}}}=E\\psi .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc73e9216faf1390c3ed550b72be21fc068ec747) With the differential operator defined by ![{\\displaystyle {\\hat {p}}\_{x}=-i\\hbar {\\frac {d}{dx}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2511885975007002c7582a9c8175689076df210a)the previous equation is evocative of the [classic kinetic energy analogue](https://en.wikipedia.org/wiki/Kinetic_energy#Kinetic_energy_of_rigid_bodies "Kinetic energy"), ![{\\displaystyle {\\frac {1}{2m}}{\\hat {p}}\_{x}^{2}=E,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/422f57813ede91b8e348e786616c6297742b4fd7) with state ![{\\displaystyle \\psi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a) in this case having energy ![{\\displaystyle E}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b) coincident with the kinetic energy of the particle. The general solutions of the Schrödinger equation for the particle in a box are ![{\\displaystyle \\psi (x)=Ae^{ikx}+Be^{-ikx}\\qquad \\qquad E={\\frac {\\hbar ^{2}k^{2}}{2m}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5fb1b2f1d5afb42edb4eb98bf89791d283c1e53) or, from [Euler's formula](https://en.wikipedia.org/wiki/Euler%27s_formula "Euler's formula"), ![{\\displaystyle \\psi (x)=C\\sin(kx)+D\\cos(kx).\\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/229ebbbe587e0c678318cb85089a74ed289c971a) The infinite potential walls of the box determine the values of ![{\\displaystyle C,D,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e153d2c75e3492a32d4fafefec88846862c3b9b) and ![{\\displaystyle k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40) at ![{\\displaystyle x=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/953917eaf52f2e1baad54c8c9e3d6f9bb3710cdc) and ![{\\displaystyle x=L}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5fe40c588800aaab69041986b49a59664cd767a) where ![{\\displaystyle \\psi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a) must be zero. Thus, at ![{\\displaystyle x=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/953917eaf52f2e1baad54c8c9e3d6f9bb3710cdc), ![{\\displaystyle \\psi (0)=0=C\\sin(0)+D\\cos(0)=D}](https://wikimedia.org/api/rest_v1/media/math/render/svg/34188e64cbf80486b3ad311b53b512fa36de59c6) and ![{\\displaystyle D=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d375dfda80ee8df1d1d7aa8b962114044e464305). At ![{\\displaystyle x=L}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5fe40c588800aaab69041986b49a59664cd767a), ![{\\displaystyle \\psi (L)=0=C\\sin(kL),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c7a5bc9ef330efb5b6eba5d89c2ee026bbeea6a8) in which ![{\\displaystyle C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029) cannot be zero as this would conflict with the postulate that ![{\\displaystyle \\psi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a) has norm 1. Therefore, since ![{\\displaystyle \\sin(kL)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5177ccdb2057c5c1be728af20b8ef3d61f79999), ![{\\displaystyle kL}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d8865c1184b2c1dff6226dae50d3be91f4f01cfe) must be an integer multiple of ![{\\displaystyle \\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a), ![{\\displaystyle k={\\frac {n\\pi }{L}}\\qquad \\qquad n=1,2,3,\\ldots .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3fd91af3024e1f59cf57e04884494fd2c55664f8) This constraint on ![{\\displaystyle k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40) implies a constraint on the energy levels, yielding ![{\\displaystyle E\_{n}={\\frac {\\hbar ^{2}\\pi ^{2}n^{2}}{2mL^{2}}}={\\frac {n^{2}h^{2}}{8mL^{2}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe507fd1f92ebfe133aeb4f1da46ea27f569f38b) A [finite potential well](https://en.wikipedia.org/wiki/Finite_potential_well "Finite potential well") is the generalization of the infinite potential well problem to potential wells having finite depth. The finite potential well problem is mathematically more complicated than the infinite particle-in-a-box problem as the wave function is not pinned to zero at the walls of the well. Instead, the wave function must satisfy more complicated mathematical boundary conditions as it is nonzero in regions outside the well. Another related problem is that of the [rectangular potential barrier](https://en.wikipedia.org/wiki/Rectangular_potential_barrier "Rectangular potential barrier"), which furnishes a model for the [quantum tunneling](https://en.wikipedia.org/wiki/Quantum_tunneling "Quantum tunneling") effect that plays an important role in the performance of modern technologies such as [flash memory](https://en.wikipedia.org/wiki/Flash_memory "Flash memory") and [scanning tunneling microscopy](https://en.wikipedia.org/wiki/Scanning_tunneling_microscopy "Scanning tunneling microscopy"). Harmonic oscillator [![](https://upload.wikimedia.org/wikipedia/commons/9/90/QuantumHarmonicOscillatorAnimation.gif)](https://en.wikipedia.org/wiki/File:QuantumHarmonicOscillatorAnimation.gif) Some trajectories of a [harmonic oscillator](https://en.wikipedia.org/wiki/Harmonic_oscillator "Harmonic oscillator") (i.e. a ball attached to a [spring](https://en.wikipedia.org/wiki/Hooke%27s_law "Hooke's law")) in [classical mechanics](https://en.wikipedia.org/wiki/Classical_mechanics "Classical mechanics") (A-B) and quantum mechanics (C-H). In quantum mechanics, the position of the ball is represented by a [wave](https://en.wikipedia.org/wiki/Wave "Wave") (called the wave function), with the [real part](https://en.wikipedia.org/wiki/Real_part "Real part") shown in blue and the [imaginary part](https://en.wikipedia.org/wiki/Imaginary_part "Imaginary part") shown in red. Some of the trajectories (such as C, D, E, and F) are [standing waves](https://en.wikipedia.org/wiki/Standing_wave "Standing wave") (or "[stationary states](https://en.wikipedia.org/wiki/Stationary_state "Stationary state")"). Each standing-wave frequency is proportional to a possible [energy level](https://en.wikipedia.org/wiki/Energy_level "Energy level") of the oscillator. This "energy quantization" does not occur in classical physics, where the oscillator can have *any* energy. As in the classical case, the potential for the quantum harmonic oscillator is given by[\[7\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Zwiebach2022-7): 234 ![{\\displaystyle V(x)={\\frac {1}{2}}m\\omega ^{2}x^{2}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac9d8735e10198c090b576c765dc2778fe458b9c) This problem can either be treated by directly solving the Schrödinger equation, which is not trivial, or by using the more elegant "ladder method" first proposed by Paul Dirac. The [eigenstates](https://en.wikipedia.org/wiki/Eigenstate "Eigenstate") are given by ![{\\displaystyle \\psi \_{n}(x)={\\sqrt {\\frac {1}{2^{n}\\,n!}}}\\cdot \\left({\\frac {m\\omega }{\\pi \\hbar }}\\right)^{1/4}\\cdot e^{-{\\frac {m\\omega x^{2}}{2\\hbar }}}\\cdot H\_{n}\\left({\\sqrt {\\frac {m\\omega }{\\hbar }}}x\\right),\\qquad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a0031b1b0c9fa4a9e4e6957718d509846522c4d) ![{\\displaystyle n=0,1,2,\\ldots .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d652f9756f144ea772e4e632cb1398652d36372c) where *Hn* are the [Hermite polynomials](https://en.wikipedia.org/wiki/Hermite_polynomials "Hermite polynomials") ![{\\displaystyle H\_{n}(x)=(-1)^{n}e^{x^{2}}{\\frac {d^{n}}{dx^{n}}}\\left(e^{-x^{2}}\\right),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0da8b4a24291ce8b32bcc0665f490f3c76afc7d3) and the corresponding energy levels are ![{\\displaystyle E\_{n}=\\hbar \\omega \\left(n+{1 \\over 2}\\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/635fb242f4c34cd5ba947f4c8132ddd47f2872c0) This is another example illustrating the discretization of energy for [bound states](https://en.wikipedia.org/wiki/Bound_state "Bound state"). Mach–Zehnder interferometer [![](https://upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Mach-Zehnder_interferometer.svg/330px-Mach-Zehnder_interferometer.svg.png)](https://en.wikipedia.org/wiki/File:Mach-Zehnder_interferometer.svg) Schematic of a Mach–Zehnder interferometer The [Mach–Zehnder interferometer](https://en.wikipedia.org/wiki/Mach%E2%80%93Zehnder_interferometer "Mach–Zehnder interferometer") (MZI) illustrates the concepts of superposition and interference with linear algebra in dimension 2, rather than differential equations. It can be seen as a simplified version of the double-slit experiment, but it is of interest in its own right, for example in the [delayed choice quantum eraser](https://en.wikipedia.org/wiki/Delayed_choice_quantum_eraser "Delayed choice quantum eraser"), the [Elitzur–Vaidman bomb tester](https://en.wikipedia.org/wiki/Elitzur%E2%80%93Vaidman_bomb_tester "Elitzur–Vaidman bomb tester"), and in studies of quantum entanglement.[\[36\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Paris1999-37)[\[37\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Haack2010-38) We can model a photon going through the interferometer by considering that at each point it can be in a superposition of only two paths: the "lower" path which starts from the left, goes straight through both beam splitters, and ends at the top, and the "upper" path which starts from the bottom, goes straight through both beam splitters, and ends at the right. The quantum state of the photon is therefore a vector ![{\\displaystyle \\psi \\in \\mathbb {C} ^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a7a51a8280039fc22cd88d90915a952f8e020f47) that is a superposition of the "lower" path ![{\\displaystyle \\psi \_{l}={\\begin{pmatrix}1\\\\0\\end{pmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/feca39f1f03b4dc63be6f7a7c2060430b1217e2f) and the "upper" path ![{\\displaystyle \\psi \_{u}={\\begin{pmatrix}0\\\\1\\end{pmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/400e9751fcd7816718398d3892bec2ad26bb4713), that is, ![{\\displaystyle \\psi =\\alpha \\psi \_{l}+\\beta \\psi \_{u}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fb88364b79f3c611ca2ca1edb5356139bf4e1085) for complex ![{\\displaystyle \\alpha ,\\beta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4b46b57cfa0011b643037751809904d915c1b48). In order to respect the postulate that ![{\\displaystyle \\langle \\psi ,\\psi \\rangle =1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d5758e7a60b4e54bc46e01b0618919c65b787a0) we require that ![{\\displaystyle \|\\alpha \|^{2}+\|\\beta \|^{2}=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/18cd7473cdb894839d10852890517b1fb687c73b). Both [beam splitters](https://en.wikipedia.org/wiki/Beam_splitter "Beam splitter") are modelled as the unitary matrix ![{\\displaystyle B={\\frac {1}{\\sqrt {2}}}{\\begin{pmatrix}1\&i\\\\i&1\\end{pmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ddf502efcb65d0cbac5bb8ef1a6f163ac9cf2145), which means that when a photon meets the beam splitter it will either stay on the same path with a probability amplitude of ![{\\displaystyle 1/{\\sqrt {2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75a0bbdb60fcb73ac67d9970a5eb0808b87fd37d), or be reflected to the other path with a probability amplitude of ![{\\displaystyle i/{\\sqrt {2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e846b6a5731f3799960a4968399d85bc0b7fb9fd). The phase shifter on the upper arm is modelled as the unitary matrix ![{\\displaystyle P={\\begin{pmatrix}1&0\\\\0\&e^{i\\Delta \\Phi }\\end{pmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3df9457946dd8035c51a39e0926be9f07c7f0a3e), which means that if the photon is on the "upper" path it will gain a relative phase of ![{\\displaystyle \\Delta \\Phi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/20cedb08e6edea3cad9b2829ef67311bbe518dd2), and it will stay unchanged if it is in the lower path. A photon that enters the interferometer from the left will then be acted upon with a beam splitter ![{\\displaystyle B}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a), a phase shifter ![{\\displaystyle P}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a), and another beam splitter ![{\\displaystyle B}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a), and so end up in the state ![{\\displaystyle BPB\\psi \_{l}=ie^{i\\Delta \\Phi /2}{\\begin{pmatrix}-\\sin(\\Delta \\Phi /2)\\\\\\cos(\\Delta \\Phi /2)\\end{pmatrix}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7927a94da54f5d57b8accffdb9ad456a1e3b5033) and the probabilities that it will be detected at the right or at the top are given respectively by ![{\\displaystyle p(u)=\|\\langle \\psi \_{u},BPB\\psi \_{l}\\rangle \|^{2}=\\cos ^{2}{\\frac {\\Delta \\Phi }{2}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/110acb8ee7dc4e309de846470778f4767fe97f8f) ![{\\displaystyle p(l)=\|\\langle \\psi \_{l},BPB\\psi \_{l}\\rangle \|^{2}=\\sin ^{2}{\\frac {\\Delta \\Phi }{2}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86880ce53051688a4f591ebfb5183fbae013deee) One can therefore use the Mach–Zehnder interferometer to estimate the [phase shift](https://en.wikipedia.org/wiki/Phase_\(waves\) "Phase (waves)") by estimating these probabilities. It is interesting to consider what would happen if the photon were definitely in either the "lower" or "upper" paths between the beam splitters. This can be accomplished by blocking one of the paths, or equivalently by removing the first beam splitter (and feeding the photon from the left or the bottom, as desired). In both cases, there will be no interference between the paths anymore, and the probabilities are given by ![{\\displaystyle p(u)=p(l)=1/2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d2a9b3426984de1a64c07261c3e14b485320fee4), independently of the phase ![{\\displaystyle \\Delta \\Phi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/20cedb08e6edea3cad9b2829ef67311bbe518dd2). From this we can conclude that the photon does not take one path or another after the first beam splitter, but rather that it is in a genuine quantum superposition of the two paths.