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Calculated Shard: 152 (from laksa001)

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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 FROM crawler3.page_info_local FINAL PREWHERE (src_root_hash, src_unparsed) IN ((getAhrefsRootHashFromUnparsed(getAhrefsUnparsedNoserviceFromURL(\'https://en.wikipedia.org/wiki/Quantum_superposition\')), getAhrefsUnparsedNoserviceFromURL(\'https://en.wikipedia.org/wiki/Quantum_superposition\'))) FORMAT JSONEachRow'

Response:

{"found_url":"https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition","crawl_time":1775361224,"first_indexed_time":1376018043,"http_code":200,"src_unparsed":"org,wikipedia!en,\/wiki\/Quantum_superposition s443","src_root_hash":"17790707453426894952","history_drop_reason":null,"meta_title":"Quantum superposition - Wikipedia","meta_descriptions":[],"attrs_boilerpipe_text":"From Wikipedia, the free encyclopedia\nQuantum superposition of states and decoherence\nQuantum superposition\nis a fundamental principle of\nquantum mechanics\nthat states that\nlinear combinations\nof\nsolutions\nto the\nSchrödinger equation\nare also solutions of the Schrödinger equation. This follows from the fact that the Schrödinger equation is a\nlinear differential equation\nin time and position. More precisely, the state of a system is given by a\nlinear combination\nof all the\neigenfunctions\nof the Schrödinger equation governing that system.\nAn example is a\nqubit\nused in\nquantum information processing\n. A qubit state is most generally a superposition of the basis states\nand\n:\nwhere\nis the\nquantum state\nof the qubit, and\n,\ndenote particular solutions to the Schrödinger equation in\nDirac notation\nweighted by the two\nprobability amplitudes\nand\nthat both are complex numbers. Here\ncorresponds to the classical 0\nbit\n, and\nto the classical 1 bit. The probabilities of measuring the system in the\nor\nstate are given by\nand\nrespectively (see the\nBorn rule\n). Before the measurement occurs the qubit is in a superposition of both states.\nThe interference fringes in the\ndouble-slit experiment\nprovide another example of the superposition principle.\nThe theory of quantum mechanics postulates that a\nwave equation\ncompletely determines the state of a quantum system at all times. Furthermore, this differential equation is restricted to be\nlinear\nand\nhomogeneous\n. These conditions mean that for any two solutions of the wave equation,\nand\n, a linear combination of those solutions also solve the wave equation:\nfor arbitrary complex coefficients\nand\n.\n[\n1\n]\n: 61 \nIf the wave equation has more than two solutions, combinations of all such solutions are again valid solutions.\nThe quantum wave equation can be solved using functions of position,\n, or using functions of momentum,\nand consequently the superposition of momentum functions are also solutions:\nThe position and momentum solutions are related by a\nlinear transformation\n, a\nFourier transformation\n.  This transformation is itself a quantum superposition and every position wave function can be represented as a superposition of momentum wave functions and vice versa. These superpositions involve an infinite number of component waves.\n[\n1\n]\n: 244 \nGeneralization to basis states\n[\nedit\n]\nOther transformations express a quantum solution as a superposition of\neigenvectors\n, each corresponding to a possible result of a measurement on the quantum system. An eigenvector\nfor a mathematical operator,\n, has the equation\nwhere\nis one possible measured quantum value for the observable\n. A superposition of these eigenvectors can represent any solution:\nThe states like\nare called basis states.\nCompact notation for superpositions\n[\nedit\n]\nImportant mathematical operations on quantum system solutions can be performed using only the coefficients of the superposition, suppressing the details of the superposed functions. This leads to quantum systems expressed in the\nDirac bra-ket notation\n:\n[\n1\n]\n: 245 \nThis approach is especially effective for systems like quantum spin with no classical coordinate analog. Such shorthand notation is very common in textbooks and papers on quantum mechanics, and superposition of basis states is a fundamental tool in quantum mechanics.\nPaul Dirac\ndescribed the superposition principle as follows:\nThe non-classical nature of the superposition process is brought out clearly if we consider the superposition of two states,\nA\nand\nB\n, such that there exists an observation which, when made on the system in state\nA\n, is certain to lead to one particular result,\na\nsay, and when made on the system in state\nB\nis certain to lead to some different result,\nb\nsay. What will be the result of the observation when made on the system in the superposed state? The answer is that the result will be sometimes\na\nand sometimes\nb\n, according to a probability law depending on the relative weights of\nA\nand\nB\nin the superposition process. It will never be different from both\na\nand\nb\n[i.e., either\na\nor\nb\n].\nThe intermediate character of the state formed by superposition thus expresses itself through the probability of a particular result for an observation being intermediate between the corresponding probabilities for the original states, not through the result itself being intermediate between the corresponding results for the original states.\n[\n2\n]\nAnton Zeilinger\n, referring to the prototypical example of the\ndouble-slit experiment\n, has elaborated regarding the creation and destruction of quantum superposition:\n\"[T]he superposition of amplitudes ... is only valid if there is no way to know, even in principle, which path the particle took. It is important to realize that this does not imply that an observer actually takes note of what happens. It is sufficient to destroy the interference pattern, if the path information is accessible in principle from the experiment or even if it is dispersed in the environment and beyond any technical possibility to be recovered, but in principle still ‘‘out there.’’ The absence of any such information is\nthe essential criterion\nfor quantum interference to appear.\n[\n3\n]\nAny quantum state can be expanded as a sum or superposition of the eigenstates of an Hermitian operator, like the Hamiltonian, because the eigenstates form a complete basis:\nwhere\nare the energy eigenstates of the Hamiltonian. For continuous variables like position eigenstates,\n:\nwhere\nis the projection of the state into the\nbasis and is called the wave function of the particle. In both instances we notice that\ncan be expanded as a superposition of an infinite number of basis states.\nGiven the Schrödinger equation\nwhere\nindexes the set of eigenstates of the Hamiltonian with energy eigenvalues\nwe see immediately that\nwhere\nis a solution of the Schrödinger equation but is not generally an eigenstate because\nand\nare not generally equal. We say that\nis made up of a superposition of energy eigenstates. Now consider the more concrete case of an\nelectron\nthat has either\nspin\nup or down. We now index the eigenstates with the\nspinors\nin the\nbasis:\nwhere\nand\ndenote spin-up and spin-down states respectively. As previously discussed, the magnitudes of the complex coefficients give the probability of finding the electron in either definite spin state:\nwhere the probability of finding the particle with either spin up or down is normalized to 1. Notice that\nand\nare complex numbers, so that\nis an example of an allowed state. We now get\nIf we consider a qubit with both position and spin, the state is a superposition of all possibilities for both:\nwhere we have a general state\nis the sum of the\ntensor products\nof the position space wave functions and spinors.\nSuccessful experiments involving superpositions of\nrelatively large\n(by the standards of quantum physics) objects have been performed.\nA\nberyllium\nion\nhas been trapped in a superposed state.\n[\n4\n]\nA\ndouble slit experiment\nhas been performed with molecules as large as\nbuckyballs\nand functionalized oligoporphyrins with up to 2000 atoms.\n[\n5\n]\n[\n6\n]\nMolecules with masses exceeding 10,000 and composed of over 810 atoms have successfully been superposed,\n[\n7\n]\nand metal clusters with masses over 170,000 \nDa\nand containing more than 7,000 atoms have also been demonstrated in quantum superposition.\n[\n8\n]\nVery sensitive magnetometers have been realized using\nsuperconducting quantum interference devices\n(SQUIDS) that operate using quantum  interference effects in superconducting circuits.\nA\npiezoelectric\n\"\ntuning fork\n\" has been constructed, which can be placed into a superposition of vibrating and non-vibrating states. The resonator comprises about 10 trillion atoms.\n[\n9\n]\nRecent research indicates that\nchlorophyll\nwithin\nplants\nappears to exploit the feature of quantum superposition to achieve greater efficiency in transporting energy, allowing pigment proteins to be spaced further apart than would otherwise be possible.\n[\n10\n]\n[\n11\n]\nIn quantum computers\n[\nedit\n]\nIn\nquantum computers\n, a\nqubit\nis the analog of the classical information\nbit\n, but rather than having one of two distinct values, qubits are a  superposition of two values.\n[\n12\n]\n: 13 \nControlling this superposition qubits is a central challenge in quantum computation. The superposition needs to be robust to unintended interactions and yet interactions are needed for computing with qubits. Qubit systems like\nnuclear spins\nwith small coupling strength are robust to outside disturbances but the same small coupling makes it difficult to readout results.\n[\n12\n]\n: 278 \nEigenstate\n – Mathematical entity to describe the probability of each possible measurement on a system\nMach–Zehnder interferometer\n – Device to determine relative phase shift\nPenrose interpretation\n – Interpretation of quantum mechanics\nPure qubit state\n – Basic unit of quantum information\nQuantum computation\n – Computer hardware technology that uses quantum mechanics\nSchrödinger's cat\n – Thought experiment in quantum mechanics\nSuperposition principle\n – Fundamental principle of physics\nWave packet\n – Short \"burst\" or \"envelope\" of restricted wave action that travels as a unit\n^\na\nb\nc\nMessiah, Albert (1976).\nQuantum mechanics. 1\n(2 ed.). Amsterdam: North-Holland.\nISBN\n \n978-0-471-59766-7\n.\n^\nP.A.M. Dirac (1947).\nThe Principles of Quantum Mechanics\n(2nd ed.). Clarendon Press. p. 12.\n^\nZeilinger A (1999). \"Experiment and the foundations of quantum physics\".\nRev. Mod. Phys\n.\n71\n(2):\nS288–\nS297.\nBibcode\n:\n1999RvMPS..71..288Z\n.\ndoi\n:\n10.1103\/revmodphys.71.s288\n.\n^\nMonroe, C.; Meekhof, D. M.; King, B. E.; Wineland, D. J. (24 May 1996).\n\"A \"Schrödinger Cat\" Superposition State of an Atom\"\n.\nScience\n.\n272\n(5265):\n1131–\n1136.\nBibcode\n:\n1996Sci...272.1131M\n.\ndoi\n:\n10.1126\/science.272.5265.1131\n.\nISSN\n \n0036-8075\n.\nPMID\n \n8662445\n.\n^\n\"Wave-particle duality of C60\"\n. 31 March 2012. Archived from the original on 31 March 2012.\n{{\ncite web\n}}\n:  CS1 maint: bot: original URL status unknown (\nlink\n)\n^\nYaakov Y. Fein; Philipp Geyer; Patrick Zwick; Filip Kiałka; Sebastian Pedalino; Marcel Mayor; Stefan Gerlich; Markus Arndt (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^\nEibenberger, S., Gerlich, S., Arndt, M., Mayor, M., Tüxen, J. (2013). \"Matter-wave interference with particles selected from a molecular library with masses exceeding 10 000 amu\",\nPhysical Chemistry Chemical Physics\n,\n15\n: 14696-14700\narXiv\n:\n1310.8343\n^\nPedalino, Sebastian; Ramírez-Galindo, Bruno E.; Ferstl, Richard; Hornberger, Klaus; Arndt, Markus; Gerlich, Stefan (22 January 2026).\n\"Probing quantum mechanics with nanoparticle matter-wave interferometry\"\n.\nNature\n.\n649\n(8098):\n866–\n870.\ndoi\n:\n10.1038\/s41586-025-09917-9\n.\nISSN\n \n0028-0836\n.\nPMC\n \n12823444\n.\nPMID\n \n41566007\n.\n^\nO’Connell, A. D.; Hofheinz, M.; Ansmann, M.; Bialczak, Radoslaw C.; Lenander, M.; Lucero, Erik; Neeley, M.; Sank, D.; Wang, H.; Weides, M.; Wenner, J.; Martinis, John M.; Cleland, A. N. (April 2010).\n\"Quantum ground state and single-phonon control of a mechanical resonator\"\n.\nNature\n.\n464\n(7289):\n697–\n703.\nBibcode\n:\n2010Natur.464..697O\n.\ndoi\n:\n10.1038\/nature08967\n.\nISSN\n \n0028-0836\n.\nPMID\n \n20237473\n.\n^\nScholes, Gregory; Elisabetta Collini; Cathy Y. Wong; Krystyna E. Wilk; Paul M. G. Curmi; Paul Brumer; Gregory D. Scholes (4 February 2010). \"Coherently wired light-harvesting in photosynthetic marine algae at ambient temperature\".\nNature\n.\n463\n(7281):\n644–\n647.\nBibcode\n:\n2010Natur.463..644C\n.\ndoi\n:\n10.1038\/nature08811\n.\nPMID\n \n20130647\n.\nS2CID\n \n4369439\n.\n^\nMoyer, Michael (September 2009).\n\"Quantum Entanglement, Photosynthesis and Better Solar Cells\"\n.\nScientific American\n. Retrieved\n12 May\n2010\n.\n^\na\nb\nNielsen, Michael A.\n;\nChuang, Isaac\n(2010).\nQuantum Computation and Quantum Information\n. Cambridge:\nCambridge University Press\n.\nISBN\n \n978-1-10700-217-3\n.\nOCLC\n \n43641333\n.\nBohr, N.\n(1927\/1928). The quantum postulate and the recent development of atomic theory,\nNature\nSupplement 14 April 1928,\n121\n: 580–590\n.\nCohen-Tannoudji, C.\n, Diu, B., Laloë, F. (1973\/1977).\nQuantum Mechanics\n, translated from the French by S. R. Hemley, N. Ostrowsky, D. Ostrowsky, second edition, volume 1, Wiley, New York,\nISBN\n \n0471164321\n.\nEinstein, A.\n(1949). Remarks concerning the essays brought together in this co-operative volume, translated from the original German by the editor, pp. 665–688 in\nSchilpp, P. A.\neditor (1949),\nAlbert Einstein: Philosopher-Scientist\n, volume\nII\n, Open Court, La Salle IL.\nFeynman, R. P.\n, Leighton, R.B., Sands, M. (1965).\nThe Feynman Lectures on Physics\n,\nvolume 3\n, Addison-Wesley, Reading, MA.\nMerzbacher, E.\n(1961\/1970).\nQuantum Mechanics\n, second edition, Wiley, New York.\nMessiah, A.\n(1961).\nQuantum Mechanics\n, volume 1, translated by G.M. Temmer from the French\nMécanique Quantique\n, North-Holland, Amsterdam.\nWheeler, J. A.\n;\nZurek, W.H.\n(1983).\nQuantum Theory and Measurement\n. Princeton NJ: Princeton University Press.\nNielsen, Michael A.\n;\nChuang, Isaac\n(2000).\nQuantum Computation and Quantum Information\n. Cambridge:\nCambridge University Press\n.\nISBN\n \n0521632358\n.\nOCLC\n \n43641333\n.\nWilliams, Colin P. (2011).\nExplorations in Quantum Computing\n.\nSpringer\n.\nISBN\n \n978-1-84628-887-6\n.\nYanofsky, Noson S.; Mannucci, Mirco (2013).\nQuantum computing for computer scientists\n.\nCambridge University Press\n.\nISBN\n \n978-0-521-87996-5\n.","attrs_markdown":"[Jump to content](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#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+superposition \"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+superposition \"You're encouraged to log in; however, it's not mandatory. [o]\")\n\nPersonal tools\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+superposition \"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+superposition \"You're encouraged to log in; however, it's not mandatory. [o]\")\n\n## Contents\nmove to sidebar\n\nhide\n\n- [(Top)](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition)\n- [1 Wave postulate](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#Wave_postulate)\n- [2 Transformation](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#Transformation)\n- [3 Generalization to basis states](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#Generalization_to_basis_states)\n- [4 Compact notation for superpositions](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#Compact_notation_for_superpositions)\n- [5 Consequences](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#Consequences)\n- [6 Theory](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#Theory)\n  Toggle Theory subsection\n  - [6\\.1 General formalism](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#General_formalism)\n  - [6\\.2 Example](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#Example)\n- [7 Experiments](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#Experiments)\n- [8 In quantum computers](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#In_quantum_computers)\n- [9 See also](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#See_also)\n- [10 References](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#References)\n- [11 Further reading](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#Further_reading)\n\nToggle the table of contents\n\n# Quantum superposition\n40 languages\n\n- [العربية](https:\/\/ar.wikipedia.org\/wiki\/%D8%AA%D8%B1%D8%A7%D9%83%D8%A8_%D9%83%D9%85%D9%8A \"تراكب كمي – Arabic\")\n- [Български](https:\/\/bg.wikipedia.org\/wiki\/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D0%BE%D0%B2%D0%B0_%D1%81%D1%83%D0%BF%D0%B5%D1%80%D0%BF%D0%BE%D0%B7%D0%B8%D1%86%D0%B8%D1%8F \"Квантова суперпозиция – Bulgarian\")\n- [Català](https:\/\/ca.wikipedia.org\/wiki\/Superposici%C3%B3_qu%C3%A0ntica \"Superposició quàntica – Catalan\")\n- [Čeština](https:\/\/cs.wikipedia.org\/wiki\/Kvantov%C3%A1_superpozice \"Kvantová superpozice – Czech\")\n- [Чӑвашла](https:\/\/cv.wikipedia.org\/wiki\/%D0%A1%D1%83%D0%BF%D0%B5%D1%80%D0%BF%D0%BE%D0%B7%D0%B8%D1%86%D0%B8_%D0%BF%D1%80%D0%B8%D0%BD%D1%86%D0%B8%D0%BF%C4%95_\\(%D0%BA%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\\) \"Суперпозици принципĕ (квантла механика) – Chuvash\")\n- [Dansk](https:\/\/da.wikipedia.org\/wiki\/Kvantemekanisk_superposition \"Kvantemekanisk superposition – Danish\")\n- [Deutsch](https:\/\/de.wikipedia.org\/wiki\/Superposition_\\(Quantenmechanik\\) \"Superposition (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_%CF%85%CF%80%CE%AD%CF%81%CE%B8%CE%B5%CF%83%CE%B7 \"Κβαντική υπέρθεση – Greek\")\n- [Español](https:\/\/es.wikipedia.org\/wiki\/Superposici%C3%B3n_cu%C3%A1ntica \"Superposición cuántica – Spanish\")\n- [Eesti](https:\/\/et.wikipedia.org\/wiki\/Superpositsioon \"Superpositsioon – Estonian\")\n- [فارسی](https:\/\/fa.wikipedia.org\/wiki\/%D8%A8%D8%B1%D9%87%D9%85%E2%80%8C%D9%86%D9%87%DB%8C_%DA%A9%D9%88%D8%A7%D9%86%D8%AA%D9%88%D9%85%DB%8C \"برهم‌نهی کوانتومی – Persian\")\n- [Suomi](https:\/\/fi.wikipedia.org\/wiki\/Kvanttisuperpositio \"Kvanttisuperpositio – Finnish\")\n- [Français](https:\/\/fr.wikipedia.org\/wiki\/Principe_de_superposition_quantique \"Principe de superposition quantique – French\")\n- [עברית](https:\/\/he.wikipedia.org\/wiki\/%D7%A1%D7%95%D7%A4%D7%A8%D7%A4%D7%95%D7%96%D7%99%D7%A6%D7%99%D7%94_%D7%A7%D7%95%D7%95%D7%A0%D7%98%D7%99%D7%AA \"סופרפוזיציה קוונטית – 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%BF%E0%A4%95_%E0%A4%85%E0%A4%A7%E0%A5%8D%E0%A4%AF%E0%A4%BE%E0%A4%B0%E0%A5%8B%E0%A4%AA%E0%A4%A3 \"प्रमात्रिक अध्यारोपण – Hindi\")\n- [Hrvatski](https:\/\/hr.wikipedia.org\/wiki\/Kvantna_superpozicija \"Kvantna superpozicija – Croatian\")\n- [Magyar](https:\/\/hu.wikipedia.org\/wiki\/Kvantum-szuperpoz%C3%ADci%C3%B3 \"Kvantum-szuperpozíció – 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%BE%D5%A5%D6%80%D5%A1%D5%A4%D6%80%D5%B8%D6%82%D5%B4 \"Քվանտային վերադրում – Armenian\")\n- [Bahasa Indonesia](https:\/\/id.wikipedia.org\/wiki\/Superposisi_kuantum \"Superposisi kuantum – Indonesian\")\n- [Italiano](https:\/\/it.wikipedia.org\/wiki\/Principio_di_sovrapposizione_\\(meccanica_quantistica\\) \"Principio di sovrapposizione (meccanica quantistica) – Italian\")\n- [日本語](https:\/\/ja.wikipedia.org\/wiki\/%E9%87%8D%E3%81%AD%E5%90%88%E3%82%8F%E3%81%9B \"重ね合わせ – Japanese\")\n- [한국어](https:\/\/ko.wikipedia.org\/wiki\/%EC%96%91%EC%9E%90_%EC%A4%91%EC%B2%A9 \"양자 중첩 – Korean\")\n- [Lietuvių](https:\/\/lt.wikipedia.org\/wiki\/Kvantin%C4%97_superpozicija \"Kvantinė 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[Talk](https:\/\/en.wikipedia.org\/wiki\/Talk:Quantum_superposition \"Discuss improvements to the content page [t]\")\n\nEnglish\n\n- [Read](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition)\n- [Edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Quantum_superposition&action=edit \"Edit this page [e]\")\n- [View history](https:\/\/en.wikipedia.org\/w\/index.php?title=Quantum_superposition&action=history \"Past revisions of this page [h]\")\n\nTools\n\nTools\n\nmove to sidebar\n\nhide\n\nActions\n\n- [Read](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition)\n- [Edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Quantum_superposition&action=edit \"Edit this page [e]\")\n- [View history](https:\/\/en.wikipedia.org\/w\/index.php?title=Quantum_superposition&action=history)\n\nGeneral\n\n- [What links here](https:\/\/en.wikipedia.org\/wiki\/Special:WhatLinksHere\/Quantum_superposition \"List of all English Wikipedia pages containing links to this page [j]\")\n- [Related 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{\\\\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 superposition** is a fundamental principle of [quantum mechanics](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics \"Quantum mechanics\") that states that [linear combinations](https:\/\/en.wikipedia.org\/wiki\/Linear_combination \"Linear combination\") of [solutions](https:\/\/en.wikipedia.org\/wiki\/Solution_\\(mathematics\\) \"Solution (mathematics)\") to the [Schrödinger equation](https:\/\/en.wikipedia.org\/wiki\/Schr%C3%B6dinger_equation \"Schrödinger equation\") are also solutions of the Schrödinger equation. This follows from the fact that the Schrödinger equation is a [linear differential equation](https:\/\/en.wikipedia.org\/wiki\/Linear_differential_equation \"Linear differential equation\") in time and position. More precisely, the state of a system is given by a [linear combination](https:\/\/en.wikipedia.org\/wiki\/Linear_combination \"Linear combination\") of all the [eigenfunctions](https:\/\/en.wikipedia.org\/wiki\/Eigenfunction \"Eigenfunction\") of the Schrödinger equation governing that system.