[\[38\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-vedral-39) Applications Quantum mechanics has had enormous success in explaining many of the features of our universe, with regard to small-scale and discrete quantities and interactions which cannot be explained by [classical methods](https://en.wikipedia.org/wiki/Classical_physics "Classical physics").[\[note 2\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-feynmanIII-40) Quantum mechanics is often the only theory that can reveal the individual behaviors of the subatomic particles that make up all forms of matter (electrons, [protons](https://en.wikipedia.org/wiki/Proton "Proton"), [neutrons](https://en.wikipedia.org/wiki/Neutron "Neutron"), [photons](https://en.wikipedia.org/wiki/Photon "Photon"), and others). [Solid-state physics](https://en.wikipedia.org/wiki/Solid-state_physics "Solid-state physics") and [materials science](https://en.wikipedia.org/wiki/Materials_science "Materials science") are dependent upon quantum mechanics.[\[39\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-marvincohen2008-41) In many aspects, modern technology operates at a scale where quantum effects are significant. Important applications of quantum theory include [quantum chemistry](https://en.wikipedia.org/wiki/Quantum_chemistry "Quantum chemistry"), [quantum optics](https://en.wikipedia.org/wiki/Quantum_optics "Quantum optics"), [quantum computing](https://en.wikipedia.org/wiki/Quantum_computing "Quantum computing"), [superconducting magnets](https://en.wikipedia.org/wiki/Superconducting_magnet "Superconducting magnet"), [light-emitting diodes](https://en.wikipedia.org/wiki/Light-emitting_diode "Light-emitting diode"), the [optical amplifier](https://en.wikipedia.org/wiki/Optical_amplifier "Optical amplifier") and the laser, the [transistor](https://en.wikipedia.org/wiki/Transistor "Transistor") and [semiconductors](https://en.wikipedia.org/wiki/Semiconductor "Semiconductor") such as the [microprocessor](https://en.wikipedia.org/wiki/Microprocessor "Microprocessor"), [medical and research imaging](https://en.wikipedia.org/wiki/Medical_imaging "Medical imaging") such as [magnetic resonance imaging](https://en.wikipedia.org/wiki/Magnetic_resonance_imaging "Magnetic resonance imaging") and [electron microscopy](https://en.wikipedia.org/wiki/Electron_microscopy "Electron microscopy").[\[40\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-42) Explanations for many biological and physical phenomena are rooted in the nature of the chemical bond, most notably the macro-molecule [DNA](https://en.wikipedia.org/wiki/DNA "DNA"). Relation to other scientific theories Classical mechanics The rules of quantum mechanics assert that the state space of a system is a Hilbert space and that observables of the system are Hermitian operators acting on vectors in that space – although they do not tell us which Hilbert space or which operators. These can be chosen appropriately in order to obtain a quantitative description of a quantum system, a necessary step in making physical predictions. An important guide for making these choices is the [correspondence principle](https://en.wikipedia.org/wiki/Correspondence_principle "Correspondence principle"), a heuristic which states that the predictions of quantum mechanics reduce to those of [classical mechanics](https://en.wikipedia.org/wiki/Classical_mechanics "Classical mechanics") in the regime of large [quantum numbers](https://en.wikipedia.org/wiki/Quantum_number "Quantum number").[\[41\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Tipler-43) One can also start from an established classical model of a particular system, and then try to guess the underlying quantum model that would give rise to the classical model in the correspondence limit. This approach is known as [quantization](https://en.wikipedia.org/wiki/Canonical_quantization "Canonical quantization").[\[42\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Peres1993-44): 299 [\[43\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-45) When quantum mechanics was originally formulated, it was applied to models whose correspondence limit was [non-relativistic](https://en.wikipedia.org/wiki/Theory_of_relativity "Theory of relativity") classical mechanics. For instance, the well-known model of the [quantum harmonic oscillator](https://en.wikipedia.org/wiki/Quantum_harmonic_oscillator "Quantum harmonic oscillator") uses an explicitly non-relativistic expression for the [kinetic energy](https://en.wikipedia.org/wiki/Kinetic_energy "Kinetic energy") of the oscillator, and is thus a quantum version of the [classical harmonic oscillator](https://en.wikipedia.org/wiki/Harmonic_oscillator "Harmonic oscillator").[\[7\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Zwiebach2022-7): 234 Complications arise with [chaotic systems](https://en.wikipedia.org/wiki/Chaotic_systems "Chaotic systems"), which do not have good quantum numbers, and [quantum chaos](https://en.wikipedia.org/wiki/Quantum_chaos "Quantum chaos") studies the relationship between classical and quantum descriptions in these systems.[\[42\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Peres1993-44): 353 [Quantum decoherence](https://en.wikipedia.org/wiki/Quantum_decoherence "Quantum decoherence") is a mechanism through which quantum systems lose [coherence](https://en.wikipedia.org/wiki/Quantum_coherence "Quantum coherence"), and thus become incapable of displaying many typically quantum effects: [quantum superpositions](https://en.wikipedia.org/wiki/Quantum_superposition "Quantum superposition") become simply probabilistic mixtures, and quantum entanglement becomes simply classical correlations.[\[7\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Zwiebach2022-7): 687–730 Quantum coherence is not typically evident at macroscopic scales, though at temperatures approaching [absolute zero](https://en.wikipedia.org/wiki/Absolute_zero "Absolute zero") quantum behavior may manifest macroscopically.[\[note 3\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-46) Many macroscopic properties of a classical system are a direct consequence of the quantum behavior of its parts. For example, the stability of bulk matter (consisting of atoms and [molecules](https://en.wikipedia.org/wiki/Molecule "Molecule") which would quickly collapse under electric forces alone), the rigidity of solids, and the mechanical, thermal, chemical, optical and magnetic properties of matter are all results of the interaction of [electric charges](https://en.wikipedia.org/wiki/Electric_charge "Electric charge") under the rules of quantum mechanics.[\[44\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-47) Special relativity and electrodynamics Early attempts to merge quantum mechanics with [special relativity](https://en.wikipedia.org/wiki/Special_relativity "Special relativity") involved the replacement of the Schrödinger equation with a covariant equation such as the [Klein–Gordon equation](https://en.wikipedia.org/wiki/Klein%E2%80%93Gordon_equation "Klein–Gordon equation") or the [Dirac equation](https://en.wikipedia.org/wiki/Dirac_equation "Dirac equation"). While these theories were successful in explaining many experimental results, they had certain unsatisfactory qualities stemming from their neglect of the relativistic creation and annihilation of particles. A fully relativistic quantum theory required the development of quantum field theory, which applies quantization to a field (rather than a fixed set of particles). The first complete quantum field theory, [quantum electrodynamics](https://en.wikipedia.org/wiki/Quantum_electrodynamics "Quantum electrodynamics"), provides a fully quantum description of the [electromagnetic interaction](https://en.wikipedia.org/wiki/Electromagnetic_interaction "Electromagnetic interaction"). Quantum electrodynamics is, along with [general relativity](https://en.wikipedia.org/wiki/General_relativity "General relativity"), one of the most accurate physical theories ever devised.[\[45\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-48)[\[46\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-49) The full apparatus of quantum field theory is often unnecessary for describing electrodynamic systems. A simpler approach, one that has been used since the inception of quantum mechanics, is to treat [charged](https://en.wikipedia.org/wiki/Electric_charge "Electric charge") particles as quantum mechanical objects being acted on by a classical [electromagnetic field](https://en.wikipedia.org/wiki/Electromagnetic_field "Electromagnetic field"). For example, the elementary quantum model of the [hydrogen atom](https://en.wikipedia.org/wiki/Hydrogen_atom "Hydrogen atom") describes the [electric field](https://en.wikipedia.org/wiki/Electric_field "Electric field") of the hydrogen atom using a classical ![{\\displaystyle \\textstyle -e^{2}/(4\\pi \\epsilon \_{\_{0}}r)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c4876353362da0a234af137590c6379d8f8c273) [Coulomb potential](https://en.wikipedia.org/wiki/Electric_potential "Electric potential").[\[7\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Zwiebach2022-7): 285 Likewise, in a [Stern–Gerlach experiment](https://en.wikipedia.org/wiki/Stern%E2%80%93Gerlach_experiment "Stern–Gerlach experiment"), a charged particle is modeled as a quantum system, while the background magnetic field is described classically.[\[42\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Peres1993-44): 26 This "semi-classical" approach fails if quantum fluctuations in the electromagnetic field play an important role, such as in the emission of photons by [charged particles](https://en.wikipedia.org/wiki/Charged_particle "Charged particle"). [Quantum field](https://en.wikipedia.org/wiki/Field_\(physics\) "Field (physics)") theories for the [strong nuclear force](https://en.wikipedia.org/wiki/Strong_nuclear_force "Strong nuclear force") and the [weak nuclear force](https://en.wikipedia.org/wiki/Weak_nuclear_force "Weak nuclear force") have also been developed. The quantum field theory of the strong nuclear force is called [quantum chromodynamics](https://en.wikipedia.org/wiki/Quantum_chromodynamics "Quantum chromodynamics"), and describes the interactions of subnuclear particles such as [quarks](https://en.wikipedia.org/wiki/Quark "Quark") and [gluons](https://en.wikipedia.org/wiki/Gluon "Gluon"). The weak nuclear force and the electromagnetic force were unified, in their quantized forms, into a single quantum field theory (known as [electroweak theory](https://en.wikipedia.org/wiki/Electroweak_theory "Electroweak theory")), by the physicists [Abdus Salam](https://en.wikipedia.org/wiki/Abdus_Salam "Abdus Salam"), [Sheldon Glashow](https://en.wikipedia.org/wiki/Sheldon_Glashow "Sheldon Glashow") and [Steven Weinberg](https://en.wikipedia.org/wiki/Steven_Weinberg "Steven Weinberg").[\[47\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-50) Relation to general relativity Even though the predictions of both quantum theory and general relativity have been supported by rigorous and repeated [empirical evidence](https://en.wikipedia.org/wiki/Empirical_evidence "Empirical evidence"), their abstract formalisms contradict each other and they have proven extremely difficult to incorporate into one consistent, cohesive model. Gravity is negligible in many areas of particle physics, so that unification between general relativity and quantum mechanics is not an urgent issue in those particular applications. However, the lack of a correct theory of [quantum gravity](https://en.wikipedia.org/wiki/Quantum_gravity "Quantum gravity") is an important issue in [physical cosmology](https://en.wikipedia.org/wiki/Physical_cosmology "Physical cosmology") and the search by physicists for an elegant "[Theory of Everything](https://en.wikipedia.org/wiki/Theory_of_Everything "Theory of Everything")" (TOE). Consequently, resolving the inconsistencies between both theories has been a major goal of 20th- and 21st-century physics. This TOE would combine not only the models of subatomic physics but also derive the four fundamental forces of nature from a single force or phenomenon.