\n\nAn example is a [qubit](https:\/\/en.wikipedia.org\/wiki\/Qubit \"Qubit\") used in [quantum information processing](https:\/\/en.wikipedia.org\/wiki\/Quantum_information_processing \"Quantum information processing\"). A qubit state is most generally a superposition of the basis states \\| 0 ⟩ {\\\\displaystyle \\|0\\\\rangle } ![{\\\\displaystyle \\|0\\\\rangle }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/ed066a3ad158da0ad6d6a421a606b1c8a35eb95b) and \\| 1 ⟩ {\\\\displaystyle \\|1\\\\rangle } ![{\\\\displaystyle \\|1\\\\rangle }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/2f53021ca18e77477ee5bd3c1523e5830189ec5c):\n\n\\|\n\nΨ\n\n⟩\n\n\\=\n\nc\n\n0\n\n\\|\n\n0\n\n⟩\n\n\\+\n\nc\n\n1\n\n\\|\n\n1\n\n⟩\n\n,\n\n{\\\\displaystyle \\|\\\\Psi \\\\rangle =c\\_{0}\\|0\\\\rangle +c\\_{1}\\|1\\\\rangle ,}\n\n![{\\\\displaystyle \\|\\\\Psi \\\\rangle =c\\_{0}\\|0\\\\rangle +c\\_{1}\\|1\\\\rangle ,}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/f10213c27e86b15bff8015bd81b1d5983d14c5a4)\n\nwhere \\| Ψ ⟩ {\\\\displaystyle \\|\\\\Psi \\\\rangle } ![{\\\\displaystyle \\|\\\\Psi \\\\rangle }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/6e77f6b1e903837c5765c9683da41dd93199621c) is the [quantum state](https:\/\/en.wikipedia.org\/wiki\/Quantum_state \"Quantum state\") of the qubit, and \\| 0 ⟩ {\\\\displaystyle \\|0\\\\rangle } ![{\\\\displaystyle \\|0\\\\rangle }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/ed066a3ad158da0ad6d6a421a606b1c8a35eb95b), \\| 1 ⟩ {\\\\displaystyle \\|1\\\\rangle } ![{\\\\displaystyle \\|1\\\\rangle }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/2f53021ca18e77477ee5bd3c1523e5830189ec5c) denote particular solutions to the Schrödinger equation in [Dirac notation](https:\/\/en.wikipedia.org\/wiki\/Bra%E2%80%93ket_notation \"Bra–ket notation\") weighted by the two [probability amplitudes](https:\/\/en.wikipedia.org\/wiki\/Probability_amplitude \"Probability amplitude\") c 0 {\\\\displaystyle c\\_{0}} ![{\\\\displaystyle c\\_{0}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/1882ba8f1dc60f0c68a642abb5af093c73910921) and c 1 {\\\\displaystyle c\\_{1}} ![{\\\\displaystyle c\\_{1}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/77b7dc6d279091d354e0b90889b463bfa7eb7247) that both are complex numbers. Here \\| 0 ⟩ {\\\\displaystyle \\|0\\\\rangle } ![{\\\\displaystyle \\|0\\\\rangle }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/ed066a3ad158da0ad6d6a421a606b1c8a35eb95b) corresponds to the classical 0 [bit](https:\/\/en.wikipedia.org\/wiki\/Bit \"Bit\"), and \\| 1 ⟩ {\\\\displaystyle \\|1\\\\rangle } ![{\\\\displaystyle \\|1\\\\rangle }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/2f53021ca18e77477ee5bd3c1523e5830189ec5c) to the classical 1 bit. The probabilities of measuring the system in the \\| 0 ⟩ {\\\\displaystyle \\|0\\\\rangle } ![{\\\\displaystyle \\|0\\\\rangle }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/ed066a3ad158da0ad6d6a421a606b1c8a35eb95b) or \\| 1 ⟩ {\\\\displaystyle \\|1\\\\rangle } ![{\\\\displaystyle \\|1\\\\rangle }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/2f53021ca18e77477ee5bd3c1523e5830189ec5c) state are given by \\| c 0 \\| 2 {\\\\displaystyle \\|c\\_{0}\\|^{2}} ![{\\\\displaystyle \\|c\\_{0}\\|^{2}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/8529f071ccb861751297a04659e8dbddca09bad8) and \\| c 1 \\| 2 {\\\\displaystyle \\|c\\_{1}\\|^{2}} ![{\\\\displaystyle \\|c\\_{1}\\|^{2}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/a9c67c5c8bf07a77edd704f880a64a35b8d9f1c5) respectively (see the [Born rule](https:\/\/en.wikipedia.org\/wiki\/Born_rule \"Born rule\")). Before the measurement occurs the qubit is in a superposition of both states.\n\nThe interference fringes in the [double-slit experiment](https:\/\/en.wikipedia.org\/wiki\/Double-slit_experiment \"Double-slit experiment\") provide another example of the superposition principle.\n\n## Wave postulate\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Quantum_superposition&action=edit&section=1 \"Edit section: Wave postulate\")\\]\n\nThe theory of quantum mechanics postulates that a [wave equation](https:\/\/en.wikipedia.org\/wiki\/Wave_equation \"Wave equation\") completely determines the state of a quantum system at all times. Furthermore, this differential equation is restricted to be [linear](https:\/\/en.wikipedia.org\/wiki\/Linear_differential_equation \"Linear differential equation\") and [homogeneous](https:\/\/en.wikipedia.org\/wiki\/Homogeneous_differential_equation \"Homogeneous differential equation\"). These conditions mean that for any two solutions of the wave equation, Ψ 1 {\\\\displaystyle \\\\Psi \\_{1}} ![{\\\\displaystyle \\\\Psi \\_{1}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/ddcd58cf9c1c4dd1eb32a29c1e3abf425ec72600) and Ψ 2 {\\\\displaystyle \\\\Psi \\_{2}} ![{\\\\displaystyle \\\\Psi \\_{2}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/1125553c0a026635c63ab2ad542b43174aca9bbe), a linear combination of those solutions also solve the wave equation: Ψ \\= c 1 Ψ 1 \\+ c 2 Ψ 2 {\\\\displaystyle \\\\Psi =c\\_{1}\\\\Psi \\_{1}+c\\_{2}\\\\Psi \\_{2}} ![{\\\\displaystyle \\\\Psi =c\\_{1}\\\\Psi \\_{1}+c\\_{2}\\\\Psi \\_{2}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/f6823dfb8e3163bbbfd61bd2d925ce8d6fdac949) for arbitrary complex coefficients c 1 {\\\\displaystyle c\\_{1}} ![{\\\\displaystyle c\\_{1}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/77b7dc6d279091d354e0b90889b463bfa7eb7247) and c 2 {\\\\displaystyle c\\_{2}} ![{\\\\displaystyle c\\_{2}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/0b30ba1b247fb8d334580cec68561e749d24aff2).[\\[1\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#cite_note-Messiah-1): 61  If the wave equation has more than two solutions, combinations of all such solutions are again valid solutions.\n\n## Transformation\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Quantum_superposition&action=edit&section=2 \"Edit section: Transformation\")\\]\n\nThe quantum wave equation can be solved using functions of position, Ψ ( r → ) {\\\\displaystyle \\\\Psi ({\\\\vec {r}})} ![{\\\\displaystyle \\\\Psi ({\\\\vec {r}})}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/860ed8dcca03bb55dda0fe92c92a079038be797b), or using functions of momentum, Φ ( p → ) {\\\\displaystyle \\\\Phi ({\\\\vec {p}})} ![{\\\\displaystyle \\\\Phi ({\\\\vec {p}})}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/96584b5a2ed608930bfa349e822c564a5298250d) and consequently the superposition of momentum functions are also solutions: Φ ( p → ) \\= d 1 Φ 1 ( p → ) \\+ d 2 Φ 2 ( p → ) {\\\\displaystyle \\\\Phi ({\\\\vec {p}})=d\\_{1}\\\\Phi \\_{1}({\\\\vec {p}})+d\\_{2}\\\\Phi \\_{2}({\\\\vec {p}})} ![{\\\\displaystyle \\\\Phi ({\\\\vec {p}})=d\\_{1}\\\\Phi \\_{1}({\\\\vec {p}})+d\\_{2}\\\\Phi \\_{2}({\\\\vec {p}})}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/f72ca07aa31123cf18829faed1ebb4abeb3a5120) The position and momentum solutions are related by a [linear transformation](https:\/\/en.wikipedia.org\/wiki\/Linear_transformation \"Linear transformation\"), a [Fourier transformation](https:\/\/en.wikipedia.org\/wiki\/Fourier_transformation \"Fourier transformation\"). This transformation is itself a quantum superposition and every position wave function can be represented as a superposition of momentum wave functions and vice versa. These superpositions involve an infinite number of component waves.[\\[1\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#cite_note-Messiah-1): 244\n\n## Generalization to basis states\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Quantum_superposition&action=edit&section=3 \"Edit section: Generalization to basis states\")\\]\n\nOther transformations express a quantum solution as a superposition of [eigenvectors](https:\/\/en.wikipedia.org\/wiki\/Eigenvectors \"Eigenvectors\"), each corresponding to a possible result of a measurement on the quantum system. An eigenvector ψ i {\\\\displaystyle \\\\psi \\_{i}} ![{\\\\displaystyle \\\\psi \\_{i}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/3e709a809cfd3cfbb647fe22c8d1f81573f4c7ae) for a mathematical operator, A ^ {\\\\displaystyle {\\\\hat {A}}} ![{\\\\displaystyle {\\\\hat {A}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/f595a6c73d1183d6a1b2ac21fe47ac28c1483821), has the equation A ^ ψ i \\= λ i ψ i {\\\\displaystyle {\\\\hat {A}}\\\\psi \\_{i}=\\\\lambda \\_{i}\\\\psi \\_{i}} ![{\\\\displaystyle {\\\\hat {A}}\\\\psi \\_{i}=\\\\lambda \\_{i}\\\\psi \\_{i}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/4f7c7cfad33ac61dd3bcde805172bb9897ac0dfe) where λ i {\\\\displaystyle \\\\lambda \\_{i}} ![{\\\\displaystyle \\\\lambda \\_{i}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/72fde940918edf84caf3d406cc7d31949166820f) is one possible measured quantum value for the observable A {\\\\displaystyle A} ![{\\\\displaystyle A}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/7daff47fa58cdfd29dc333def748ff5fa4c923e3). A superposition of these eigenvectors can represent any solution: Ψ \\= ∑ n a i ψ i . {\\\\displaystyle \\\\Psi =\\\\sum \\_{n}a\\_{i}\\\\psi \\_{i}.} ![{\\\\displaystyle \\\\Psi =\\\\sum \\_{n}a\\_{i}\\\\psi \\_{i}.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/da5aeb99c104b07a8842ea74cc61a45a859aa2a3) The states like ψ i {\\\\displaystyle \\\\psi \\_{i}} ![{\\\\displaystyle \\\\psi \\_{i}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/3e709a809cfd3cfbb647fe22c8d1f81573f4c7ae) are called basis states.\n\n## Compact notation for superpositions\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Quantum_superposition&action=edit&section=4 \"Edit section: Compact notation for superpositions\")\\]\n\nImportant mathematical operations on quantum system solutions can be performed using only the coefficients of the superposition, suppressing the details of the superposed functions. This leads to quantum systems expressed in the [Dirac bra-ket notation](https:\/\/en.wikipedia.org\/wiki\/Bra-ket_notation \"Bra-ket notation\"):[\\[1\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#cite_note-Messiah-1): 245  \\| v ⟩ \\= d 1 \\| 1 ⟩ \\+ d 2 \\| 2 ⟩ {\\\\displaystyle \\|v\\\\rangle =d\\_{1}\\|1\\\\rangle +d\\_{2}\\|2\\\\rangle } ![{\\\\displaystyle \\|v\\\\rangle =d\\_{1}\\|1\\\\rangle +d\\_{2}\\|2\\\\rangle }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/acc7ac146085e844fd0f2988bed9f23c30af497a) This approach is especially effective for systems like quantum spin with no classical coordinate analog. Such shorthand notation is very common in textbooks and papers on quantum mechanics, and superposition of basis states is a fundamental tool in quantum mechanics.\n\n## Consequences\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Quantum_superposition&action=edit&section=5 \"Edit section: Consequences\")\\]\n\n[Paul Dirac](https:\/\/en.wikipedia.org\/wiki\/Paul_Dirac \"Paul Dirac\") described the superposition principle as follows:\n\n> The non-classical nature of the superposition process is brought out clearly if we consider the superposition of two states, *A* and *B*, such that there exists an observation which, when made on the system in state *A*, is certain to lead to one particular result, *a* say, and when made on the system in state *B* is certain to lead to some different result, *b* say. What will be the result of the observation when made on the system in the superposed state? The answer is that the result will be sometimes *a* and sometimes *b*, according to a probability law depending on the relative weights of *A* and *B* in the superposition process. It will never be different from both *a* and *b* \\[i.e., either *a* or *b*\\]. *The intermediate character of the state formed by superposition thus expresses itself through the probability of a particular result for an observation being intermediate between the corresponding probabilities for the original states, not through the result itself being intermediate between the corresponding results for the original states.*[\\[2\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#cite_note-Dirac1947-2)\n\n[Anton Zeilinger](https:\/\/en.wikipedia.org\/wiki\/Anton_Zeilinger \"Anton Zeilinger\"), referring to the prototypical example of the [double-slit experiment](https:\/\/en.wikipedia.org\/wiki\/Double-slit_experiment \"Double-slit experiment\"), has elaborated regarding the creation and destruction of quantum superposition:\n\n> \"\\[T\\]he superposition of amplitudes ... is only valid if there is no way to know, even in principle, which path the particle took. It is important to realize that this does not imply that an observer actually takes note of what happens. It is sufficient to destroy the interference pattern, if the path information is accessible in principle from the experiment or even if it is dispersed in the environment and beyond any technical possibility to be recovered, but in principle still ‘‘out there.’’ The absence of any such information is *the essential criterion* for quantum interference to appear.[\\[3\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#cite_note-Zeilinger-3)\n\n## Theory\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Quantum_superposition&action=edit&section=6 \"Edit section: Theory\")\\]\n\n### General formalism\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Quantum_superposition&action=edit&section=7 \"Edit section: General formalism\")\\]\n\nAny quantum state can be expanded as a sum or superposition of the eigenstates of an Hermitian operator, like the Hamiltonian, because the eigenstates form a complete basis:\n\n\\|\n\nα\n\n⟩\n\n\\=\n\n∑\n\nn\n\nc\n\nn\n\n\\|\n\nn\n\n⟩\n\n,\n\n{\\\\displaystyle \\|\\\\alpha \\\\rangle =\\\\sum \\_{n}c\\_{n}\\|n\\\\rangle ,}\n\n![{\\\\displaystyle \\|\\\\alpha \\\\rangle =\\\\sum \\_{n}c\\_{n}\\|n\\\\rangle ,}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/df393c7be36c417cc59f5a1a605aea04beef2e7b)\n\nwhere \\| n ⟩ {\\\\displaystyle \\|n\\\\rangle } ![{\\\\displaystyle \\|n\\\\rangle }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/89c45bda6ac852da785c06476f2d3e3b6cba812a) are the energy eigenstates of the Hamiltonian. For continuous variables like position eigenstates, \\| x ⟩ {\\\\displaystyle \\|x\\\\rangle } ![{\\\\displaystyle \\|x\\\\rangle }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/48004887d8f9dfc489bd2bc793780b7f1d8039ad):\n\n\\|\n\nα\n\n⟩\n\n\\=\n\n∫\n\nd\n\nx\n\n′\n\n\\|\n\nx\n\n′\n\n⟩\n\n⟨\n\nx\n\n′\n\n\\|\n\nα\n\n⟩\n\n,\n\n{\\\\displaystyle \\|\\\\alpha \\\\rangle =\\\\int dx'\\|x'\\\\rangle \\\\langle x'\\|\\\\alpha \\\\rangle ,}\n\n![{\\\\displaystyle \\|\\\\alpha \\\\rangle =\\\\int dx'\\|x'\\\\rangle \\\\langle x'\\|\\\\alpha \\\\rangle ,}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/9495e664ba18698edce2bfcc82c345de5ceaf6c6)\n\nwhere ϕ α ( x ) \\= ⟨ x \\| α ⟩ {\\\\displaystyle \\\\phi \\_{\\\\alpha }(x)=\\\\langle x\\|\\\\alpha \\\\rangle } ![{\\\\displaystyle \\\\phi \\_{\\\\alpha }(x)=\\\\langle x\\|\\\\alpha \\\\rangle }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/2ef30d5158d3fe8417e7fe37540e5370e46d9887) is the projection of the state into the \\| x ⟩ {\\\\displaystyle \\|x\\\\rangle } ![{\\\\displaystyle \\|x\\\\rangle }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/48004887d8f9dfc489bd2bc793780b7f1d8039ad) basis and is called the wave function of the particle. In both instances we notice that \\| α ⟩ {\\\\displaystyle \\|\\\\alpha \\\\rangle } ![{\\\\displaystyle \\|\\\\alpha \\\\rangle }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/f42032e642ee1c9d27adb318d34c7cc85f7a95d5) can be expanded as a superposition of an infinite number of basis states.\n\n### Example\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Quantum_superposition&action=edit&section=8 \"Edit section: Example\")\\]\n\nGiven the Schrödinger equation\n\nH\n\n^\n\n\\|\n\nn\n\n⟩\n\n\\=\n\nE\n\nn\n\n\\|\n\nn\n\n⟩\n\n,\n\n{\\\\displaystyle {\\\\hat {H}}\\|n\\\\rangle =E\\_{n}\\|n\\\\rangle ,}\n\n![{\\\\displaystyle {\\\\hat {H}}\\|n\\\\rangle =E\\_{n}\\|n\\\\rangle ,}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/8a6afb6d354132f7d244ee60cf28e4608aa6e922)\n\nwhere \\| n ⟩ {\\\\displaystyle \\|n\\\\rangle } ![{\\\\displaystyle \\|n\\\\rangle }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/89c45bda6ac852da785c06476f2d3e3b6cba812a) indexes the set of eigenstates of the Hamiltonian with energy eigenvalues E n , {\\\\displaystyle E\\_{n},} ![{\\\\displaystyle E\\_{n},}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/578d4bb07de7b29269be95abb6a214848c09aea7) we see immediately that\n\nH\n\n^\n\n(\n\n\\|\n\nn\n\n⟩\n\n\\+\n\n\\|\n\nn\n\n′\n\n⟩\n\n)\n\n\\=\n\nE\n\nn\n\n\\|\n\nn\n\n⟩\n\n\\+\n\nE\n\nn\n\n′\n\n\\|\n\nn\n\n′\n\n⟩\n\n,\n\n{\\\\displaystyle {\\\\hat {H}}{\\\\big (}\\|n\\\\rangle +\\|n'\\\\rangle {\\\\big )}=E\\_{n}\\|n\\\\rangle +E\\_{n'}\\|n'\\\\rangle ,}\n\n![{\\\\displaystyle {\\\\hat {H}}{\\\\big (}\\|n\\\\rangle +\\|n'\\\\rangle {\\\\big )}=E\\_{n}\\|n\\\\rangle +E\\_{n'}\\|n'\\\\rangle ,}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/806bff6ca40452045baac80fd266eb3c453e7340)\n\nwhere\n\n\\|\n\nΨ\n\n⟩\n\n\\=\n\n\\|\n\nn\n\n⟩\n\n\\+\n\n\\|\n\nn\n\n′\n\n⟩\n\n{\\\\displaystyle \\|\\\\Psi \\\\rangle =\\|n\\\\rangle +\\|n'\\\\rangle }\n\n![{\\\\displaystyle \\|\\\\Psi \\\\rangle =\\|n\\\\rangle +\\|n'\\\\rangle }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/99dd060b4c8a46870132855acbe5a64eefdf0ea3)\n\nis a solution of the Schrödinger equation but is not generally an eigenstate because E n {\\\\displaystyle E\\_{n}} ![{\\\\displaystyle E\\_{n}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/ad6b82f2a00af6c9efd4c16d4e99329605645c0c) and E n ′ {\\\\displaystyle E\\_{n'}} ![{\\\\displaystyle E\\_{n'}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/9d618f0a54e5627c8baa3cfb77b3e2dfac8f31bc) are not generally equal. We say that \\| Ψ ⟩ {\\\\displaystyle \\|\\\\Psi \\\\rangle } ![{\\\\displaystyle \\|\\\\Psi \\\\rangle }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/6e77f6b1e903837c5765c9683da41dd93199621c) is made up of a superposition of energy eigenstates. Now consider the more concrete case of an [electron](https:\/\/en.wikipedia.org\/wiki\/Electron \"Electron\") that has either [spin](https:\/\/en.wikipedia.org\/wiki\/Spin_\\(physics\\) \"Spin (physics)\") up or down. We now index the eigenstates with the [spinors](https:\/\/en.wikipedia.org\/wiki\/Spinor \"Spinor\") in the z ^ {\\\\displaystyle {\\\\hat {z}}} ![{\\\\displaystyle {\\\\hat {z}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/722665b45e05afe79f4395a3de0237d8ce856273) basis:\n\n\\|\n\nΨ\n\n⟩\n\n\\=\n\nc\n\n1\n\n\\|\n\n↑\n\n⟩\n\n\\+\n\nc\n\n2\n\n\\|\n\n↓\n\n⟩\n\n,\n\n{\\\\displaystyle \\|\\\\Psi \\\\rangle =c\\_{1}\\|{\\\\uparrow }\\\\rangle +c\\_{2}\\|{\\\\downarrow }\\\\rangle ,}\n\n![{\\\\displaystyle \\|\\\\Psi \\\\rangle =c\\_{1}\\|{\\\\uparrow }\\\\rangle +c\\_{2}\\|{\\\\downarrow }\\\\rangle ,}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/42f17d6ba89011bea1b2f5f2747b88a7482c681e)\n\nwhere \\| ↑ ⟩ {\\\\displaystyle \\|{\\\\uparrow }\\\\rangle } ![{\\\\displaystyle \\|{\\\\uparrow }\\\\rangle }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/6e81fba5ac4e54a40f5b9d63e66548328b7ffb6e) and \\| ↓ ⟩ {\\\\displaystyle \\|{\\\\downarrow }\\\\rangle } ![{\\\\displaystyle \\|{\\\\downarrow }\\\\rangle }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/622f3f29fa4b88926ffa22e3da1daa5f2893d17c) denote spin-up and spin-down states respectively. As previously discussed, the magnitudes of the complex coefficients give the probability of finding the electron in either definite spin state:\n\nP\n\n(\n\n\\|\n\n↑\n\n⟩\n\n)\n\n\\=\n\n\\|\n\nc\n\n1\n\n\\|\n\n2\n\n,\n\n{\\\\displaystyle P{\\\\big (}\\|{\\\\uparrow }\\\\rangle {\\\\big )}=\\|c\\_{1}\\|^{2},}\n\n![{\\\\displaystyle P{\\\\big (}\\|{\\\\uparrow }\\\\rangle {\\\\big )}=\\|c\\_{1}\\|^{2},}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/4e31d0a1610fddb7d86d8548a0ae43a9dff80529)\n\nP\n\n(\n\n\\|\n\n↓\n\n⟩\n\n)\n\n\\=\n\n\\|\n\nc\n\n2\n\n\\|\n\n2\n\n,\n\n{\\\\displaystyle P{\\\\big (}\\|{\\\\downarrow }\\\\rangle {\\\\big )}=\\|c\\_{2}\\|^{2},}\n\n![{\\\\displaystyle P{\\\\big (}\\|{\\\\downarrow }\\\\rangle {\\\\big )}=\\|c\\_{2}\\|^{2},}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/8ad7f70c0f22f0405876a0e5c22cfdf96bcf05a9)\n\nP\n\ntotal\n\n\\=\n\nP\n\n(\n\n\\|\n\n↑\n\n⟩\n\n)\n\n\\+\n\nP\n\n(\n\n\\|\n\n↓\n\n⟩\n\n)\n\n\\=\n\n\\|\n\nc\n\n1\n\n\\|\n\n2\n\n\\+\n\n\\|\n\nc\n\n2\n\n\\|\n\n2\n\n\\=\n\n1\n\n,\n\n{\\\\displaystyle P\\_{\\\\text{total}}=P{\\\\big (}\\|{\\\\uparrow }\\\\rangle {\\\\big )}+P{\\\\big (}\\|{\\\\downarrow }\\\\rangle {\\\\big )}=\\|c\\_{1}\\|^{2}+\\|c\\_{2}\\|^{2}=1,}\n\n![{\\\\displaystyle P\\_{\\\\text{total}}=P{\\\\big (}\\|{\\\\uparrow }\\\\rangle {\\\\big )}+P{\\\\big (}\\|{\\\\downarrow }\\\\rangle {\\\\big )}=\\|c\\_{1}\\|^{2}+\\|c\\_{2}\\|^{2}=1,}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/5fd7a45939635ea1e22247c9c7293c938999cefe)\n\nwhere the probability of finding the particle with either spin up or down is normalized to 1. Notice that c 1 {\\\\displaystyle c\\_{1}} ![{\\\\displaystyle c\\_{1}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/77b7dc6d279091d354e0b90889b463bfa7eb7247) and c 2 {\\\\displaystyle c\\_{2}} ![{\\\\displaystyle c\\_{2}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/0b30ba1b247fb8d334580cec68561e749d24aff2) are complex numbers, so that\n\n\\|\n\nΨ\n\n⟩\n\n\\=\n\n3\n\n5\n\ni\n\n\\|\n\n↑\n\n⟩\n\n\\+\n\n4\n\n5\n\n\\|\n\n↓\n\n⟩\n\n.\n\n{\\\\displaystyle \\|\\\\Psi \\\\rangle ={\\\\frac {3}{5}}i\\|{\\\\uparrow }\\\\rangle +{\\\\frac {4}{5}}\\|{\\\\downarrow }\\\\rangle .}\n\n![{\\\\displaystyle \\|\\\\Psi \\\\rangle ={\\\\frac {3}{5}}i\\|{\\\\uparrow }\\\\rangle +{\\\\frac {4}{5}}\\|{\\\\downarrow }\\\\rangle .}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/1751456a557a43b4c870d3dd5351854eb5556233)\n\nis an example of an allowed state. We now get\n\nP\n\n(\n\n\\|\n\n↑\n\n⟩\n\n)\n\n\\=\n\n\\|\n\n3\n\ni\n\n5\n\n\\|\n\n2\n\n\\=\n\n9\n\n25\n\n,\n\n{\\\\displaystyle P{\\\\big (}\\|{\\\\uparrow }\\\\rangle {\\\\big )}=\\\\left\\|{\\\\frac {3i}{5}}\\\\right\\|^{2}={\\\\frac {9}{25}},}\n\n![{\\\\displaystyle P{\\\\big (}\\|{\\\\uparrow }\\\\rangle {\\\\big )}=\\\\left\\|{\\\\frac {3i}{5}}\\\\right\\|^{2}={\\\\frac {9}{25}},}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/eeca39a54f3fc393e54a5fdc7feeb5aeb0b527ac)\n\nP\n\n(\n\n\\|\n\n↓\n\n⟩\n\n)\n\n\\=\n\n\\|\n\n4\n\n5\n\n\\|\n\n2\n\n\\=\n\n16\n\n25\n\n,\n\n{\\\\displaystyle P{\\\\big (}\\|{\\\\downarrow }\\\\rangle {\\\\big )}=\\\\left\\|{\\\\frac {4}{5}}\\\\right\\|^{2}={\\\\frac {16}{25}},}\n\n![{\\\\displaystyle P{\\\\big (}\\|{\\\\downarrow }\\\\rangle {\\\\big )}=\\\\left\\|{\\\\frac {4}{5}}\\\\right\\|^{2}={\\\\frac {16}{25}},}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/863ea6a63d777554353bd1f448c5d4d989136162)\n\nP\n\ntotal\n\n\\=\n\nP\n\n(\n\n\\|\n\n↑\n\n⟩\n\n)\n\n\\+\n\nP\n\n(\n\n\\|\n\n↓\n\n⟩\n\n)\n\n\\=\n\n9\n\n25\n\n\\+\n\n16\n\n25\n\n\\=\n\n1\\.\n\n{\\\\displaystyle P\\_{\\\\text{total}}=P{\\\\big (}\\|{\\\\uparrow }\\\\rangle {\\\\big )}+P{\\\\big (}\\|{\\\\downarrow }\\\\rangle {\\\\big )}={\\\\frac {9}{25}}+{\\\\frac {16}{25}}=1.}\n\n![{\\\\displaystyle P\\_{\\\\text{total}}=P{\\\\big (}\\|{\\\\uparrow }\\\\rangle {\\\\big )}+P{\\\\big (}\\|{\\\\downarrow }\\\\rangle {\\\\big )}={\\\\frac {9}{25}}+{\\\\frac {16}{25}}=1.