[\[48\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-NYT-20221010-51) One proposal for doing so is [string theory](https://en.wikipedia.org/wiki/String_theory "String theory"), which posits that the [point-like particles](https://en.wikipedia.org/wiki/Point_particle "Point particle") of [particle physics](https://en.wikipedia.org/wiki/Particle_physics "Particle physics") are replaced by [one-dimensional](https://en.wikipedia.org/wiki/Dimension_\(mathematics_and_physics\) "Dimension (mathematics and physics)") objects called [strings](https://en.wikipedia.org/wiki/String_\(physics\) "String (physics)"). String theory describes how these strings propagate through space and interact with each other. On distance scales larger than the string scale, a string looks just like an ordinary particle, with its [mass](https://en.wikipedia.org/wiki/Mass "Mass"), [charge](https://en.wikipedia.org/wiki/Charge_\(physics\) "Charge (physics)"), and other properties determined by the [vibrational](https://en.wikipedia.org/wiki/Vibration "Vibration") state of the string. In string theory, one of the many vibrational states of the string corresponds to the [graviton](https://en.wikipedia.org/wiki/Graviton "Graviton"), a quantum mechanical particle that carries gravitational force.[\[49\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-52)[\[50\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-53) Another popular theory is [loop quantum gravity](https://en.wikipedia.org/wiki/Loop_quantum_gravity "Loop quantum gravity") (LQG), which describes quantum properties of gravity and is thus a theory of [quantum spacetime](https://en.wikipedia.org/wiki/Quantum_spacetime "Quantum spacetime"). LQG is an attempt to merge and adapt standard quantum mechanics and standard general relativity. This theory describes space as an extremely fine fabric "woven" of finite loops called [spin networks](https://en.wikipedia.org/wiki/Spin_network "Spin network"). The evolution of a spin network over time is called a [spin foam](https://en.wikipedia.org/wiki/Spin_foam "Spin foam"). The characteristic length scale of a spin foam is the [Planck length](https://en.wikipedia.org/wiki/Planck_length "Planck length"), approximately 1.616×10−35 m, and so lengths shorter than the Planck length are not physically meaningful in LQG.[\[51\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-54) Philosophical implications Unsolved problem in physics Is there a preferred interpretation of quantum mechanics? How does the quantum description of reality, which includes elements such as the "[superposition](https://en.wikipedia.org/wiki/Superposition_principle "Superposition principle") of states" and "[wave function collapse](https://en.wikipedia.org/wiki/Wave_function_collapse "Wave function collapse")", give rise to the reality we perceive? Since its inception, the many counter-intuitive aspects and results of quantum mechanics have provoked strong [philosophical](https://en.wikipedia.org/wiki/Philosophical "Philosophical") debates and many [interpretations](https://en.wikipedia.org/wiki/Interpretations_of_quantum_mechanics "Interpretations of quantum mechanics"). The arguments centre on the probabilistic nature of quantum mechanics, the difficulties with [wavefunction collapse](https://en.wikipedia.org/wiki/Wavefunction_collapse "Wavefunction collapse") and the related [measurement problem](https://en.wikipedia.org/wiki/Measurement_problem "Measurement problem"), and [quantum nonlocality](https://en.wikipedia.org/wiki/Quantum_nonlocality "Quantum nonlocality"). Perhaps the only consensus that exists about these issues is that there is no consensus. [Richard Feynman](https://en.wikipedia.org/wiki/Richard_Feynman "Richard Feynman") once said, "I think I can safely say that nobody understands quantum mechanics."[\[52\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-55) According to [Steven Weinberg](https://en.wikipedia.org/wiki/Steven_Weinberg "Steven Weinberg"), "There is now in my opinion no entirely satisfactory interpretation of quantum mechanics."[\[53\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-56) The views of [Niels Bohr](https://en.wikipedia.org/wiki/Niels_Bohr "Niels Bohr"), Werner Heisenberg and other physicists are often grouped together as the "[Copenhagen interpretation](https://en.wikipedia.org/wiki/Copenhagen_interpretation "Copenhagen interpretation")".[\[54\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-57)[\[55\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-58) According to these views, the probabilistic nature of quantum mechanics is not a *temporary* feature which will eventually be replaced by a deterministic theory, but is instead a *final* renunciation of the classical idea of "causality". Bohr in particular emphasized that any well-defined application of the quantum mechanical formalism must always make reference to the experimental arrangement, due to the [complementary](https://en.wikipedia.org/wiki/Complementarity_\(physics\) "Complementarity (physics)") nature of evidence obtained under different experimental situations. Copenhagen-type interpretations were adopted by Nobel laureates in quantum physics, including Bohr,[\[56\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-BohrComo-59) Heisenberg,[\[57\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-60) Schrödinger,[\[58\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-61) Feynman,[\[2\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Feynman-2) and [Zeilinger](https://en.wikipedia.org/wiki/Anton_Zeilinger "Anton Zeilinger")[\[59\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-MaKoflerZeilinger-62) as well as 21st-century researchers in quantum foundations.[\[60\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-:25-63) [Albert Einstein](https://en.wikipedia.org/wiki/Albert_Einstein "Albert Einstein"), himself one of the founders of [quantum theory](https://en.wikipedia.org/wiki/Old_quantum_theory "Old quantum theory"), was troubled by its apparent failure to respect some cherished metaphysical principles, such as [determinism](https://en.wikipedia.org/wiki/Determinism "Determinism") and [locality](https://en.wikipedia.org/wiki/Principle_of_locality "Principle of locality"). Einstein's long-running exchanges with Bohr about the meaning and status of quantum mechanics are now known as the [Bohr–Einstein debates](https://en.wikipedia.org/wiki/Bohr%E2%80%93Einstein_debates "Bohr–Einstein debates"). Einstein believed that underlying quantum mechanics must be a theory that explicitly forbids [action at a distance](https://en.wikipedia.org/wiki/Action_at_a_distance "Action at a distance"). He argued that quantum mechanics was incomplete, a theory that was valid but not fundamental, analogous to how [thermodynamics](https://en.wikipedia.org/wiki/Thermodynamics "Thermodynamics") is valid, but the fundamental theory behind it is [statistical mechanics](https://en.wikipedia.org/wiki/Statistical_mechanics "Statistical mechanics"). In 1935, Einstein and his collaborators [Boris Podolsky](https://en.wikipedia.org/wiki/Boris_Podolsky "Boris Podolsky") and [Nathan Rosen](https://en.wikipedia.org/wiki/Nathan_Rosen "Nathan Rosen") published an argument that the principle of locality implies the incompleteness of quantum mechanics, a [thought experiment](https://en.wikipedia.org/wiki/Thought_experiment "Thought experiment") later termed the [Einstein–Podolsky–Rosen paradox](https://en.wikipedia.org/wiki/Einstein%E2%80%93Podolsky%E2%80%93Rosen_paradox "Einstein–Podolsky–Rosen paradox").[\[note 4\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-68) In 1964, [John Bell](https://en.wikipedia.org/wiki/John_Stewart_Bell "John Stewart Bell") showed that EPR's principle of locality, together with determinism, was actually incompatible with quantum mechanics: they implied constraints on the correlations produced by distance systems, now known as [Bell inequalities](https://en.wikipedia.org/wiki/Bell_inequalities "Bell inequalities"), that can be violated by entangled particles.[\[65\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-69) Since then [several experiments](https://en.wikipedia.org/wiki/Bell_test "Bell test") have been performed to obtain these correlations, with the result that they do in fact violate Bell inequalities, and thus falsify the conjunction of locality with determinism.[\[16\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-wiseman15-16)[\[17\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-wolchover17-17) [Bohmian mechanics](https://en.wikipedia.org/wiki/Bohmian_mechanics "Bohmian mechanics") shows that it is possible to reformulate quantum mechanics to make it deterministic, at the price of making it explicitly nonlocal. It attributes not only a wave function to a physical system, but in addition a real position, that evolves deterministically under a nonlocal guiding equation. The evolution of a physical system is given at all times by the Schrödinger equation together with the guiding equation; there is never a collapse of the wave function. This solves the measurement problem.[\[66\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-70) [![](https://upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Schroedingers_cat_film.svg/250px-Schroedingers_cat_film.svg.png)](https://en.wikipedia.org/wiki/File:Schroedingers_cat_film.svg) The [Schrödinger's cat](https://en.wikipedia.org/wiki/Schr%C3%B6dinger%27s_cat "Schrödinger's cat") thought experiment can be used to visualize the many-worlds interpretation of quantum mechanics, where a branching of the universe occurs through a superposition of two quantum mechanical states. Everett's [many-worlds interpretation](https://en.wikipedia.org/wiki/Many-worlds_interpretation "Many-worlds interpretation"), formulated in 1956, holds that *all* the possibilities described by quantum theory *simultaneously* occur in a multiverse composed of mostly independent parallel universes.[\[67\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-71) This is a consequence of removing the axiom of the collapse of the wave packet. All possible states of the measured system and the measuring apparatus, together with the observer, are present in a real physical quantum superposition. While the multiverse is deterministic, we perceive non-deterministic behavior governed by probabilities, because we do not observe the multiverse as a whole, but only one parallel universe at a time. Exactly how this is supposed to work has been the subject of much debate. Several attempts have been made to make sense of this and derive the Born rule,[\[68\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-dewitt73-72)[\[69\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-wallace2003-73) with no consensus on whether they have been successful.[\[70\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-ballentine1973-74)[\[71\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-75)[\[72\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-kent2009-76) [Relational quantum mechanics](https://en.wikipedia.org/wiki/Relational_quantum_mechanics "Relational quantum mechanics") appeared in the late 1990s as a modern derivative of Copenhagen-type ideas,[\[73\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-77)[\[74\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-78) and [QBism](https://en.wikipedia.org/wiki/QBism "QBism") was developed some years later.[\[75\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-:23-79)[\[76\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-80) History Quantum mechanics was developed in the early decades of the 20th century, driven by the need to explain phenomena that, in some cases, had been observed in earlier times. Scientific inquiry into the wave nature of light began in the 17th and 18th centuries, when scientists such as [Robert Hooke](https://en.wikipedia.org/wiki/Robert_Hooke "Robert Hooke"), [Christiaan Huygens](https://en.wikipedia.org/wiki/Christiaan_Huygens "Christiaan Huygens") and [Leonhard Euler](https://en.wikipedia.org/wiki/Leonhard_Euler "Leonhard Euler") proposed a wave theory of light based on experimental observations.[\[77\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Born_&_Wolf-81) In 1803 English [polymath](https://en.wikipedia.org/wiki/Polymath "Polymath") [Thomas Young](https://en.wikipedia.org/wiki/Thomas_Young_\(scientist\) "Thomas Young (scientist)") described the famous [double-slit experiment](https://en.wikipedia.org/wiki/Young%27s_interference_experiment "Young's interference experiment").[\[78\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-82) This experiment played a major role in the general acceptance of the [wave theory of light](https://en.wikipedia.org/wiki/Wave_theory_of_light "Wave theory of light"). During the early 19th century, [chemical](https://en.wikipedia.org/wiki/Chemistry "Chemistry") research by [John Dalton](https://en.wikipedia.org/wiki/John_Dalton "John Dalton") and [Amedeo Avogadro](https://en.wikipedia.org/wiki/Amedeo_Avogadro "Amedeo Avogadro") lent weight to the [atomic theory](https://en.wikipedia.org/wiki/Atomic_theory "Atomic theory") of matter, an idea that [James Clerk Maxwell](https://en.wikipedia.org/wiki/James_Clerk_Maxwell "James Clerk Maxwell"), [Ludwig Boltzmann](https://en.wikipedia.org/wiki/Ludwig_Boltzmann "Ludwig Boltzmann") and others built upon to establish the [kinetic theory of gases](https://en.wikipedia.org/wiki/Kinetic_theory_of_gases "Kinetic theory of gases"). The successes of kinetic theory gave further credence to the idea that matter is composed of atoms, yet the theory also had shortcomings that would only be resolved by the development of quantum mechanics.