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/845feda8d1aa925d760f2a5e4b88e350100ccfa1)\n\nIf we consider a qubit with both position and spin, the state is a superposition of all possibilities for both:\n\nΨ\n\n\\=\n\nψ\n\n\\+\n\n(\n\nx\n\n)\n\n⊗\n\n\\|\n\n↑\n\n⟩\n\n\\+\n\nψ\n\n−\n\n(\n\nx\n\n)\n\n⊗\n\n\\|\n\n↓\n\n⟩\n\n,\n\n{\\\\displaystyle \\\\Psi =\\\\psi \\_{+}(x)\\\\otimes \\|{\\\\uparrow }\\\\rangle +\\\\psi \\_{-}(x)\\\\otimes \\|{\\\\downarrow }\\\\rangle ,}\n\n![{\\\\displaystyle \\\\Psi =\\\\psi \\_{+}(x)\\\\otimes \\|{\\\\uparrow }\\\\rangle +\\\\psi \\_{-}(x)\\\\otimes \\|{\\\\downarrow }\\\\rangle ,}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/4ece787cabf4ceba45438338ca75ca96b78c13f4)\n\nwhere we have a general state Ψ {\\\\displaystyle \\\\Psi } ![{\\\\displaystyle \\\\Psi }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/f5471531a3fe80741a839bc98d49fae862a6439a) is the sum of the [tensor products](https:\/\/en.wikipedia.org\/wiki\/Tensor_products \"Tensor products\") of the position space wave functions and spinors.\n\n## Experiments\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Quantum_superposition&action=edit&section=9 \"Edit section: Experiments\")\\]\n\nSuccessful experiments involving superpositions of [relatively large](https:\/\/en.wikipedia.org\/wiki\/Mesoscopic \"Mesoscopic\") (by the standards of quantum physics) objects have been performed.\n\n- A [beryllium](https:\/\/en.wikipedia.org\/wiki\/Beryllium \"Beryllium\") [ion](https:\/\/en.wikipedia.org\/wiki\/Ion \"Ion\") has been trapped in a superposed state.[\\[4\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#cite_note-4)\n- A [double slit experiment](https:\/\/en.wikipedia.org\/wiki\/Double_slit_experiment \"Double slit experiment\") has been performed with molecules as large as [buckyballs](https:\/\/en.wikipedia.org\/wiki\/Buckminsterfullerene \"Buckminsterfullerene\") and functionalized oligoporphyrins with up to 2000 atoms.[\\[5\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#cite_note-5)[\\[6\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#cite_note-6)\n- Molecules with masses exceeding 10,000 and composed of over 810 atoms have successfully been superposed,[\\[7\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#cite_note-7) and metal clusters with masses over 170,000 [Da](https:\/\/en.wikipedia.org\/wiki\/Dalton_\\(unit\\) \"Dalton (unit)\") and containing more than 7,000 atoms have also been demonstrated in quantum superposition.[\\[8\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#cite_note-8)\n- Very sensitive magnetometers have been realized using [superconducting quantum interference devices](https:\/\/en.wikipedia.org\/wiki\/SQUID \"SQUID\") (SQUIDS) that operate using quantum interference effects in superconducting circuits.\n\n- A [piezoelectric](https:\/\/en.wikipedia.org\/wiki\/Piezoelectric \"Piezoelectric\") \"[tuning fork](https:\/\/en.wikipedia.org\/wiki\/Tuning_fork \"Tuning fork\")\" has been constructed, which can be placed into a superposition of vibrating and non-vibrating states. The resonator comprises about 10 trillion atoms.[\\[9\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#cite_note-9)\n- Recent research indicates that [chlorophyll](https:\/\/en.wikipedia.org\/wiki\/Chlorophyll \"Chlorophyll\") within [plants](https:\/\/en.wikipedia.org\/wiki\/Plants \"Plants\") appears to exploit the feature of quantum superposition to achieve greater efficiency in transporting energy, allowing pigment proteins to be spaced further apart than would otherwise be possible.[\\[10\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#cite_note-doi:10.1038\/nature08811-10)[\\[11\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#cite_note-11)\n\n## In quantum computers\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Quantum_superposition&action=edit&section=10 \"Edit section: In quantum computers\")\\]\n\nIn [quantum computers](https:\/\/en.wikipedia.org\/wiki\/Quantum_computers \"Quantum computers\"), a [qubit](https:\/\/en.wikipedia.org\/wiki\/Qubit \"Qubit\") is the analog of the classical information [bit](https:\/\/en.wikipedia.org\/wiki\/Bit \"Bit\"), but rather than having one of two distinct values, qubits are a superposition of two values.[\\[12\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#cite_note-Nielsen-Chuang-12): 13  Controlling this superposition qubits is a central challenge in quantum computation. The superposition needs to be robust to unintended interactions and yet interactions are needed for computing with qubits. Qubit systems like [nuclear spins](https:\/\/en.wikipedia.org\/wiki\/Nuclear_spin \"Nuclear spin\") with small coupling strength are robust to outside disturbances but the same small coupling makes it difficult to readout results.[\\[12\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#cite_note-Nielsen-Chuang-12): 278\n\n## See also\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Quantum_superposition&action=edit&section=11 \"Edit section: See also\")\\]\n\n- [Eigenstate](https:\/\/en.wikipedia.org\/wiki\/Eigenstate \"Eigenstate\") – Mathematical entity to describe the probability of each possible measurement on a systemPages displaying short descriptions of redirect targets\n- [Mach–Zehnder interferometer](https:\/\/en.wikipedia.org\/wiki\/Mach%E2%80%93Zehnder_interferometer \"Mach–Zehnder interferometer\") – Device to determine relative phase shift\n- [Penrose interpretation](https:\/\/en.wikipedia.org\/wiki\/Penrose_interpretation \"Penrose interpretation\") – Interpretation of quantum mechanics\n- [Pure qubit state](https:\/\/en.wikipedia.org\/wiki\/Pure_qubit_state \"Pure qubit state\") – Basic unit of quantum informationPages displaying short descriptions of redirect targets\n- [Quantum computation](https:\/\/en.wikipedia.org\/wiki\/Quantum_computation \"Quantum computation\") – Computer hardware technology that uses quantum mechanicsPages displaying short descriptions of redirect targets\n- [Schrödinger's cat](https:\/\/en.wikipedia.org\/wiki\/Schr%C3%B6dinger%27s_cat \"Schrödinger's cat\") – Thought experiment in quantum mechanics\n- [Superposition principle](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle \"Superposition principle\") – Fundamental principle of physics\n- [Wave packet](https:\/\/en.wikipedia.org\/wiki\/Wave_packet \"Wave packet\") – Short \"burst\" or \"envelope\" of restricted wave action that travels as a unit\n\n## References\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Quantum_superposition&action=edit&section=12 \"Edit section: References\")\\]\n\n1. ^ [***a***](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#cite_ref-Messiah_1-0) [***b***](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#cite_ref-Messiah_1-1) [***c***](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#cite_ref-Messiah_1-2)\n   Messiah, Albert (1976). *Quantum mechanics. 1* (2 ed.). Amsterdam: North-Holland. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n   \n   [978-0-471-59766-7](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-471-59766-7 \"Special:BookSources\/978-0-471-59766-7\")\n   \n   .\n2. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#cite_ref-Dirac1947_2-0)**\n   P.A.M. Dirac (1947). *The Principles of Quantum Mechanics* (2nd ed.). Clarendon Press. p. 12.\n3. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#cite_ref-Zeilinger_3-0)**\n   Zeilinger A (1999). \"Experiment and the foundations of quantum physics\". *Rev. Mod. Phys*. **71** (2): S288–S297. [Bibcode](https:\/\/en.wikipedia.org\/wiki\/Bibcode_\\(identifier\\) \"Bibcode (identifier)\"):[1999RvMPS..71..288Z](https:\/\/ui.adsabs.harvard.edu\/abs\/1999RvMPS..71..288Z). [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1103\/revmodphys.71.s288](https:\/\/doi.org\/10.1103%2Frevmodphys.71.s288).\n4. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#cite_ref-4)**\n   Monroe, C.; Meekhof, D. M.; King, B. E.; Wineland, D. J. (24 May 1996). [\"A \"Schrödinger Cat\" Superposition State of an Atom\"](https:\/\/www.science.org\/doi\/10.1126\/science.272.5265.1131). *Science*. **272** (5265): 1131–1136\\. [Bibcode](https:\/\/en.wikipedia.org\/wiki\/Bibcode_\\(identifier\\) \"Bibcode (identifier)\"):[1996Sci...272.1131M](https:\/\/ui.adsabs.harvard.edu\/abs\/1996Sci...272.1131M). [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1126\/science.272.5265.1131](https:\/\/doi.org\/10.1126%2Fscience.272.5265.1131). [ISSN](https:\/\/en.wikipedia.org\/wiki\/ISSN_\\(identifier\\) \"ISSN (identifier)\") [0036-8075](https:\/\/search.worldcat.org\/issn\/0036-8075). [PMID](https:\/\/en.wikipedia.org\/wiki\/PMID_\\(identifier\\) \"PMID (identifier)\") [8662445](https:\/\/pubmed.ncbi.nlm.nih.gov\/8662445).\n5. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#cite_ref-5)**\n   [\"Wave-particle duality of C60\"](https:\/\/web.archive.org\/web\/20120331115055\/http:\/\/www.quantum.at\/research\/molecule-interferometry-foundations\/wave-particle-duality-of-c60.html). 31 March 2012. Archived from the original on 31 March 2012.\n   \n   `{{cite web}}`: CS1 maint: bot: original URL status unknown ([link](https:\/\/en.wikipedia.org\/wiki\/Category:CS1_maint:_bot:_original_URL_status_unknown \"Category:CS1 maint: bot: original URL status unknown\"))\n6. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#cite_ref-6)**\n   Yaakov Y. Fein; Philipp Geyer; Patrick Zwick; Filip Kiałka; Sebastian Pedalino; Marcel Mayor; Stefan Gerlich; Markus Arndt (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).\n7. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#cite_ref-7)**\n   Eibenberger, S., Gerlich, S., Arndt, M., Mayor, M., Tüxen, J. (2013). \"Matter-wave interference with particles selected from a molecular library with masses exceeding 10 000 amu\", *Physical Chemistry Chemical Physics*, **15**: 14696-14700\n   \n   [arXiv](https:\/\/en.wikipedia.org\/wiki\/ArXiv_\\(identifier\\) \"ArXiv (identifier)\"):[1310\\.8343](https:\/\/arxiv.org\/abs\/1310.8343)\n8. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#cite_ref-8)**\n   Pedalino, Sebastian; Ramírez-Galindo, Bruno E.; Ferstl, Richard; Hornberger, Klaus; Arndt, Markus; Gerlich, Stefan (22 January 2026). [\"Probing quantum mechanics with nanoparticle matter-wave interferometry\"](https:\/\/www.nature.com\/articles\/s41586-025-09917-9). *Nature*. **649** (8098): 866–870\\. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1038\/s41586-025-09917-9](https:\/\/doi.org\/10.1038%2Fs41586-025-09917-9). [ISSN](https:\/\/en.wikipedia.org\/wiki\/ISSN_\\(identifier\\) \"ISSN (identifier)\") [0028-0836](https:\/\/search.worldcat.org\/issn\/0028-0836). [PMC](https:\/\/en.wikipedia.org\/wiki\/PMC_\\(identifier\\) \"PMC (identifier)\") [12823444](https:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC12823444). [PMID](https:\/\/en.wikipedia.org\/wiki\/PMID_\\(identifier\\) \"PMID (identifier)\") [41566007](https:\/\/pubmed.ncbi.nlm.nih.gov\/41566007).\n9. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#cite_ref-9)**\n   O’Connell, A. D.; Hofheinz, M.; Ansmann, M.; Bialczak, Radoslaw C.; Lenander, M.; Lucero, Erik; Neeley, M.; Sank, D.; Wang, H.; Weides, M.; Wenner, J.; Martinis, John M.; Cleland, A. N. (April 2010). [\"Quantum ground state and single-phonon control of a mechanical resonator\"](https:\/\/www.nature.com\/articles\/nature08967). *Nature*. **464** (7289): 697–703\\. [Bibcode](https:\/\/en.wikipedia.org\/wiki\/Bibcode_\\(identifier\\) \"Bibcode (identifier)\"):[2010Natur.464..697O](https:\/\/ui.adsabs.harvard.edu\/abs\/2010Natur.464..697O). [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1038\/nature08967](https:\/\/doi.org\/10.1038%2Fnature08967). [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)\") [20237473](https:\/\/pubmed.ncbi.nlm.nih.gov\/20237473).\n10. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#cite_ref-doi:10.1038\/nature08811_10-0)**\n    Scholes, Gregory; Elisabetta Collini; Cathy Y. Wong; Krystyna E. Wilk; Paul M. G. Curmi; Paul Brumer; Gregory D. Scholes (4 February 2010). \"Coherently wired light-harvesting in photosynthetic marine algae at ambient temperature\". *[Nature](https:\/\/en.wikipedia.org\/wiki\/Nature_\\(journal\\) \"Nature (journal)\")*. **463** (7281): 644–647\\. [Bibcode](https:\/\/en.wikipedia.org\/wiki\/Bibcode_\\(identifier\\) \"Bibcode (identifier)\"):[2010Natur.463..644C](https:\/\/ui.adsabs.harvard.edu\/abs\/2010Natur.463..644C). [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1038\/nature08811](https:\/\/doi.org\/10.1038%2Fnature08811). [PMID](https:\/\/en.wikipedia.org\/wiki\/PMID_\\(identifier\\) \"PMID (identifier)\") [20130647](https:\/\/pubmed.ncbi.nlm.nih.gov\/20130647). [S2CID](https:\/\/en.wikipedia.org\/wiki\/S2CID_\\(identifier\\) \"S2CID (identifier)\") [4369439](https:\/\/api.semanticscholar.org\/CorpusID:4369439).\n11. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#cite_ref-11)**\n    Moyer, Michael (September 2009). [\"Quantum Entanglement, Photosynthesis and Better Solar Cells\"](http:\/\/www.scientificamerican.com\/article.cfm?id=quantum-entanglement-and-photo). *Scientific American*. Retrieved 12 May 2010.\n12. ^ [***a***](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#cite_ref-Nielsen-Chuang_12-0) [***b***](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#cite_ref-Nielsen-Chuang_12-1)\n    [Nielsen, Michael A.](https:\/\/en.wikipedia.org\/wiki\/Michael_Nielsen \"Michael Nielsen\"); [Chuang, Isaac](https:\/\/en.wikipedia.org\/wiki\/Isaac_Chuang \"Isaac Chuang\") (2010). [*Quantum Computation and Quantum Information*](https:\/\/www.cambridge.org\/9781107002173). Cambridge: [Cambridge University Press](https:\/\/en.wikipedia.org\/wiki\/Cambridge_University_Press \"Cambridge University Press\"). [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-1-10700-217-3](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-1-10700-217-3 \"Special:BookSources\/978-1-10700-217-3\")\n    \n    . [OCLC](https:\/\/en.wikipedia.org\/wiki\/OCLC_\\(identifier\\) \"OCLC (identifier)\") [43641333](https:\/\/search.worldcat.org\/oclc\/43641333).\n\n## Further reading\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Quantum_superposition&action=edit&section=13 \"Edit section: Further reading\")\\]\n\n- [Bohr, N.](https:\/\/en.wikipedia.org\/wiki\/Niels_Bohr \"Niels Bohr\") (1927\/1928). The quantum postulate and the recent development of atomic theory, [*Nature* Supplement 14 April 1928, **121**: 580–590](http:\/\/www.nature.com\/nature\/journal\/v121\/n3050\/abs\/121580a0.html).\n- [Cohen-Tannoudji, C.](https:\/\/en.wikipedia.org\/wiki\/Claude_Cohen-Tannoudji \"Claude Cohen-Tannoudji\"), Diu, B., Laloë, F. (1973\/1977). *Quantum Mechanics*, translated from the French by S. R. Hemley, N. Ostrowsky, D. Ostrowsky, second edition, volume 1, Wiley, New York,\n  [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  [0471164321](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/0471164321 \"Special:BookSources\/0471164321\")\n  .\n- [Einstein, A.](https:\/\/en.wikipedia.org\/wiki\/Albert_Einstein \"Albert Einstein\") (1949). Remarks concerning the essays brought together in this co-operative volume, translated from the original German by the editor, pp. 665–688 in [Schilpp, P. A.](https:\/\/en.wikipedia.org\/wiki\/Paul_Arthur_Schilpp \"Paul Arthur Schilpp\") editor (1949), [*Albert Einstein: Philosopher-Scientist*](https:\/\/www.worldcat.org\/oclc\/311439), volume II, Open Court, La Salle IL.\n- [Feynman, R. P.](https:\/\/en.wikipedia.org\/wiki\/Richard_Feynman \"Richard Feynman\"), Leighton, R.B., Sands, M. (1965). *The Feynman Lectures on Physics*, [volume 3](https:\/\/feynmanlectures.caltech.edu\/III_toc.html), Addison-Wesley, Reading, MA.\n- [Merzbacher, E.](https:\/\/en.wikipedia.org\/wiki\/Eugen_Merzbacher \"Eugen Merzbacher\") (1961\/1970). *Quantum Mechanics*, second edition, Wiley, New York.\n- [Messiah, A.](https:\/\/en.wikipedia.org\/wiki\/Albert_Messiah \"Albert Messiah\") (1961). *Quantum Mechanics*, volume 1, translated by G.M. Temmer from the French *Mécanique Quantique*, North-Holland, Amsterdam.\n- [Wheeler, J. A.](https:\/\/en.wikipedia.org\/wiki\/John_Archibald_Wheeler \"John Archibald Wheeler\"); [Zurek, W.H.](https:\/\/en.wikipedia.org\/wiki\/Wojciech_H._Zurek \"Wojciech H. Zurek\") (1983). *Quantum Theory and Measurement*. Princeton NJ: Princeton University Press.\n- [Nielsen, Michael A.](https:\/\/en.wikipedia.org\/wiki\/Michael_Nielsen \"Michael Nielsen\"); [Chuang, Isaac](https:\/\/en.wikipedia.org\/wiki\/Isaac_Chuang \"Isaac Chuang\") (2000). *Quantum Computation and Quantum Information*. Cambridge: [Cambridge University Press](https:\/\/en.wikipedia.org\/wiki\/Cambridge_University_Press \"Cambridge University Press\"). [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  \n  [0521632358](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/0521632358 \"Special:BookSources\/0521632358\")\n  \n  . [OCLC](https:\/\/en.wikipedia.org\/wiki\/OCLC_\\(identifier\\) \"OCLC (identifier)\") [43641333](https:\/\/search.worldcat.org\/oclc\/43641333).\n- Williams, Colin P. (2011). *Explorations in Quantum Computing*. [Springer](https:\/\/en.wikipedia.org\/wiki\/Springer_Science%2BBusiness_Media \"Springer Science+Business Media\"). [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  \n  [978-1-84628-887-6](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-1-84628-887-6 \"Special:BookSources\/978-1-84628-887-6\")\n  \n  .\n- Yanofsky, Noson S.; Mannucci, Mirco (2013). *Quantum computing for computer scientists*. [Cambridge University Press](https:\/\/en.wikipedia.org\/wiki\/Cambridge_University_Press \"Cambridge University Press\"). [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  \n  [978-0-521-87996-5](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-521-87996-5 \"Special:BookSources\/978-0-521-87996-5\")\n  \n  .\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](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics \"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]() [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|   |   |\n|---|---|\n| [Authority control databases](https:\/\/en.wikipedia.org\/wiki\/Help:Authority_control \"Help:Authority control\") [![Edit this at Wikidata](https:\/\/upload.wikimedia.org\/wikipedia\/en\/thumb\/8\/8a\/OOjs_UI_icon_edit-ltr-progressive.svg\/20px-OOjs_UI_icon_edit-ltr-progressive.svg.png)](https:\/\/www.wikidata.org\/wiki\/Q830791#identifiers \"Edit this at Wikidata\") | [GND](https:\/\/d-nb.info\/gnd\/4346012-4) |\n\n![](https:\/\/en.wikipedia.org\/wiki\/Special:CentralAutoLogin\/start?useformat=desktop&type=1x1&usesul3=1)\n\nRetrieved from 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Wikipedia® is a registered trademark of the [Wikimedia Foundation, Inc.](https:\/\/wikimediafoundation.org\/), a non-profit organization.\n\n- [Privacy policy](https:\/\/foundation.wikimedia.org\/wiki\/Special:MyLanguage\/Policy:Privacy_policy)\n- [About Wikipedia](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:About)\n- [Disclaimers](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:General_disclaimer)\n- [Contact Wikipedia](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:Contact_us)\n- [Legal & safety contacts](https:\/\/foundation.wikimedia.org\/wiki\/Special:MyLanguage\/Legal:Wikimedia_Foundation_Legal_and_Safety_Contact_Information)\n- [Code of Conduct](https:\/\/foundation.wikimedia.org\/wiki\/Special:MyLanguage\/Policy:Universal_Code_of_Conduct)\n- [Developers](https:\/\/developer.wikimedia.org\/)\n- [Statistics](https:\/\/stats.wikimedia.org\/#\/en.wikipedia.org)\n- [Cookie statement](https:\/\/foundation.wikimedia.org\/wiki\/Special:MyLanguage\/Policy:Cookie_statement)\n- [Mobile view](https:\/\/en.wikipedia.org\/w\/index.php?title=Quantum_superposition&mobileaction=toggle_view_mobile)\n\n- [![Wikimedia Foundation](https:\/\/en.wikipedia.org\/static\/images\/footer\/wikimedia.svg)](https:\/\/www.wikimedia.org\/)\n- [![Powered by MediaWiki](https:\/\/en.wikipedia.org\/w\/resources\/assets\/mediawiki_compact.svg)](https:\/\/www.mediawiki.org\/)\n\nSearch\n\nToggle the table of contents\n\nQuantum superposition\n\n40 languages\n\n[Add topic](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition)","attrs_readable_markdown":"From Wikipedia, the free encyclopedia\n\nQuantum superposition of states and decoherence\n\n**Quantum superposition** is a fundamental principle of [quantum mechanics](https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics \"Quantum mechanics\") that states that [linear combinations](https:\/\/en.wikipedia.org\/wiki\/Linear_combination \"Linear combination\") of [solutions](https:\/\/en.wikipedia.org\/wiki\/Solution_\\(mathematics\\) \"Solution (mathematics)\") to the [Schrödinger equation](https:\/\/en.wikipedia.org\/wiki\/Schr%C3%B6dinger_equation \"Schrödinger equation\") are also solutions of the Schrödinger equation. This follows from the fact that the Schrödinger equation is a [linear differential equation](https:\/\/en.wikipedia.org\/wiki\/Linear_differential_equation \"Linear differential equation\") in time and position. More precisely, the state of a system is given by a [linear combination](https:\/\/en.wikipedia.org\/wiki\/Linear_combination \"Linear combination\") of all the [eigenfunctions](https:\/\/en.wikipedia.org\/wiki\/Eigenfunction \"Eigenfunction\") of the Schrödinger equation governing that system.\n\nAn example is a [qubit](https:\/\/en.wikipedia.org\/wiki\/Qubit \"Qubit\") used in [quantum information processing](https:\/\/en.wikipedia.org\/wiki\/Quantum_information_processing \"Quantum information processing\"). A qubit state is most generally a superposition of the basis states ![{\\\\displaystyle \\|0\\\\rangle }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/ed066a3ad158da0ad6d6a421a606b1c8a35eb95b) and ![{\\\\displaystyle \\|1\\\\rangle }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/2f53021ca18e77477ee5bd3c1523e5830189ec5c):\n\n![{\\\\displaystyle \\|\\\\Psi \\\\rangle =c\\_{0}\\|0\\\\rangle +c\\_{1}\\|1\\\\rangle ,}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/f10213c27e86b15bff8015bd81b1d5983d14c5a4)\n\nwhere ![{\\\\displaystyle \\|\\\\Psi \\\\rangle }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/6e77f6b1e903837c5765c9683da41dd93199621c) is the [quantum state](https:\/\/en.wikipedia.org\/wiki\/Quantum_state \"Quantum state\") of the qubit, and ![{\\\\displaystyle \\|0\\\\rangle }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/ed066a3ad158da0ad6d6a421a606b1c8a35eb95b), ![{\\\\displaystyle \\|1\\\\rangle }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/2f53021ca18e77477ee5bd3c1523e5830189ec5c) denote particular solutions to the Schrödinger equation in [Dirac notation](https:\/\/en.wikipedia.org\/wiki\/Bra%E2%80%93ket_notation \"Bra–ket notation\") weighted by the two [probability amplitudes](https:\/\/en.wikipedia.org\/wiki\/Probability_amplitude \"Probability amplitude\") ![{\\\\displaystyle c\\_{0}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/1882ba8f1dc60f0c68a642abb5af093c73910921) and ![{\\\\displaystyle c\\_{1}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/77b7dc6d279091d354e0b90889b463bfa7eb7247) that both are complex numbers. Here ![{\\\\displaystyle \\|0\\\\rangle }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/ed066a3ad158da0ad6d6a421a606b1c8a35eb95b) corresponds to the classical 0 [bit](https:\/\/en.wikipedia.org\/wiki\/Bit \"Bit\"), and ![{\\\\displaystyle \\|1\\\\rangle }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/2f53021ca18e77477ee5bd3c1523e5830189ec5c) to the classical 1 bit. The probabilities of measuring the system in the ![{\\\\displaystyle \\|0\\\\rangle }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/ed066a3ad158da0ad6d6a421a606b1c8a35eb95b) or ![{\\\\displaystyle \\|1\\\\rangle }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/2f53021ca18e77477ee5bd3c1523e5830189ec5c) state are given by ![{\\\\displaystyle \\|c\\_{0}\\|^{2}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/8529f071ccb861751297a04659e8dbddca09bad8) and ![{\\\\displaystyle \\|c\\_{1}\\|^{2}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/a9c67c5c8bf07a77edd704f880a64a35b8d9f1c5) respectively (see the [Born rule](https:\/\/en.wikipedia.org\/wiki\/Born_rule \"Born rule\")). Before the measurement occurs the qubit is in a superposition of both states.\n\nThe interference fringes in the [double-slit experiment](https:\/\/en.wikipedia.org\/wiki\/Double-slit_experiment \"Double-slit experiment\") provide another example of the superposition principle.\n\nThe theory of quantum mechanics postulates that a [wave equation](https:\/\/en.wikipedia.org\/wiki\/Wave_equation \"Wave equation\") completely determines the state of a quantum system at all times. Furthermore, this differential equation is restricted to be [linear](https:\/\/en.wikipedia.org\/wiki\/Linear_differential_equation \"Linear differential equation\") and [homogeneous](https:\/\/en.wikipedia.org\/wiki\/Homogeneous_differential_equation \"Homogeneous differential equation\"). These conditions mean that for any two solutions of the wave equation, ![{\\\\displaystyle \\\\Psi \\_{1}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/ddcd58cf9c1c4dd1eb32a29c1e3abf425ec72600) and ![{\\\\displaystyle \\\\Psi \\_{2}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/1125553c0a026635c63ab2ad542b43174aca9bbe), a linear combination of those solutions also solve the wave equation: ![{\\\\displaystyle \\\\Psi =c\\_{1}\\\\Psi \\_{1}+c\\_{2}\\\\Psi \\_{2}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/f6823dfb8e3163bbbfd61bd2d925ce8d6fdac949) for arbitrary complex coefficients ![{\\\\displaystyle c\\_{1}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/77b7dc6d279091d354e0b90889b463bfa7eb7247) and ![{\\\\displaystyle c\\_{2}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/0b30ba1b247fb8d334580cec68561e749d24aff2).[\\[1\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#cite_note-Messiah-1): 61  If the wave equation has more than two solutions, combinations of all such solutions are again valid solutions.\n\nThe quantum wave equation can be solved using functions of position, ![{\\\\displaystyle \\\\Psi ({\\\\vec {r}})}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/860ed8dcca03bb55dda0fe92c92a079038be797b), or using functions of momentum, ![{\\\\displaystyle \\\\Phi ({\\\\vec {p}})}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/96584b5a2ed608930bfa349e822c564a5298250d) and consequently the superposition of momentum functions are also solutions: ![{\\\\displaystyle \\\\Phi ({\\\\vec {p}})=d\\_{1}\\\\Phi \\_{1}({\\\\vec {p}})+d\\_{2}\\\\Phi \\_{2}({\\\\vec {p}})}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/f72ca07aa31123cf18829faed1ebb4abeb3a5120) The position and momentum solutions are related by a [linear transformation](https:\/\/en.wikipedia.org\/wiki\/Linear_transformation \"Linear transformation\"), a [Fourier transformation](https:\/\/en.wikipedia.org\/wiki\/Fourier_transformation \"Fourier transformation\"). This transformation is itself a quantum superposition and every position wave function can be represented as a superposition of momentum wave functions and vice versa. These superpositions involve an infinite number of component waves.[\\[1\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#cite_note-Messiah-1): 244\n\n## Generalization to basis states\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Quantum_superposition&action=edit&section=3 \"Edit section: Generalization to basis states\")\\]\n\nOther transformations express a quantum solution as a superposition of [eigenvectors](https:\/\/en.wikipedia.org\/wiki\/Eigenvectors \"Eigenvectors\"), each corresponding to a possible result of a measurement on the quantum system. An eigenvector ![{\\\\displaystyle \\\\psi \\_{i}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/3e709a809cfd3cfbb647fe22c8d1f81573f4c7ae) for a mathematical operator, ![{\\\\displaystyle {\\\\hat {A}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/f595a6c73d1183d6a1b2ac21fe47ac28c1483821), has the equation ![{\\\\displaystyle {\\\\hat {A}}\\\\psi \\_{i}=\\\\lambda \\_{i}\\\\psi \\_{i}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/4f7c7cfad33ac61dd3bcde805172bb9897ac0dfe) where ![{\\\\displaystyle \\\\lambda \\_{i}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/72fde940918edf84caf3d406cc7d31949166820f) is one possible measured quantum value for the observable ![{\\\\displaystyle A}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/7daff47fa58cdfd29dc333def748ff5fa4c923e3). A superposition of these eigenvectors can represent any solution: ![{\\\\displaystyle \\\\Psi =\\\\sum \\_{n}a\\_{i}\\\\psi \\_{i}.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/da5aeb99c104b07a8842ea74cc61a45a859aa2a3) The states like ![{\\\\displaystyle \\\\psi \\_{i}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/3e709a809cfd3cfbb647fe22c8d1f81573f4c7ae) are called basis states.\n\n## Compact notation for superpositions\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Quantum_superposition&action=edit&section=4 \"Edit section: Compact notation for superpositions\")\\]\n\nImportant mathematical operations on quantum system solutions can be performed using only the coefficients of the superposition, suppressing the details of the superposed functions. This leads to quantum systems expressed in the [Dirac bra-ket notation](https:\/\/en.wikipedia.org\/wiki\/Bra-ket_notation \"Bra-ket notation\"):[\\[1\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#cite_note-Messiah-1): 245  ![{\\\\displaystyle \\|v\\\\rangle =d\\_{1}\\|1\\\\rangle +d\\_{2}\\|2\\\\rangle }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/acc7ac146085e844fd0f2988bed9f23c30af497a) This approach is especially effective for systems like quantum spin with no classical coordinate analog. Such shorthand notation is very common in textbooks and papers on quantum mechanics, and superposition of basis states is a fundamental tool in quantum mechanics.\n\n[Paul Dirac](https:\/\/en.wikipedia.org\/wiki\/Paul_Dirac \"Paul Dirac\") described the superposition principle as follows:\n\n> The non-classical nature of the superposition process is brought out clearly if we consider the superposition of two states, *A* and *B*, such that there exists an observation which, when made on the system in state *A*, is certain to lead to one particular result, *a* say, and when made on the system in state *B* is certain to lead to some different result, *b* say. What will be the result of the observation when made on the system in the superposed state? The answer is that the result will be sometimes *a* and sometimes *b*, according to a probability law depending on the relative weights of *A* and *B* in the superposition process. It will never be different from both *a* and *b* \\[i.e., either *a* or *b*\\]. *The intermediate character of the state formed by superposition thus expresses itself through the probability of a particular result for an observation being intermediate between the corresponding probabilities for the original states, not through the result itself being intermediate between the corresponding results for the original states.*[\\[2\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#cite_note-Dirac1947-2)\n\n[Anton Zeilinger](https:\/\/en.wikipedia.org\/wiki\/Anton_Zeilinger \"Anton Zeilinger\"), referring to the prototypical example of the [double-slit experiment](https:\/\/en.wikipedia.org\/wiki\/Double-slit_experiment \"Double-slit experiment\"), has elaborated regarding the creation and destruction of quantum superposition:\n\n> \"\\[T\\]he superposition of amplitudes ... is only valid if there is no way to know, even in principle, which path the particle took. It is important to realize that this does not imply that an observer actually takes note of what happens. It is sufficient to destroy the interference pattern, if the path information is accessible in principle from the experiment or even if it is dispersed in the environment and beyond any technical possibility to be recovered, but in principle still ‘‘out there.’’ The absence of any such information is *the essential criterion* for quantum interference to appear.[\\[3\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#cite_note-Zeilinger-3)\n\nAny quantum state can be expanded as a sum or superposition of the eigenstates of an Hermitian operator, like the Hamiltonian, because the eigenstates form a complete basis:\n\n![{\\\\displaystyle \\|\\\\alpha \\\\rangle =\\\\sum \\_{n}c\\_{n}\\|n\\\\rangle ,}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/df393c7be36c417cc59f5a1a605aea04beef2e7b)\n\nwhere ![{\\\\displaystyle \\|n\\\\rangle }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/89c45bda6ac852da785c06476f2d3e3b6cba812a) are the energy eigenstates of the Hamiltonian. For continuous variables like position eigenstates, ![{\\\\displaystyle \\|x\\\\rangle }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/48004887d8f9dfc489bd2bc793780b7f1d8039ad):\n\n![{\\\\displaystyle \\|\\\\alpha \\\\rangle =\\\\int dx'\\|x'\\\\rangle \\\\langle x'\\|\\\\alpha \\\\rangle ,}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/9495e664ba18698edce2bfcc82c345de5ceaf6c6)\n\nwhere ![{\\\\displaystyle \\\\phi \\_{\\\\alpha }(x)=\\\\langle x\\|\\\\alpha \\\\rangle }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/2ef30d5158d3fe8417e7fe37540e5370e46d9887) is the projection of the state into the ![{\\\\displaystyle \\|x\\\\rangle }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/48004887d8f9dfc489bd2bc793780b7f1d8039ad) basis and is called the wave function of the particle. In both instances we notice that ![{\\\\displaystyle \\|\\\\alpha \\\\rangle }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/f42032e642ee1c9d27adb318d34c7cc85f7a95d5) can be expanded as a superposition of an infinite number of basis states.