[\[79\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Feynman-kinetic-theory-83) While the early conception of atoms from [Greek philosophy](https://en.wikipedia.org/wiki/Greek_philosophy "Greek philosophy") had been that they were indivisible units – the word "atom" deriving from the [Greek](https://en.wikipedia.org/wiki/Greek_language "Greek language") for 'uncuttable' – the 19th century saw the formulation of hypotheses about subatomic structure. One important discovery in that regard was [Michael Faraday](https://en.wikipedia.org/wiki/Michael_Faraday "Michael Faraday")'s 1838 observation of a glow caused by an electrical discharge inside a glass tube containing gas at low pressure. [Julius Plücker](https://en.wikipedia.org/wiki/Julius_Pl%C3%BCcker "Julius Plücker"), [Johann Wilhelm Hittorf](https://en.wikipedia.org/wiki/Johann_Wilhelm_Hittorf "Johann Wilhelm Hittorf") and [Eugen Goldstein](https://en.wikipedia.org/wiki/Eugen_Goldstein "Eugen Goldstein") carried on and improved upon Faraday's work, leading to the identification of [cathode rays](https://en.wikipedia.org/wiki/Cathode_rays "Cathode rays"), which [J. J. Thomson](https://en.wikipedia.org/wiki/J._J._Thomson "J. J. Thomson") found to consist of subatomic particles that would be called electrons.[\[80\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-84)[\[81\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-85) [![](https://upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Max_Planck_%281858-1947%29.jpg/250px-Max_Planck_%281858-1947%29.jpg)](https://en.wikipedia.org/wiki/File:Max_Planck_\(1858-1947\).jpg) [Max Planck](https://en.wikipedia.org/wiki/Max_Planck "Max Planck") is considered the father of the quantum theory. The [black-body radiation](https://en.wikipedia.org/wiki/Black-body_radiation "Black-body radiation") problem was discovered by [Gustav Kirchhoff](https://en.wikipedia.org/wiki/Gustav_Kirchhoff "Gustav Kirchhoff") in 1859. In 1900, Max Planck proposed the hypothesis that energy is radiated and absorbed in discrete "quanta" (or energy packets), yielding a calculation that precisely matched the observed patterns of black-body radiation.[\[82\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-86) The word *quantum* derives from the [Latin](https://en.wikipedia.org/wiki/Latin "Latin"), meaning "how great" or "how much".[\[83\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-87) According to Planck, quantities of energy could be thought of as divided into "elements" whose size (*E*) would be proportional to their [frequency](https://en.wikipedia.org/wiki/Frequency "Frequency") (*ν*): ![{\\displaystyle E=h\\nu \\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/24767663bb931daecf1eb628be4fea46f32e3622), where *h* is the [Planck constant](https://en.wikipedia.org/wiki/Planck_constant "Planck constant"). Planck cautiously insisted that this was only an aspect of the processes of absorption and emission of radiation and was not the *physical reality* of the radiation.[\[84\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-88) In fact, he considered his quantum hypothesis a mathematical trick to get the right answer rather than a sizable discovery.[\[85\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Kragh-89) However, in 1905 Albert Einstein interpreted Planck's quantum hypothesis [realistically](https://en.wikipedia.org/wiki/Local_realism "Local realism") and used it to explain the [photoelectric effect](https://en.wikipedia.org/wiki/Photoelectric_effect "Photoelectric effect"), in which shining light on certain materials can eject electrons from the material. Niels Bohr then developed Planck's ideas about radiation into a [model of the hydrogen atom](https://en.wikipedia.org/wiki/Bohr_model "Bohr model") that successfully predicted the [spectral lines](https://en.wikipedia.org/wiki/Spectral_line "Spectral line") of hydrogen.[\[86\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-90) Einstein further developed this idea to show that an [electromagnetic wave](https://en.wikipedia.org/wiki/Electromagnetic_wave "Electromagnetic wave") such as light could also be described as a particle (later called the photon), with a discrete amount of energy that depends on its frequency.[\[87\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-91) In his paper "On the Quantum Theory of Radiation", Einstein expanded on the interaction between energy and matter to explain the absorption and emission of energy by atoms. Although overshadowed at the time by his general theory of relativity, this paper articulated the mechanism underlying the stimulated emission of radiation,[\[88\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-92) which became the basis of the laser.[\[89\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-93) [![](https://upload.wikimedia.org/wikipedia/commons/thumb/6/6e/Solvay_conference_1927.jpg/500px-Solvay_conference_1927.jpg)](https://en.wikipedia.org/wiki/File:Solvay_conference_1927.jpg) The 1927 [Solvay Conference](https://en.wikipedia.org/wiki/Solvay_Conference "Solvay Conference") in [Brussels](https://en.wikipedia.org/wiki/Brussels "Brussels") was the fifth world physics conference. This phase is known as the [old quantum theory](https://en.wikipedia.org/wiki/Old_quantum_theory "Old quantum theory"). Never complete or self-consistent, the old quantum theory was rather a set of [heuristic](https://en.wikipedia.org/wiki/Heuristic "Heuristic") corrections to classical mechanics.[\[90\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-terHaar-94)[\[91\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-95) The theory is now understood as a [semi-classical approximation](https://en.wikipedia.org/wiki/WKB_approximation#Application_to_the_Schr%C3%B6dinger_equation "WKB approximation") to modern quantum mechanics.[\[92\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-96)[\[93\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-97) Notable results from this period include, in addition to the work of Planck, Einstein and Bohr mentioned above, Einstein and [Peter Debye](https://en.wikipedia.org/wiki/Peter_Debye "Peter Debye")'s work on the [specific heat](https://en.wikipedia.org/wiki/Specific_heat "Specific heat") of solids, Bohr and [Hendrika Johanna van Leeuwen](https://en.wikipedia.org/wiki/Hendrika_Johanna_van_Leeuwen "Hendrika Johanna van Leeuwen")'s [proof](https://en.wikipedia.org/wiki/Bohr%E2%80%93Van_Leeuwen_theorem "Bohr–Van Leeuwen theorem") that classical physics cannot account for [diamagnetism](https://en.wikipedia.org/wiki/Diamagnetism "Diamagnetism"), and [Arnold Sommerfeld](https://en.wikipedia.org/wiki/Arnold_Sommerfeld "Arnold Sommerfeld")'s extension of the Bohr model to include special-relativistic effects.[\[90\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-terHaar-94)[\[94\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Aharoni-98) In the mid-1920s quantum mechanics was developed to become the standard formulation for atomic physics. In 1923, the French physicist [Louis de Broglie](https://en.wikipedia.org/wiki/Louis_de_Broglie "Louis de Broglie") put forward his theory of matter waves by stating that particles can exhibit wave characteristics and vice versa. Building on de Broglie's approach, modern quantum mechanics was born in 1925, when the German physicists Werner Heisenberg, Max Born, and [Pascual Jordan](https://en.wikipedia.org/wiki/Pascual_Jordan "Pascual Jordan")[\[95\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Edwards79-99)[\[96\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Edwards81-100) developed [matrix mechanics](https://en.wikipedia.org/wiki/Matrix_mechanics "Matrix mechanics") and the Austrian physicist Erwin Schrödinger invented [wave mechanics](https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation "Schrödinger equation"). Born introduced the probabilistic interpretation of Schrödinger's wave function in July 1926.[\[97\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-101) Thus, the entire field of quantum physics emerged, leading to its wider acceptance at the Fifth [Solvay Conference](https://en.wikipedia.org/wiki/Solvay_Conference "Solvay Conference") in 1927.[\[98\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-pais1997-102) By 1930, quantum mechanics had been further unified and formalized by [David Hilbert](https://en.wikipedia.org/wiki/David_Hilbert "David Hilbert"), Paul Dirac and [John von Neumann](https://en.wikipedia.org/wiki/John_von_Neumann "John von Neumann")[\[99\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-103) with greater emphasis on [measurement](https://en.wikipedia.org/wiki/Measurement_in_quantum_mechanics "Measurement in quantum mechanics"), the statistical nature of our knowledge of reality, and [philosophical speculation about the 'observer'](https://en.wikipedia.org/wiki/Interpretations_of_quantum_mechanics "Interpretations of quantum mechanics"). It has since permeated many disciplines, including quantum chemistry, [quantum electronics](https://en.wikipedia.org/wiki/Quantum_electronics "Quantum electronics"), [quantum optics](https://en.wikipedia.org/wiki/Quantum_optics "Quantum optics"), and [quantum information science](https://en.wikipedia.org/wiki/Quantum_information_science "Quantum information science"). It also provides a useful framework for many features of the modern [periodic table of elements](https://en.wikipedia.org/wiki/Periodic_table_of_elements "Periodic table of elements"), and describes the behaviors of [atoms](https://en.wikipedia.org/wiki/Atoms "Atoms") during [chemical bonding](https://en.wikipedia.org/wiki/Chemical_bond "Chemical bond") and the flow of electrons in computer [semiconductors](https://en.wikipedia.org/wiki/Semiconductor "Semiconductor"), and therefore plays a crucial role in many modern technologies. While quantum mechanics was constructed to describe the world of the very small, it is also needed to explain some [macroscopic](https://en.wikipedia.org/wiki/Macroscopic "Macroscopic") phenomena such as [superconductors](https://en.wikipedia.org/wiki/Superconductors "Superconductors")[\[100\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-feynman2015-104) and [superfluids](https://en.wikipedia.org/wiki/Superfluid "Superfluid").[\[101\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-105) See also - [Bra–ket notation](https://en.wikipedia.org/wiki/Bra%E2%80%93ket_notation "Bra–ket notation") - [Einstein's thought experiments](https://en.wikipedia.org/wiki/Einstein%27s_thought_experiments "Einstein's thought experiments") - [List of textbooks on classical mechanics and quantum mechanics](https://en.wikipedia.org/wiki/List_of_textbooks_on_classical_mechanics_and_quantum_mechanics "List of textbooks on classical mechanics and quantum mechanics") - [Macroscopic quantum phenomena](https://en.wikipedia.org/wiki/Macroscopic_quantum_phenomena "Macroscopic quantum phenomena") - [Phase-space formulation](https://en.wikipedia.org/wiki/Phase-space_formulation "Phase-space formulation") - [Regularization (physics)](https://en.wikipedia.org/wiki/Regularization_\(physics\) "Regularization (physics)") - [Two-state quantum system](https://en.wikipedia.org/wiki/Two-state_quantum_system "Two-state quantum system") Explanatory notes 1. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-35)** A momentum eigenstate would be a perfectly monochromatic wave of infinite extent, which is not square-integrable. Likewise, a position eigenstate would be a [Dirac delta distribution](https://en.wikipedia.org/wiki/Dirac_delta_distribution "Dirac delta distribution"), not square-integrable and technically not a function at all. Consequently, neither can belong to the particle's Hilbert space. Physicists sometimes introduce fictitious "bases" for a Hilbert space comprising elements outside that space. These are invented for calculational convenience and do not represent physical states.[\[26\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-Cohen-Tannoudji-26): 100–105 2. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-feynmanIII_40-0)** See, for example, [the Feynman Lectures on Physics](https://en.wikipedia.org/wiki/The_Feynman_Lectures_on_Physics "The Feynman Lectures on Physics") for some of the technological applications which use quantum mechanics, e.g., [transistors](https://en.wikipedia.org/wiki/Transistor "Transistor") (vol **III**, pp. 14–11 ff), [integrated circuits](https://en.wikipedia.org/wiki/Integrated_circuit "Integrated circuit"), which are follow-on technology in solid-state physics (vol **II**, pp. 8–6), and [lasers](https://en.wikipedia.org/wiki/Laser "Laser") (vol **III**, pp. 9–13). 3. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-46)** See *[Macroscopic quantum phenomena](https://en.wikipedia.org/wiki/Macroscopic_quantum_phenomena "Macroscopic quantum phenomena")*, *[Bose–Einstein condensate](https://en.wikipedia.org/wiki/Bose%E2%80%93Einstein_condensate "Bose–Einstein condensate")*, and *[Quantum machine](https://en.wikipedia.org/wiki/Quantum_machine "Quantum machine")* 4. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-68)** The published form of the EPR argument was due to Podolsky, and Einstein himself was not satisfied with it. In his own publications and correspondence, Einstein used a different argument to insist that quantum mechanics is an incomplete theory.[\[61\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-spekkens-64)[\[62\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-howard-65)[\[63\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-66)[\[64\]](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-67) References 1. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Born1926_1-0)** [Born, M.](https://en.wikipedia.org/wiki/Max_Born "Max Born") (1926). "Zur Quantenmechanik der Stoßvorgänge" \[On the Quantum Mechanics of Collision Processes\]. *Zeitschrift für Physik* (in German). **37** (12): 863–867\. [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[1926ZPhy...37..863B](https://ui.adsabs.harvard.edu/abs/1926ZPhy...37..863B). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1007/BF01397477](https://doi.org/10.1007%2FBF01397477). [ISSN](https://en.wikipedia.org/wiki/ISSN_\(identifier\) "ISSN (identifier)") [1434-6001](https://search.worldcat.org/issn/1434-6001). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [119896026](https://api.semanticscholar.org/CorpusID:119896026). 2. ^ [***a***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Feynman_2-0) [***b***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Feynman_2-1) [***c***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Feynman_2-2) [***d***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Feynman_2-3) Feynman, Richard; Leighton, Robert; Sands, Matthew (1964). [*The Feynman Lectures on Physics*](https://feynmanlectures.caltech.edu/III_01.html). Vol. 3. California Institute of Technology. Retrieved 19 December 2020. Reprinted, Addison-Wesley, 1989, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-201-50064-6](https://en.wikipedia.org/wiki/Special:BookSources/978-0-201-50064-6 "Special:BookSources/978-0-201-50064-6") 3. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-3)** Jaeger, Gregg (September 2014). "What in the (quantum) world is macroscopic?". *American Journal of Physics*. **82** (9): 896–905\. [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[2014AmJPh..82..896J](https://ui.adsabs.harvard.edu/abs/2014AmJPh..82..896J). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1119/1.4878358](https://doi.org/10.1119%2F1.4878358). 4. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-4)** Fein, Yaakov Y.; Geyer, Philipp; Zwick, Patrick; Kiałka, Filip; Pedalino, Sebastian; Mayor, Marcel; Gerlich, Stefan; Arndt, Markus (September 2019). "Quantum superposition of molecules beyond 25 kDa". *Nature Physics*. **15** (12): 1242–1245\. [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[2019NatPh..15.1242F](https://ui.adsabs.harvard.edu/abs/2019NatPh..15.1242F). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1038/s41567-019-0663-9](https://doi.org/10.1038%2Fs41567-019-0663-9). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [203638258](https://api.semanticscholar.org/CorpusID:203638258). 5. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-5)** Bojowald, Martin (2015). "Quantum cosmology: a review". *Reports on Progress in Physics*. **78** (2) 023901. [arXiv](https://en.wikipedia.org/wiki/ArXiv_\(identifier\) "ArXiv (identifier)"):[1501\.04899](https://arxiv.org/abs/1501.04899). [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[2015RPPh...78b3901B](https://ui.adsabs.harvard.edu/abs/2015RPPh...78b3901B). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1088/0034-4885/78/2/023901](https://doi.org/10.1088%2F0034-4885%2F78%2F2%2F023901). [PMID](https://en.wikipedia.org/wiki/PMID_\(identifier\) "PMID (identifier)") [25582917](https://pubmed.ncbi.nlm.nih.gov/25582917). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [18463042](https://api.semanticscholar.org/CorpusID:18463042). 6. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-6)** 7. ^ [***a***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Zwiebach2022_7-0) [***b***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Zwiebach2022_7-1) [***c***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Zwiebach2022_7-2) [***d***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Zwiebach2022_7-3) [***e***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Zwiebach2022_7-4) [***f***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Zwiebach2022_7-5) [***g***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Zwiebach2022_7-6) [***h***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Zwiebach2022_7-7) [***i***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Zwiebach2022_7-8) [***j***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Zwiebach2022_7-9) [Zwiebach, Barton](https://en.wikipedia.org/wiki/Barton_Zwiebach "Barton Zwiebach") (2022). *Mastering Quantum Mechanics: Essentials, Theory, and Applications*. MIT Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-262-04613-8](https://en.wikipedia.org/wiki/Special:BookSources/978-0-262-04613-8 "Special:BookSources/978-0-262-04613-8") . 8. ^ [***a***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Lederman_8-0) [***b***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Lederman_8-1) [***c***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Lederman_8-2) Lederman, Leon M.; Hill, Christopher T. (2011). [*Quantum Physics for Poets*](https://books.google.com/books?id=qY_yOwHg_WYC&pg=PA102). US: Prometheus Books. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-1-61614-281-0](https://en.wikipedia.org/wiki/Special:BookSources/978-1-61614-281-0 "Special:BookSources/978-1-61614-281-0") . 9. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-M%C3%BCller-Kirsten_9-0)** Müller-Kirsten, H. J. W. (2006). [*Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral*](https://books.google.com/books?id=p1_Z81Le58MC&pg=PA14). US: World Scientific. p. 14. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-981-256-691-1](https://en.wikipedia.org/wiki/Special:BookSources/978-981-256-691-1 "Special:BookSources/978-981-256-691-1") . 10. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Plotnitsky_10-0)** Plotnitsky, Arkady (2012). [*Niels Bohr and Complementarity: An Introduction*](https://books.google.com/books?id=dmdUp97S4AYC&pg=PA75). US: Springer. pp. 75–76\. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-1-4614-4517-3](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4614-4517-3 "Special:BookSources/978-1-4614-4517-3") . 11. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-11)** [Griffiths, David J.](https://en.wikipedia.org/wiki/David_J._Griffiths "David J. Griffiths") (1995). [*Introduction to Quantum Mechanics*](https://en.wikipedia.org/wiki/Introduction_to_Quantum_Mechanics_\(book\) "Introduction to Quantum Mechanics (book)"). Prentice Hall. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [0-13-124405-1](https://en.wikipedia.org/wiki/Special:BookSources/0-13-124405-1 "Special:BookSources/0-13-124405-1") . 12. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Trixler2013_12-0)** Trixler, F. (2013). ["Quantum tunnelling to the origin and evolution of life"](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3768233). *Current Organic Chemistry*. **17** (16): 1758–1770\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.2174/13852728113179990083](https://doi.org/10.2174%2F13852728113179990083). [PMC](https://en.wikipedia.org/wiki/PMC_\(identifier\) "PMC (identifier)") [3768233](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3768233). [PMID](https://en.wikipedia.org/wiki/PMID_\(identifier\) "PMID (identifier)") [24039543](https://pubmed.ncbi.nlm.nih.gov/24039543). 13. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-13)** Phifer, Arnold (2012-03-27). ["Developing more energy-efficient transistors through quantum tunneling"](https://news.nd.edu/news/developing-more-energy-efficient-transistors-through-quantum-tunneling/). *Notre Dame News*. Retrieved 2024-06-07. 14. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-14)** [Bub, Jeffrey](https://en.wikipedia.org/wiki/Jeffrey_Bub "Jeffrey Bub") (2019). ["Quantum entanglement"](https://plato.stanford.edu/entries/qt-entangle/). In Zalta, Edward N. (ed.). *Stanford Encyclopedia of Philosophy*. Metaphysics Research Lab, Stanford University. 15. ^ [***a***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Caves_15-0) [***b***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Caves_15-1) [Caves, Carlton M.](https://en.wikipedia.org/wiki/Carlton_M._Caves "Carlton M. Caves") (2015). "Quantum Information Science: Emerging No More". In Kelley, Paul; Agrawal, Govind; Bass, Mike; Hecht, Jeff; Stroud, Carlos (eds.). *OSA Century of Optics*. [The Optical Society](https://en.wikipedia.org/wiki/The_Optical_Society "The Optical Society"). pp. 320–323\. [arXiv](https://en.wikipedia.org/wiki/ArXiv_\(identifier\) "ArXiv (identifier)"):[1302\.1864](https://arxiv.org/abs/1302.1864). [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[2013arXiv1302.1864C](https://ui.adsabs.harvard.edu/abs/2013arXiv1302.1864C). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-1-943580-04-0](https://en.wikipedia.org/wiki/Special:BookSources/978-1-943580-04-0 "Special:BookSources/978-1-943580-04-0") . 16. ^ [***a***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-wiseman15_16-0) [***b***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-wiseman15_16-1) [Wiseman, Howard](https://en.wikipedia.org/wiki/Howard_M._Wiseman "Howard M. Wiseman") (October 2015). ["Death by experiment for local realism"](https://doi.org/10.1038%2Fnature15631). *[Nature](https://en.wikipedia.org/wiki/Nature_\(journal\) "Nature (journal)")*. **526** (7575): 649–650\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1038/nature15631](https://doi.org/10.1038%2Fnature15631). [ISSN](https://en.wikipedia.org/wiki/ISSN_\(identifier\) "ISSN (identifier)") [0028-0836](https://search.worldcat.org/issn/0028-0836). [PMID](https://en.wikipedia.org/wiki/PMID_\(identifier\) "PMID (identifier)") [26503054](https://pubmed.ncbi.nlm.nih.gov/26503054). 17. ^ [***a***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-wolchover17_17-0) [***b***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-wolchover17_17-1) [Wolchover, Natalie](https://en.wikipedia.org/wiki/Natalie_Wolchover "Natalie Wolchover") (7 February 2017). ["Experiment Reaffirms Quantum Weirdness"](https://www.quantamagazine.org/20170207-bell-test-quantum-loophole/). *[Quanta Magazine](https://en.wikipedia.org/wiki/Quanta_Magazine "Quanta Magazine")*. Retrieved 8 February 2020. 18. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-18)** [Baez, John C.](https://en.wikipedia.org/wiki/John_C._Baez "John C. Baez") (20 March 2020). ["How to Learn Math and Physics"](https://math.ucr.edu/home/baez/books.html). *University of California, Riverside*. Retrieved 19 December 2020. "there's no way to understand the interpretation of quantum mechanics without also being able to *solve quantum mechanics problems* – to understand the theory, you need to be able to use it (and vice versa)" 19. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-19)** [Sagan, Carl](https://en.wikipedia.org/wiki/Carl_Sagan "Carl Sagan") (1996). [*The Demon-Haunted World: Science as a Candle in the Dark*](https://en.wikipedia.org/wiki/The_Demon-Haunted_World "The Demon-Haunted World"). Ballantine Books. p. 249. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [0-345-40946-9](https://en.wikipedia.org/wiki/Special:BookSources/0-345-40946-9 "Special:BookSources/0-345-40946-9") . ""For most physics students, (the "mathematical underpinning" of quantum mechanics) might occupy them from, say, third grade to early graduate school – roughly 15 years. ... The job of the popularizer of science, trying to get across some idea of quantum mechanics to a general audience that has not gone through these initiation rites, is daunting. Indeed, there are no successful popularizations of quantum mechanics in my opinion – partly for this reason." 20. ^ [***a***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Holevo2001_20-0) [***b***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Holevo2001_20-1) [***c***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Holevo2001_20-2) [Holevo, Alexander S.](https://en.wikipedia.org/wiki/Alexander_Holevo "Alexander Holevo") (2001). *Statistical Structure of Quantum Theory*. Lecture Notes in Physics. Springer. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [3-540-42082-7](https://en.wikipedia.org/wiki/Special:BookSources/3-540-42082-7 "Special:BookSources/3-540-42082-7") . [OCLC](https://en.wikipedia.org/wiki/OCLC_\(identifier\) "OCLC (identifier)") [318268606](https://search.worldcat.org/oclc/318268606). 21. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-21)** Hall, Brian C. (2013). *Quantum Theory for Mathematicians*. [Graduate Texts in Mathematics](https://en.wikipedia.org/wiki/Graduate_Texts_in_Mathematics "Graduate Texts in Mathematics"). Vol. 267. Springer. p. 125. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-1-4614-7115-8](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4614-7115-8 "Special:BookSources/978-1-4614-7115-8") . 22. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-22)** Weinberg, Steven (2010). [*Dreams Of A Final Theory: The Search for The Fundamental Laws of Nature*](https://books.google.com/books?id=OLrZkgPsZR0C). Random House. p. [82](https://books.google.com/books?id=OLrZkgPsZR0C&pg=PT82). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-1-4070-6396-6](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4070-6396-6 "Special:BookSources/978-1-4070-6396-6") . 23. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-23)** Zhang, Ruiqin; Deng, Conghao (1993). "Exact solutions of the Schrödinger equation for some quantum-mechanical many-body systems". *Physical Review A*. **47** (1): 71–77\. [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[1993PhRvA..47...71Z](https://ui.adsabs.harvard.edu/abs/1993PhRvA..47...71Z). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1103/PhysRevA.47.71](https://doi.org/10.1103%2FPhysRevA.47.71). [ISSN](https://en.wikipedia.org/wiki/ISSN_\(identifier\) "ISSN (identifier)") [1050-2947](https://search.worldcat.org/issn/1050-2947). [PMID](https://en.wikipedia.org/wiki/PMID_\(identifier\) "PMID (identifier)") [9908895](https://pubmed.ncbi.nlm.nih.gov/9908895). 24. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-24)** Li, Jing; Drummond, N. D.; Schuck, Peter; Olevano, Valerio (2019-04-01). ["Comparing many-body approaches against the helium atom exact solution"](https://doi.org/10.21468%2FSciPostPhys.6.4.040). *SciPost Physics*. **6** (4): 40. [arXiv](https://en.wikipedia.org/wiki/ArXiv_\(identifier\) "ArXiv (identifier)"):[1801\.09977](https://arxiv.org/abs/1801.09977). [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[2019ScPP....6...40L](https://ui.adsabs.harvard.edu/abs/2019ScPP....6...40L). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.21468/SciPostPhys.6.4.040](https://doi.org/10.21468%2FSciPostPhys.6.4.040). [ISSN](https://en.wikipedia.org/wiki/ISSN_\(identifier\) "ISSN (identifier)") [2542-4653](https://search.worldcat.org/issn/2542-4653). 25. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-25)** Drake, Gordon W. F. (2023). "High Precision Calculations for Helium". In Drake, Gordon W. F. (ed.). *Springer Handbook of Atomic, Molecular, and Optical Physics*. Springer Handbooks. Cham: Springer International Publishing. pp. 199–216\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1007/978-3-030-73893-8\_12](https://doi.org/10.1007%2F978-3-030-73893-8_12). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-3-030-73892-1](https://en.wikipedia.org/wiki/Special:BookSources/978-3-030-73892-1 "Special:BookSources/978-3-030-73892-1") . 26. ^ [***a***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Cohen-Tannoudji_26-0) [***b***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Cohen-Tannoudji_26-1) [***c***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Cohen-Tannoudji_26-2) [***d***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Cohen-Tannoudji_26-3) [Cohen-Tannoudji, Claude](https://en.wikipedia.org/wiki/Claude_Cohen-Tannoudji "Claude Cohen-Tannoudji"); Diu, Bernard; Laloë, Franck (2005). *Quantum Mechanics*. Translated by Hemley, Susan Reid; Ostrowsky, Nicole; Ostrowsky, Dan. John Wiley & Sons. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [0-471-16433-X](https://en.wikipedia.org/wiki/Special:BookSources/0-471-16433-X "Special:BookSources/0-471-16433-X") . 27. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-L&L_27-0)** [Landau, Lev D.](https://en.wikipedia.org/wiki/Lev_Landau "Lev Landau"); [Lifschitz, Evgeny M.](https://en.wikipedia.org/wiki/Evgeny_Lifshitz "Evgeny Lifshitz") (1977). [*Quantum Mechanics: Non-Relativistic Theory*](https://archive.org/details/QuantumMechanics_104). Vol. 3 (3rd ed.). [Pergamon Press](https://en.wikipedia.org/wiki/Pergamon_Press "Pergamon Press"). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-08-020940-1](https://en.wikipedia.org/wiki/Special:BookSources/978-0-08-020940-1 "Special:BookSources/978-0-08-020940-1") . [OCLC](https://en.wikipedia.org/wiki/OCLC_\(identifier\) "OCLC (identifier)") [2284121](https://search.worldcat.org/oclc/2284121). 28. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-ballentine1970_28-0)** Section 3.2 of Ballentine, Leslie E. (1970), "The Statistical Interpretation of Quantum Mechanics", *Reviews of Modern Physics*, **42** (4): 358–381, [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[1970RvMP...42..358B](https://ui.adsabs.harvard.edu/abs/1970RvMP...42..358B), [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1103/RevModPhys.42.358](https://doi.org/10.1103%2FRevModPhys.42.358), [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [120024263](https://api.semanticscholar.org/CorpusID:120024263) . This fact is experimentally well-known for example in quantum optics; see e.g. chap. 2 and Fig. 2.1 Leonhardt, Ulf (1997), [*Measuring the Quantum State of Light*](https://archive.org/details/measuringquantum0000leon), Cambridge: Cambridge University Press, [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[1997mqsl.book.....L](https://ui.adsabs.harvard.edu/abs/1997mqsl.book.....L), [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [0-521-49730-2](https://en.wikipedia.org/wiki/Special:BookSources/0-521-49730-2 "Special:BookSources/0-521-49730-2") . 29. ^ [***a***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-:0_29-0) [***b***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-:0_29-1) [***c***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-:0_29-2) [Nielsen, Michael A.](https://en.wikipedia.org/wiki/Michael_Nielsen "Michael Nielsen"); [Chuang, Isaac L.](https://en.wikipedia.org/wiki/Isaac_Chuang "Isaac Chuang") (2010). *Quantum Computation and Quantum Information* (2nd ed.). Cambridge University Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-1-107-00217-3](https://en.wikipedia.org/wiki/Special:BookSources/978-1-107-00217-3 "Special:BookSources/978-1-107-00217-3") . [OCLC](https://en.wikipedia.org/wiki/OCLC_\(identifier\) "OCLC (identifier)") [844974180](https://search.worldcat.org/oclc/844974180). 30. ^ [***a***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-:1_30-0) [***b***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-:1_30-1) [Rieffel, Eleanor G.](https://en.wikipedia.org/wiki/Eleanor_Rieffel "Eleanor Rieffel"); Polak, Wolfgang H. (2011). [*Quantum Computing: A Gentle Introduction*](https://en.wikipedia.org/wiki/Quantum_Computing:_A_Gentle_Introduction "Quantum Computing: A Gentle Introduction"). MIT Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-262-01506-6](https://en.wikipedia.org/wiki/Special:BookSources/978-0-262-01506-6 "Special:BookSources/978-0-262-01506-6") . 31. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-wilde_31-0)** Wilde, Mark M. (2017). *Quantum Information Theory* (2nd ed.). Cambridge University Press. [arXiv](https://en.wikipedia.org/wiki/ArXiv_\(identifier\) "ArXiv (identifier)"):[1106\.1445](https://arxiv.org/abs/1106.1445). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1017/9781316809976.001](https://doi.org/10.1017%2F9781316809976.001). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-1-107-17616-4](https://en.wikipedia.org/wiki/Special:BookSources/978-1-107-17616-4 "Special:BookSources/978-1-107-17616-4") . [OCLC](https://en.wikipedia.org/wiki/OCLC_\(identifier\) "OCLC (identifier)") [973404322](https://search.worldcat.org/oclc/973404322). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [2515538](https://api.semanticscholar.org/CorpusID:2515538). 32. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-32)** Schlosshauer, Maximilian (October 2019). "Quantum decoherence". *Physics Reports*. **831**: 1–57\. [arXiv](https://en.wikipedia.org/wiki/ArXiv_\(identifier\) "ArXiv (identifier)"):[1911\.06282](https://arxiv.org/abs/1911.06282). [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[2019PhR...831....1S](https://ui.adsabs.harvard.edu/abs/2019PhR...831....1S). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1016/j.physrep.2019.10.001](https://doi.org/10.1016%2Fj.physrep.2019.10.001). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [208006050](https://api.semanticscholar.org/CorpusID:208006050). 33. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-33)** [Rechenberg, Helmut](https://en.wikipedia.org/wiki/Helmut_Rechenberg "Helmut Rechenberg") (1987). ["Erwin Schrödinger and the creation of wave mechanics"](http://www.actaphys.uj.edu.pl/fulltext?series=Reg&vol=19&page=683) (PDF). *[Acta Physica Polonica B](https://en.wikipedia.org/wiki/Acta_Physica_Polonica_B "Acta Physica Polonica B")*. **19** (8): 683–695. Retrieved 13 June 2016. 34. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-34)** Feynman, Richard P.; Hibbs, Albert R. (2005). Steyer, Daniel F. (ed.). *Quantum Mechanics and Path Integrals* (Emended ed.). McGraw-Hill. pp. v–vii. 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"Entanglement and visibility at the output of a Mach–Zehnder interferometer". *[Physical Review A](https://en.wikipedia.org/wiki/Physical_Review_A "Physical Review A")*. **59** (2): 1615–1621\. [arXiv](https://en.wikipedia.org/wiki/ArXiv_\(identifier\) "ArXiv (identifier)"):[quant-ph/9811078](https://arxiv.org/abs/quant-ph/9811078). [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[1999PhRvA..59.1615P](https://ui.adsabs.harvard.edu/abs/1999PhRvA..59.1615P). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1103/PhysRevA.59.1615](https://doi.org/10.1103%2FPhysRevA.59.1615). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [13963928](https://api.semanticscholar.org/CorpusID:13963928). 37. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Haack2010_38-0)** Haack, G. R.; Förster, H.; Büttiker, M. (2010). "Parity detection and entanglement with a Mach-Zehnder interferometer". *[Physical Review B](https://en.wikipedia.org/wiki/Physical_Review_B "Physical Review B")*. **82** (15) 155303. [arXiv](https://en.wikipedia.org/wiki/ArXiv_\(identifier\) "ArXiv (identifier)"):[1005\.3976](https://arxiv.org/abs/1005.3976). [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[2010PhRvB..82o5303H](https://ui.adsabs.harvard.edu/abs/2010PhRvB..82o5303H). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1103/PhysRevB.82.155303](https://doi.org/10.1103%2FPhysRevB.82.155303). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [119261326](https://api.semanticscholar.org/CorpusID:119261326). 38. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-vedral_39-0)** [Vedral, Vlatko](https://en.wikipedia.org/wiki/Vlatko_Vedral "Vlatko Vedral") (2006). *Introduction to Quantum Information Science*. Oxford University Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-19-921570-6](https://en.wikipedia.org/wiki/Special:BookSources/978-0-19-921570-6 "Special:BookSources/978-0-19-921570-6") . [OCLC](https://en.wikipedia.org/wiki/OCLC_\(identifier\) "OCLC (identifier)") [442351498](https://search.worldcat.org/oclc/442351498). 39. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-marvincohen2008_41-0)** Cohen, Marvin L. (2008). ["Essay: Fifty Years of Condensed Matter Physics"](http://prl.aps.org/edannounce/PhysRevLett.101.250001). *Physical Review Letters*. **101** (25) 250001. [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[2008PhRvL.101y0001C](https://ui.adsabs.harvard.edu/abs/2008PhRvL.101y0001C). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1103/PhysRevLett.101.250001](https://doi.org/10.1103%2FPhysRevLett.101.250001). [PMID](https://en.wikipedia.org/wiki/PMID_\(identifier\) "PMID (identifier)") [19113681](https://pubmed.ncbi.nlm.nih.gov/19113681). Retrieved 31 March 2012. 40. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-42)** Matson, John. ["What Is Quantum Mechanics Good for?"](http://www.scientificamerican.com/article/everyday-quantum-physics/). *Scientific American*. Retrieved 18 May 2016. 41. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Tipler_43-0)** Tipler, Paul; Llewellyn, Ralph (2008). *Modern Physics* (5th ed.). W. H. Freeman and Company. pp. 160–161\. 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[*The Nature of Space and Time*](https://books.google.com/books?id=6a-agBFWuyQC&pg=PA61). Princeton University Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-1-4008-3474-7](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4008-3474-7 "Special:BookSources/978-1-4008-3474-7") . 46. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-49)** Aoyama, Tatsumi; Hayakawa, Masashi; Kinoshita, Toichiro; Nio, Makiko (2012). "Tenth-Order QED Contribution to the Electron g-2 and an Improved Value of the Fine Structure Constant". *[Physical Review Letters](https://en.wikipedia.org/wiki/Physical_Review_Letters "Physical Review Letters")*. **109** (11) 111807. [arXiv](https://en.wikipedia.org/wiki/ArXiv_\(identifier\) "ArXiv (identifier)"):[1205\.5368](https://arxiv.org/abs/1205.5368). 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[OCLC](https://en.wikipedia.org/wiki/OCLC_\(identifier\) "OCLC (identifier)") [1151058062](https://search.worldcat.org/oclc/1151058062). 78. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-82)** Scheider, Walter (April 1986). ["Bringing one of the great moments of science to the classroom"](http://www.cavendishscience.org/phys/tyoung/tyoung.htm). *[The Physics Teacher](https://en.wikipedia.org/wiki/The_Physics_Teacher "The Physics Teacher")*. **24** (4): 217–219\. [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[1986PhTea..24..217S](https://ui.adsabs.harvard.edu/abs/1986PhTea..24..217S). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1119/1.2341987](https://doi.org/10.1119%2F1.2341987). [ISSN](https://en.wikipedia.org/wiki/ISSN_\(identifier\) "ISSN (identifier)") [0031-921X](https://search.worldcat.org/issn/0031-921X). 79. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Feynman-kinetic-theory_83-0)** Feynman, Richard; Leighton, Robert; Sands, Matthew (1964). [*The Feynman Lectures on Physics*](https://feynmanlectures.caltech.edu/I_40.html). Vol. 1. California Institute of Technology. Retrieved 30 September 2021. Reprinted, Addison-Wesley, 1989, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-201-50064-6](https://en.wikipedia.org/wiki/Special:BookSources/978-0-201-50064-6 "Special:BookSources/978-0-201-50064-6") 80. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-84)** Martin, Andre (1986), "Cathode Ray Tubes for Industrial and Military Applications", in Hawkes, Peter (ed.), *Advances in Electronics and Electron Physics, Volume 67*, Academic Press, p. 183, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-08-057733-3](https://en.wikipedia.org/wiki/Special:BookSources/978-0-08-057733-3 "Special:BookSources/978-0-08-057733-3") , "Evidence for the existence of "cathode-rays" was first found by Plücker and Hittorf ..." 81. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-85)** Dahl, Per F. (1997). [*Flash of the Cathode Rays: A History of J. J. Thomson's Electron*](https://books.google.com/books?id=xUzaWGocMdMC). CRC Press. pp. 47–57\. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-7503-0453-5](https://en.wikipedia.org/wiki/Special:BookSources/978-0-7503-0453-5 "Special:BookSources/978-0-7503-0453-5") . 82. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-86)** [Mehra, J.](https://en.wikipedia.org/wiki/Jagdish_Mehra "Jagdish Mehra"); Rechenberg, H. (1982). *The Historical Development of Quantum Theory, Vol. 1: The Quantum Theory of Planck, Einstein, Bohr and Sommerfeld. Its Foundation and the Rise of Its Difficulties (1900–1925)*. New York: Springer-Verlag. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-387-90642-3](https://en.wikipedia.org/wiki/Special:BookSources/978-0-387-90642-3 "Special:BookSources/978-0-387-90642-3") . 83. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-87)** ["Quantum"](http://www.merriam-webster.com/dictionary/quantum). *Merriam-Webster Dictionary*. [Archived](https://web.archive.org/web/20121026104456/http://www.merriam-webster.com/dictionary/quantum) from the original on Oct 26, 2012. Retrieved 18 August 2012. 84. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-88)** [Kuhn, T. S.](https://en.wikipedia.org/wiki/Thomas_Samuel_Kuhn "Thomas Samuel Kuhn") (1978). *Black-body theory and the quantum discontinuity 1894–1912*. Oxford: Clarendon Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-19-502383-1](https://en.wikipedia.org/wiki/Special:BookSources/978-0-19-502383-1 "Special:BookSources/978-0-19-502383-1") . 85. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Kragh_89-0)** [Kragh, Helge](https://en.wikipedia.org/wiki/Helge_Kragh "Helge Kragh") (1 December 2000). ["Max Planck: the reluctant revolutionary"](https://physicsworld.com/a/max-planck-the-reluctant-revolutionary/). *[Physics World](https://en.wikipedia.org/wiki/Physics_World "Physics World")*. Retrieved 12 December 2020. 86. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-90)** [Stachel, John](https://en.wikipedia.org/wiki/John_Stachel "John Stachel") (2009). "Bohr and the Photon". *Quantum Reality, Relativistic Causality, and Closing the Epistemic Circle*. The Western Ontario Series in Philosophy of Science. Vol. 73. Dordrecht: Springer. pp. 69–83\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1007/978-1-4020-9107-0\_5](https://doi.org/10.1007%2F978-1-4020-9107-0_5). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-1-4020-9106-3](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4020-9106-3 "Special:BookSources/978-1-4020-9106-3") . 87. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-91)** Einstein, Albert (1905). ["Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt"](https://doi.org/10.1002%2Fandp.19053220607) \[On a heuristic point of view concerning the production and transformation of light\]. *[Annalen der Physik](https://en.wikipedia.org/wiki/Annalen_der_Physik "Annalen der Physik")* (in German). **17** (6): 132–148\. [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[1905AnP...322..132E](https://ui.adsabs.harvard.edu/abs/1905AnP...322..132E). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1002/andp.19053220607](https://doi.org/10.1002%2Fandp.19053220607). Reprinted in [Stachel, John](https://en.wikipedia.org/wiki/John_Stachel "John Stachel"), ed. (1989). *The Collected Papers of Albert Einstein* (in German). Vol. 2. Princeton University Press. pp. 149–166\. See also "Einstein's early work on the quantum hypothesis", ibid. pp. 134–148. 88. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-92)** [Einstein, Albert](https://en.wikipedia.org/wiki/Albert_Einstein "Albert Einstein") (1917). "Zur Quantentheorie der Strahlung" \[On the Quantum Theory of Radiation\]. *[Physikalische Zeitschrift](https://en.wikipedia.org/wiki/Physikalische_Zeitschrift "Physikalische Zeitschrift")* (in German). **18**: 121–128\. [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[1917PhyZ...18..121E](https://ui.adsabs.harvard.edu/abs/1917PhyZ...18..121E). Translated in Einstein, A. (1967). "On the Quantum Theory of Radiation". *The Old Quantum Theory*. Elsevier. pp. 167–183\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1016/b978-0-08-012102-4.50018-8](https://doi.org/10.1016%2Fb978-0-08-012102-4.50018-8). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-08-012102-4](https://en.wikipedia.org/wiki/Special:BookSources/978-0-08-012102-4 "Special:BookSources/978-0-08-012102-4") . 89. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-93)** [Ball, Philip](https://en.wikipedia.org/wiki/Philip_Ball "Philip Ball") (2017-08-31). ["A century ago Einstein sparked the notion of the laser"](https://physicsworld.com/a/a-century-ago-einstein-sparked-the-notion-of-the-laser/). *[Physics World](https://en.wikipedia.org/wiki/Physics_World "Physics World")*. Retrieved 2024-03-23. 90. ^ [***a***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-terHaar_94-0) [***b***](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-terHaar_94-1) ter Haar, D. (1967). [*The Old Quantum Theory*](https://archive.org/details/oldquantumtheory0000haar). Pergamon Press. pp. 3–75\. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-08-012101-7](https://en.wikipedia.org/wiki/Special:BookSources/978-0-08-012101-7 "Special:BookSources/978-0-08-012101-7") . [LCCN](https://en.wikipedia.org/wiki/LCCN_\(identifier\) "LCCN (identifier)") [66-29628](https://lccn.loc.gov/66-29628). 91. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-95)** Bokulich, Alisa; Bokulich, Peter (2020-08-13). ["Bohr's Correspondence Principle"](https://plato.stanford.edu/entries/bohr-correspondence/). In [Zalta, Edward N.](https://en.wikipedia.org/wiki/Edward_N._Zalta "Edward N. Zalta") (ed.). *[Stanford Encyclopedia of Philosophy](https://en.wikipedia.org/wiki/Stanford_Encyclopedia_of_Philosophy "Stanford Encyclopedia of Philosophy")*. [ISSN](https://en.wikipedia.org/wiki/ISSN_\(identifier\) "ISSN (identifier)") [1095-5054](https://search.worldcat.org/issn/1095-5054). [OCLC](https://en.wikipedia.org/wiki/OCLC_\(identifier\) "OCLC (identifier)") [429049174](https://search.worldcat.org/oclc/429049174). 92. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-96)** ["Semi-classical approximation"](https://web.archive.org/web/20221007190530/https://encyclopediaofmath.org/index.php?title=Semi-classical_approximation). *Encyclopedia of Mathematics*. Archived from [the original](https://www.encyclopediaofmath.org/index.php?title=Semi-classical_approximation) on 7 October 2022. Retrieved 1 February 2020. 93. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-97)** [Sakurai, J. J.](https://en.wikipedia.org/wiki/J._J._Sakurai "J. J. Sakurai"); Napolitano, J. (2014). "Quantum Dynamics". [*Modern Quantum Mechanics*](https://en.wikipedia.org/wiki/Modern_Quantum_Mechanics "Modern Quantum Mechanics"). Pearson. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-1-292-02410-3](https://en.wikipedia.org/wiki/Special:BookSources/978-1-292-02410-3 "Special:BookSources/978-1-292-02410-3") . [OCLC](https://en.wikipedia.org/wiki/OCLC_\(identifier\) "OCLC (identifier)") [929609283](https://search.worldcat.org/oclc/929609283). 94. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Aharoni_98-0)** [Aharoni, Amikam](https://en.wikipedia.org/wiki/Amikam_Aharoni "Amikam Aharoni") (1996). [*Introduction to the Theory of Ferromagnetism*](https://archive.org/details/introductiontoth00ahar/page/6). [Clarendon Press](https://en.wikipedia.org/wiki/Clarendon_Press "Clarendon Press"). pp. [6–7](https://archive.org/details/introductiontoth00ahar/page/6). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [0-19-851791-2](https://en.wikipedia.org/wiki/Special:BookSources/0-19-851791-2 "Special:BookSources/0-19-851791-2") . 95. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Edwards79_99-0)** David Edwards, "The Mathematical Foundations of Quantum Mechanics", *Synthese*, Volume 42, Number 1/September, 1979, pp. 1–70. 96. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-Edwards81_100-0)** David Edwards, "The Mathematical Foundations of Quantum Field Theory: Fermions, Gauge Fields, and Super-symmetry, Part I: Lattice Field Theories", *International Journal of Theoretical Physics*, Vol. 20, No. 7 (1981). 97. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-101)** [Bernstein, Jeremy](https://en.wikipedia.org/wiki/Jeremy_Bernstein "Jeremy Bernstein") (November 2005). ["Max Born and the quantum theory"](https://doi.org/10.1119%2F1.2060717). *[American Journal of Physics](https://en.wikipedia.org/wiki/American_Journal_of_Physics "American Journal of Physics")*. **73** (11): 999–1008\. [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[2005AmJPh..73..999B](https://ui.adsabs.harvard.edu/abs/2005AmJPh..73..999B). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1119/1.2060717](https://doi.org/10.1119%2F1.2060717). [ISSN](https://en.wikipedia.org/wiki/ISSN_\(identifier\) "ISSN (identifier)") [0002-9505](https://search.worldcat.org/issn/0002-9505). 98. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-pais1997_102-0)** [Pais, Abraham](https://en.wikipedia.org/wiki/Abraham_Pais "Abraham Pais") (1997). [*A Tale of Two Continents: A Physicist's Life in a Turbulent World*](https://archive.org/details/taleoftwocontine00pais). Princeton, New Jersey: Princeton University Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [0-691-01243-1](https://en.wikipedia.org/wiki/Special:BookSources/0-691-01243-1 "Special:BookSources/0-691-01243-1") . 99. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-103)** Van Hove, Leon (1958). ["Von Neumann's contributions to quantum mechanics"](https://www.ams.org/journals/bull/1958-64-03/S0002-9904-1958-10206-2/S0002-9904-1958-10206-2.pdf) (PDF). *[Bulletin of the American Mathematical Society](https://en.wikipedia.org/wiki/Bulletin_of_the_American_Mathematical_Society "Bulletin of the American Mathematical Society")*. **64** (3): Part 2:95–99. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1090/s0002-9904-1958-10206-2](https://doi.org/10.1090%2Fs0002-9904-1958-10206-2). [Archived](https://web.archive.org/web/20240120073106/https://www.ams.org/journals/bull/1958-64-03/S0002-9904-1958-10206-2/S0002-9904-1958-10206-2.pdf) (PDF) from the original on Jan 20, 2024. 100. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-feynman2015_104-0)** [Feynman, Richard](https://en.wikipedia.org/wiki/Richard_Feynman "Richard Feynman"). ["The Feynman Lectures on Physics Vol. III Ch. 21: The Schrödinger Equation in a Classical Context: A Seminar on Superconductivity, 21-4"](https://feynmanlectures.caltech.edu/III_21.html#Ch21-S5). [California Institute of Technology](https://en.wikipedia.org/wiki/California_Institute_of_Technology "California Institute of Technology"). Retrieved 24 November 2015. "...it was long believed that the wave function of the Schrödinger equation would never have a macroscopic representation analogous to the macroscopic representation of the amplitude for photons. On the other hand, it is now realized that the phenomena of superconductivity presents us with just this situation." `{{cite web}}`: CS1 maint: url-status ([link](https://en.wikipedia.org/wiki/Category:CS1_maint:_url-status "Category:CS1 maint: url-status")) 101. **[^](https://en.wikipedia.org/wiki/Quantum_mechanics#cite_ref-105)** Packard, Richard (2006). ["Berkeley Experiments on Superfluid Macroscopic Quantum Effects"](https://web.archive.org/web/20151125112132/http://research.physics.berkeley.edu/packard/publications/Articles/LT24_Berk_expts_on_macro_sup_effects.pdf) (PDF). Physics Department, University of California, Berkeley. Archived from [the original](http://physics.berkeley.edu/sites/default/files/_/lt24_berk_expts_on_macro_sup_effects.pdf) (PDF) on 25 November 2015. Retrieved 24 November 2015. Further reading The following titles, all by working physicists, attempt to communicate quantum theory to lay people, using a minimum of technical apparatus: - [Chester, Marvin](https://en.wikipedia.org/wiki/Marvin_Chester "Marvin Chester") (1987). *Primer of Quantum Mechanics*. John Wiley. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [0-486-42878-8](https://en.wikipedia.org/wiki/Special:BookSources/0-486-42878-8 "Special:BookSources/0-486-42878-8") - [Cox, Brian](https://en.wikipedia.org/wiki/Brian_Cox_\(physicist\) "Brian Cox (physicist)"); [Forshaw, Jeff](https://en.wikipedia.org/wiki/Jeff_Forshaw "Jeff Forshaw") (2011). [*The Quantum Universe: Everything That Can Happen Does Happen*](https://en.wikipedia.org/wiki/The_Quantum_Universe "The Quantum Universe"). Allen Lane. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-1-84614-432-5](https://en.wikipedia.org/wiki/Special:BookSources/978-1-84614-432-5 "Special:BookSources/978-1-84614-432-5") . - [Richard Feynman](https://en.wikipedia.org/wiki/Richard_Feynman "Richard Feynman"), 1985. *[QED: The Strange Theory of Light and Matter](https://en.wikipedia.org/wiki/QED:_The_Strange_Theory_of_Light_and_Matter "QED: The Strange Theory of Light and Matter")*, Princeton University Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [0-691-08388-6](https://en.wikipedia.org/wiki/Special:BookSources/0-691-08388-6 "Special:BookSources/0-691-08388-6") . Four elementary lectures on quantum electrodynamics and [quantum field theory](https://en.wikipedia.org/wiki/Quantum_field_theory "Quantum field theory"), yet containing many insights for the expert. - [Ghirardi, GianCarlo](https://en.wikipedia.org/wiki/Giancarlo_Ghirardi "Giancarlo Ghirardi"), 2004. *Sneaking a Look at God's Cards*, Gerald Malsbary, trans. Princeton Univ. Press. The most technical of the works cited here. Passages using [algebra](https://en.wikipedia.org/wiki/Algebra "Algebra"), [trigonometry](https://en.wikipedia.org/wiki/Trigonometry "Trigonometry"), and [bra–ket notation](https://en.wikipedia.org/wiki/Bra%E2%80%93ket_notation "Bra–ket notation") can be passed over on a first reading. - [N. David Mermin](https://en.wikipedia.org/wiki/N._David_Mermin "N. David Mermin"), 1990, "Spooky actions at a distance: mysteries of the QT" in his *Boojums All the Way Through*. Cambridge University Press: 110–76. - [Victor Stenger](https://en.wikipedia.org/wiki/Victor_Stenger "Victor Stenger"), 2000. *Timeless Reality: Symmetry, Simplicity, and Multiple Universes*. Buffalo, New York: Prometheus Books. Chpts. 5–8. Includes cosmological and philosophical considerations. More technical: - [Bernstein, Jeremy](https://en.wikipedia.org/wiki/Jeremy_Bernstein "Jeremy Bernstein") (2009). [*Quantum Leaps*](https://books.google.com/books?id=j0Me3brYOL0C). Cambridge, Massachusetts: Belknap Press of Harvard University Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-674-03541-6](https://en.wikipedia.org/wiki/Special:BookSources/978-0-674-03541-6 "Special:BookSources/978-0-674-03541-6") . - [Bohm, David](https://en.wikipedia.org/wiki/David_Bohm "David Bohm") (1989). [*Quantum Theory*](https://archive.org/details/quantumtheory0000bohm). Dover Publications. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-486-65969-5](https://en.wikipedia.org/wiki/Special:BookSources/978-0-486-65969-5 "Special:BookSources/978-0-486-65969-5") . - [Binney, James](https://en.wikipedia.org/wiki/James_Binney "James Binney"); Skinner, David (2008). *The Physics of Quantum Mechanics*. Oxford University Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-19-968857-9](https://en.wikipedia.org/wiki/Special:BookSources/978-0-19-968857-9 "Special:BookSources/978-0-19-968857-9") . - Eisberg, Robert; [Resnick, Robert](https://en.wikipedia.org/wiki/Robert_Resnick "Robert Resnick") (1985). [*Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles*](https://archive.org/details/quantumphysicsof00eisb) (2nd ed.). Wiley. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-471-87373-0](https://en.wikipedia.org/wiki/Special:BookSources/978-0-471-87373-0 "Special:BookSources/978-0-471-87373-0") . - [Bryce DeWitt](https://en.wikipedia.org/wiki/Bryce_DeWitt "Bryce DeWitt"), R. Neill Graham, eds., 1973. *The Many-Worlds Interpretation of Quantum Mechanics*, Princeton Series in Physics, Princeton University Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [0-691-08131-X](https://en.wikipedia.org/wiki/Special:BookSources/0-691-08131-X "Special:BookSources/0-691-08131-X") - [Everett, Hugh](https://en.wikipedia.org/wiki/Hugh_Everett "Hugh Everett") (1957). "Relative State Formulation of Quantum Mechanics". *Reviews of Modern Physics*. **29** (3): 454–462\. [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[1957RvMP...29..454E](https://ui.adsabs.harvard.edu/abs/1957RvMP...29..454E). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1103/RevModPhys.29.454](https://doi.org/10.1103%2FRevModPhys.29.454). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [17178479](https://api.semanticscholar.org/CorpusID:17178479). - [Feynman, Richard P.](https://en.wikipedia.org/wiki/Richard_Feynman "Richard Feynman"); [Leighton, Robert B.](https://en.wikipedia.org/wiki/Robert_B._Leighton "Robert B. Leighton"); Sands, Matthew (1965). [*The Feynman Lectures on Physics*](https://en.wikipedia.org/wiki/The_Feynman_Lectures_on_Physics "The Feynman Lectures on Physics"). Vol. 1–3\. Addison-Wesley. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-7382-0008-8](https://en.wikipedia.org/wiki/Special:BookSources/978-0-7382-0008-8 "Special:BookSources/978-0-7382-0008-8") . - [D. Greenberger](https://en.wikipedia.org/wiki/Daniel_Greenberger "Daniel Greenberger"), [K. Hentschel](https://en.wikipedia.org/wiki/Klaus_Hentschel "Klaus Hentschel"), F. Weinert, eds., 2009. *Compendium of quantum physics, Concepts, experiments, history and philosophy*, Springer-Verlag, Berlin, Heidelberg. Short articles on many QM topics. - [Griffiths, David J.](https://en.wikipedia.org/wiki/David_J._Griffiths "David J. Griffiths") (2004). *[Introduction to Quantum Mechanics](https://en.wikipedia.org/wiki/Introduction_to_Quantum_Mechanics_\(book\) "Introduction to Quantum Mechanics (book)")* (2nd ed.). Prentice Hall. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-13-111892-8](https://en.wikipedia.org/wiki/Special:BookSources/978-0-13-111892-8 "Special:BookSources/978-0-13-111892-8") . [OCLC](https://en.wikipedia.org/wiki/OCLC_\(identifier\) "OCLC (identifier)") [40251748](https://search.worldcat.org/oclc/40251748). A standard undergraduate text. - [Max Jammer](https://en.wikipedia.org/wiki/Max_Jammer "Max Jammer"), 1966. *The Conceptual Development of Quantum Mechanics*. McGraw Hill. - [Hagen Kleinert](https://en.wikipedia.org/wiki/Hagen_Kleinert "Hagen Kleinert"), 2004. *Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets*, 3rd ed. Singapore: World Scientific. [Draft of 4th edition.](http://www.physik.fu-berlin.de/~kleinert/b5) [Archived](https://web.archive.org/web/20080615134934/http://www.physik.fu-berlin.de/~kleinert/b5) 2008-06-15 at the [Wayback Machine](https://en.wikipedia.org/wiki/Wayback_Machine "Wayback Machine") - Landau, L. D.; Lifshitz, E. M. (1977). *Quantum Mechanics: Non-Relativistic Theory*. Vol. 3 (3rd ed.). [Pergamon Press](https://en.wikipedia.org/wiki/Pergamon_Press "Pergamon Press"). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-08-020940-1](https://en.wikipedia.org/wiki/Special:BookSources/978-0-08-020940-1 "Special:BookSources/978-0-08-020940-1") . [Online copy](https://archive.org/details/QuantumMechanics_104) - [Liboff, Richard L.](https://en.wikipedia.org/wiki/Liboff,_Richard "Liboff, Richard") (2002). *Introductory Quantum Mechanics*. Addison-Wesley. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-8053-8714-8](https://en.wikipedia.org/wiki/Special:BookSources/978-0-8053-8714-8 "Special:BookSources/978-0-8053-8714-8") . - Gunther Ludwig, 1968. *Wave Mechanics*. London: Pergamon Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [0-08-203204-1](https://en.wikipedia.org/wiki/Special:BookSources/0-08-203204-1 "Special:BookSources/0-08-203204-1") - [George Mackey](https://en.wikipedia.org/wiki/George_Mackey "George Mackey") (2004). *The mathematical foundations of quantum mechanics*. Dover Publications. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [0-486-43517-2](https://en.wikipedia.org/wiki/Special:BookSources/0-486-43517-2 "Special:BookSources/0-486-43517-2") . - [Merzbacher, Eugen](https://en.wikipedia.org/wiki/Eugen_Merzbacher "Eugen Merzbacher") (1998). *Quantum Mechanics*. Wiley, John & Sons, Inc. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-471-88702-7](https://en.wikipedia.org/wiki/Special:BookSources/978-0-471-88702-7 "Special:BookSources/978-0-471-88702-7") . - [Albert Messiah](https://en.wikipedia.org/wiki/Albert_Messiah "Albert Messiah"), 1966. *Quantum Mechanics* (Vol. I), English translation from French by G. M. Temmer. North Holland, John Wiley & Sons. Cf. chpt. IV, section III. - [Omnès, Roland](https://en.wikipedia.org/wiki/Roland_Omn%C3%A8s "Roland Omnès") (1999). [*Understanding Quantum Mechanics*](https://archive.org/details/understandingqua00omne). Princeton University Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-691-00435-8](https://en.wikipedia.org/wiki/Special:BookSources/978-0-691-00435-8 "Special:BookSources/978-0-691-00435-8") . [OCLC](https://en.wikipedia.org/wiki/OCLC_\(identifier\) "OCLC (identifier)") [39849482](https://search.worldcat.org/oclc/39849482). - [Scerri, Eric. R.](https://en.wikipedia.org/wiki/Eric_R._Scerri "Eric R. Scerri") (2006). *The Periodic Table: Its Story and Its Significance*. Oxford University Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [0-19-530573-6](https://en.wikipedia.org/wiki/Special:BookSources/0-19-530573-6 "Special:BookSources/0-19-530573-6") . Considers the extent to which chemistry and the periodic system have been reduced to quantum mechanics. - [Schiff, Leonard I.](https://en.wikipedia.org/wiki/Leonard_I._Schiff "Leonard I. Schiff") (1955). *Quantum Mechanics*. McGraw Hill. - [Shankar, R.](https://en.wikipedia.org/wiki/Ramamurti_Shankar "Ramamurti Shankar") (1994). *Principles of Quantum Mechanics*. Springer. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-306-44790-7](https://en.wikipedia.org/wiki/Special:BookSources/978-0-306-44790-7 "Special:BookSources/978-0-306-44790-7") . - [Stone, A. Douglas](https://en.wikipedia.org/wiki/A._Douglas_Stone "A. Douglas Stone") (2013). [*Einstein and the Quantum*](https://archive.org/details/einsteinquantumq0000ston). Princeton University Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-691-13968-5](https://en.wikipedia.org/wiki/Special:BookSources/978-0-691-13968-5 "Special:BookSources/978-0-691-13968-5") . - *What is Quantum Mechanics? A Physics Adventure*. Boston: [Transnational College](https://en.wikipedia.org/w/index.php?title=Transnational_College&action=edit&redlink=1 "Transnational College (page does not exist)"), Language Research Foundation. 1996. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-9643504-1-0](https://en.wikipedia.org/wiki/Special:BookSources/978-0-9643504-1-0 "Special:BookSources/978-0-9643504-1-0") . [OCLC](https://en.wikipedia.org/wiki/OCLC_\(identifier\) "OCLC (identifier)") [34661512](https://search.worldcat.org/oclc/34661512). - [Veltman, Martinus J. G.](https://en.wikipedia.org/wiki/Martinus_J._G._Veltman "Martinus J. G. Veltman") (2003), *Facts and Mysteries in Elementary Particle Physics*. External links - [Introduction to quantum mechanics by Timon Idema](https://interactivetextbooks.tudelft.nl/introduction-to-quantum-mechanics/) - [Quantum Physics Made Relatively Simple](https://bethe.cornell.edu/): three video lectures by [Hans Bethe](https://en.wikipedia.org/wiki/Hans_Bethe "Hans Bethe"). **Course material** - [Quantum Cook Book](http://oyc.yale.edu/sites/default/files/notes_quantum_cookbook.pdf) and [PHYS 201: Fundamentals of Physics II](http://oyc.yale.edu/physics/phys-201#sessions) by [Ramamurti Shankar](https://en.wikipedia.org/wiki/Ramamurti_Shankar "Ramamurti Shankar"), Yale OpenCourseware. - *[Modern Physics: With waves, thermodynamics, and optics](https://lightandmatter.com/mod/)* – an online textbook. - [MIT OpenCourseWare](https://en.wikipedia.org/wiki/MIT_OpenCourseWare "MIT OpenCourseWare"): [Chemistry](https://ocw.mit.edu/courses/chemistry/) and [Physics](https://ocw.mit.edu/courses/physics/). See [8\.04](https://ocw.mit.edu/courses/physics/8-04-quantum-physics-i-spring-2016/), [8\.05](https://ocw.mit.edu/courses/physics/8-05-quantum-physics-ii-fall-2013/index.htm) and [8\.06](https://ocw.mit.edu/courses/physics/8-06-quantum-physics-iii-spring-2018/index.htm). - [⁠5\+1/2⁠ Examples in Quantum Mechanics](http://physics.csbsju.edu/QM/). **Philosophy** - Ismael, Jenann. ["Quantum Mechanics"](https://plato.stanford.edu/entries/qm/). In [Zalta, Edward N.](https://en.wikipedia.org/wiki/Edward_N._Zalta "Edward N. Zalta") (ed.). *[Stanford Encyclopedia of Philosophy](https://en.wikipedia.org/wiki/Stanford_Encyclopedia_of_Philosophy "Stanford Encyclopedia of Philosophy")*. [ISSN](https://en.wikipedia.org/wiki/ISSN_\(identifier\) "ISSN (identifier)") [1095-5054](https://search.worldcat.org/issn/1095-5054). [OCLC](https://en.wikipedia.org/wiki/OCLC_\(identifier\) "OCLC (identifier)") [429049174](https://search.worldcat.org/oclc/429049174). - [Zalta, Edward N.](https://en.wikipedia.org/wiki/Edward_N._Zalta "Edward N. Zalta") (ed.). ["Philosophical Issues in Quantum Theory"](https://plato.stanford.edu/entries/qt-issues/). *[Stanford Encyclopedia of Philosophy](https://en.wikipedia.org/wiki/Stanford_Encyclopedia_of_Philosophy "Stanford Encyclopedia of Philosophy")*. [ISSN](https://en.wikipedia.org/wiki/ISSN_\(identifier\) "ISSN (identifier)") [1095-5054](https://search.worldcat.org/issn/1095-5054). [OCLC](https://en.wikipedia.org/wiki/OCLC_\(identifier\) "OCLC (identifier)") [429049174](https://search.worldcat.org/oclc/429049174).

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