\n\nGiven the Schrödinger equation\n\n![{\\\\displaystyle {\\\\hat {H}}\\|n\\\\rangle =E\\_{n}\\|n\\\\rangle ,}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/8a6afb6d354132f7d244ee60cf28e4608aa6e922)\n\nwhere ![{\\\\displaystyle \\|n\\\\rangle }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/89c45bda6ac852da785c06476f2d3e3b6cba812a) indexes the set of eigenstates of the Hamiltonian with energy eigenvalues ![{\\\\displaystyle E\\_{n},}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/578d4bb07de7b29269be95abb6a214848c09aea7) we see immediately that\n\n![{\\\\displaystyle {\\\\hat {H}}{\\\\big (}\\|n\\\\rangle +\\|n'\\\\rangle {\\\\big )}=E\\_{n}\\|n\\\\rangle +E\\_{n'}\\|n'\\\\rangle ,}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/806bff6ca40452045baac80fd266eb3c453e7340)\n\nwhere\n\n![{\\\\displaystyle \\|\\\\Psi \\\\rangle =\\|n\\\\rangle +\\|n'\\\\rangle }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/99dd060b4c8a46870132855acbe5a64eefdf0ea3)\n\nis a solution of the Schrödinger equation but is not generally an eigenstate because ![{\\\\displaystyle E\\_{n}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/ad6b82f2a00af6c9efd4c16d4e99329605645c0c) and ![{\\\\displaystyle E\\_{n'}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/9d618f0a54e5627c8baa3cfb77b3e2dfac8f31bc) are not generally equal. We say that ![{\\\\displaystyle \\|\\\\Psi \\\\rangle }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/6e77f6b1e903837c5765c9683da41dd93199621c) is made up of a superposition of energy eigenstates. Now consider the more concrete case of an [electron](https:\/\/en.wikipedia.org\/wiki\/Electron \"Electron\") that has either [spin](https:\/\/en.wikipedia.org\/wiki\/Spin_\\(physics\\) \"Spin (physics)\") up or down. We now index the eigenstates with the [spinors](https:\/\/en.wikipedia.org\/wiki\/Spinor \"Spinor\") in the ![{\\\\displaystyle {\\\\hat {z}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/722665b45e05afe79f4395a3de0237d8ce856273) basis:\n\n![{\\\\displaystyle \\|\\\\Psi \\\\rangle =c\\_{1}\\|{\\\\uparrow }\\\\rangle +c\\_{2}\\|{\\\\downarrow }\\\\rangle ,}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/42f17d6ba89011bea1b2f5f2747b88a7482c681e)\n\nwhere ![{\\\\displaystyle \\|{\\\\uparrow }\\\\rangle }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/6e81fba5ac4e54a40f5b9d63e66548328b7ffb6e) and ![{\\\\displaystyle \\|{\\\\downarrow }\\\\rangle }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/622f3f29fa4b88926ffa22e3da1daa5f2893d17c) denote spin-up and spin-down states respectively. As previously discussed, the magnitudes of the complex coefficients give the probability of finding the electron in either definite spin state:\n\n![{\\\\displaystyle P{\\\\big (}\\|{\\\\uparrow }\\\\rangle {\\\\big )}=\\|c\\_{1}\\|^{2},}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/4e31d0a1610fddb7d86d8548a0ae43a9dff80529)\n\n![{\\\\displaystyle P{\\\\big (}\\|{\\\\downarrow }\\\\rangle {\\\\big )}=\\|c\\_{2}\\|^{2},}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/8ad7f70c0f22f0405876a0e5c22cfdf96bcf05a9)\n\n![{\\\\displaystyle P\\_{\\\\text{total}}=P{\\\\big (}\\|{\\\\uparrow }\\\\rangle {\\\\big )}+P{\\\\big (}\\|{\\\\downarrow }\\\\rangle {\\\\big )}=\\|c\\_{1}\\|^{2}+\\|c\\_{2}\\|^{2}=1,}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/5fd7a45939635ea1e22247c9c7293c938999cefe)\n\nwhere the probability of finding the particle with either spin up or down is normalized to 1. Notice that ![{\\\\displaystyle c\\_{1}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/77b7dc6d279091d354e0b90889b463bfa7eb7247) and ![{\\\\displaystyle c\\_{2}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/0b30ba1b247fb8d334580cec68561e749d24aff2) are complex numbers, so that\n\n![{\\\\displaystyle \\|\\\\Psi \\\\rangle ={\\\\frac {3}{5}}i\\|{\\\\uparrow }\\\\rangle +{\\\\frac {4}{5}}\\|{\\\\downarrow }\\\\rangle .}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/1751456a557a43b4c870d3dd5351854eb5556233)\n\nis an example of an allowed state. We now get\n\n![{\\\\displaystyle P{\\\\big (}\\|{\\\\uparrow }\\\\rangle {\\\\big )}=\\\\left\\|{\\\\frac {3i}{5}}\\\\right\\|^{2}={\\\\frac {9}{25}},}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/eeca39a54f3fc393e54a5fdc7feeb5aeb0b527ac)\n\n![{\\\\displaystyle P{\\\\big (}\\|{\\\\downarrow }\\\\rangle {\\\\big )}=\\\\left\\|{\\\\frac {4}{5}}\\\\right\\|^{2}={\\\\frac {16}{25}},}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/863ea6a63d777554353bd1f448c5d4d989136162)\n\n![{\\\\displaystyle P\\_{\\\\text{total}}=P{\\\\big (}\\|{\\\\uparrow }\\\\rangle {\\\\big )}+P{\\\\big (}\\|{\\\\downarrow }\\\\rangle {\\\\big )}={\\\\frac {9}{25}}+{\\\\frac {16}{25}}=1.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/845feda8d1aa925d760f2a5e4b88e350100ccfa1)\n\nIf we consider a qubit with both position and spin, the state is a superposition of all possibilities for both:\n\n![{\\\\displaystyle \\\\Psi =\\\\psi \\_{+}(x)\\\\otimes \\|{\\\\uparrow }\\\\rangle +\\\\psi \\_{-}(x)\\\\otimes \\|{\\\\downarrow }\\\\rangle ,}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/4ece787cabf4ceba45438338ca75ca96b78c13f4)\n\nwhere we have a general state ![{\\\\displaystyle \\\\Psi }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/f5471531a3fe80741a839bc98d49fae862a6439a) is the sum of the [tensor products](https:\/\/en.wikipedia.org\/wiki\/Tensor_products \"Tensor products\") of the position space wave functions and spinors.\n\nSuccessful experiments involving superpositions of [relatively large](https:\/\/en.wikipedia.org\/wiki\/Mesoscopic \"Mesoscopic\") (by the standards of quantum physics) objects have been performed.\n\n- A [beryllium](https:\/\/en.wikipedia.org\/wiki\/Beryllium \"Beryllium\") [ion](https:\/\/en.wikipedia.org\/wiki\/Ion \"Ion\") has been trapped in a superposed state.[\\[4\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#cite_note-4)\n- A [double slit experiment](https:\/\/en.wikipedia.org\/wiki\/Double_slit_experiment \"Double slit experiment\") has been performed with molecules as large as [buckyballs](https:\/\/en.wikipedia.org\/wiki\/Buckminsterfullerene \"Buckminsterfullerene\") and functionalized oligoporphyrins with up to 2000 atoms.[\\[5\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#cite_note-5)[\\[6\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#cite_note-6)\n- Molecules with masses exceeding 10,000 and composed of over 810 atoms have successfully been superposed,[\\[7\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#cite_note-7) and metal clusters with masses over 170,000 [Da](https:\/\/en.wikipedia.org\/wiki\/Dalton_\\(unit\\) \"Dalton (unit)\") and containing more than 7,000 atoms have also been demonstrated in quantum superposition.[\\[8\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#cite_note-8)\n- Very sensitive magnetometers have been realized using [superconducting quantum interference devices](https:\/\/en.wikipedia.org\/wiki\/SQUID \"SQUID\") (SQUIDS) that operate using quantum interference effects in superconducting circuits.\n\n- A [piezoelectric](https:\/\/en.wikipedia.org\/wiki\/Piezoelectric \"Piezoelectric\") \"[tuning fork](https:\/\/en.wikipedia.org\/wiki\/Tuning_fork \"Tuning fork\")\" has been constructed, which can be placed into a superposition of vibrating and non-vibrating states. The resonator comprises about 10 trillion atoms.[\\[9\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#cite_note-9)\n- Recent research indicates that [chlorophyll](https:\/\/en.wikipedia.org\/wiki\/Chlorophyll \"Chlorophyll\") within [plants](https:\/\/en.wikipedia.org\/wiki\/Plants \"Plants\") appears to exploit the feature of quantum superposition to achieve greater efficiency in transporting energy, allowing pigment proteins to be spaced further apart than would otherwise be possible.[\\[10\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#cite_note-doi:10.1038\/nature08811-10)[\\[11\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#cite_note-11)\n\n## In quantum computers\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Quantum_superposition&action=edit&section=10 \"Edit section: In quantum computers\")\\]\n\nIn [quantum computers](https:\/\/en.wikipedia.org\/wiki\/Quantum_computers \"Quantum computers\"), a [qubit](https:\/\/en.wikipedia.org\/wiki\/Qubit \"Qubit\") is the analog of the classical information [bit](https:\/\/en.wikipedia.org\/wiki\/Bit \"Bit\"), but rather than having one of two distinct values, qubits are a superposition of two values.[\\[12\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#cite_note-Nielsen-Chuang-12): 13  Controlling this superposition qubits is a central challenge in quantum computation. The superposition needs to be robust to unintended interactions and yet interactions are needed for computing with qubits. Qubit systems like [nuclear spins](https:\/\/en.wikipedia.org\/wiki\/Nuclear_spin \"Nuclear spin\") with small coupling strength are robust to outside disturbances but the same small coupling makes it difficult to readout results.[\\[12\\]](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#cite_note-Nielsen-Chuang-12): 278\n\n- [Eigenstate](https:\/\/en.wikipedia.org\/wiki\/Eigenstate \"Eigenstate\") – Mathematical entity to describe the probability of each possible measurement on a system\n- [Mach–Zehnder interferometer](https:\/\/en.wikipedia.org\/wiki\/Mach%E2%80%93Zehnder_interferometer \"Mach–Zehnder interferometer\") – Device to determine relative phase shift\n- [Penrose interpretation](https:\/\/en.wikipedia.org\/wiki\/Penrose_interpretation \"Penrose interpretation\") – Interpretation of quantum mechanics\n- [Pure qubit state](https:\/\/en.wikipedia.org\/wiki\/Pure_qubit_state \"Pure qubit state\") – Basic unit of quantum information\n- [Quantum computation](https:\/\/en.wikipedia.org\/wiki\/Quantum_computation \"Quantum computation\") – Computer hardware technology that uses quantum mechanics\n- [Schrödinger's cat](https:\/\/en.wikipedia.org\/wiki\/Schr%C3%B6dinger%27s_cat \"Schrödinger's cat\") – Thought experiment in quantum mechanics\n- [Superposition principle](https:\/\/en.wikipedia.org\/wiki\/Superposition_principle \"Superposition principle\") – Fundamental principle of physics\n- [Wave packet](https:\/\/en.wikipedia.org\/wiki\/Wave_packet \"Wave packet\") – Short \"burst\" or \"envelope\" of restricted wave action that travels as a unit\n\n1. ^ [***a***](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#cite_ref-Messiah_1-0) [***b***](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#cite_ref-Messiah_1-1) [***c***](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#cite_ref-Messiah_1-2)\n   Messiah, Albert (1976). *Quantum mechanics. 1* (2 ed.). Amsterdam: North-Holland. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n   \n   [978-0-471-59766-7](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-471-59766-7 \"Special:BookSources\/978-0-471-59766-7\")\n   \n   .\n2. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#cite_ref-Dirac1947_2-0)**\n   P.A.M. Dirac (1947). *The Principles of Quantum Mechanics* (2nd ed.). Clarendon Press. p. 12.\n3. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#cite_ref-Zeilinger_3-0)**\n   Zeilinger A (1999). \"Experiment and the foundations of quantum physics\". *Rev. Mod. Phys*. **71** (2): S288–S297. [Bibcode](https:\/\/en.wikipedia.org\/wiki\/Bibcode_\\(identifier\\) \"Bibcode (identifier)\"):[1999RvMPS..71..288Z](https:\/\/ui.adsabs.harvard.edu\/abs\/1999RvMPS..71..288Z). [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1103\/revmodphys.71.s288](https:\/\/doi.org\/10.1103%2Frevmodphys.71.s288).\n4. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#cite_ref-4)**\n   Monroe, C.; Meekhof, D. M.; King, B. E.; Wineland, D. J. (24 May 1996). [\"A \"Schrödinger Cat\" Superposition State of an Atom\"](https:\/\/www.science.org\/doi\/10.1126\/science.272.5265.1131). *Science*. **272** (5265): 1131–1136\\. [Bibcode](https:\/\/en.wikipedia.org\/wiki\/Bibcode_\\(identifier\\) \"Bibcode (identifier)\"):[1996Sci...272.1131M](https:\/\/ui.adsabs.harvard.edu\/abs\/1996Sci...272.1131M). [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1126\/science.272.5265.1131](https:\/\/doi.org\/10.1126%2Fscience.272.5265.1131). [ISSN](https:\/\/en.wikipedia.org\/wiki\/ISSN_\\(identifier\\) \"ISSN (identifier)\") [0036-8075](https:\/\/search.worldcat.org\/issn\/0036-8075). [PMID](https:\/\/en.wikipedia.org\/wiki\/PMID_\\(identifier\\) \"PMID (identifier)\") [8662445](https:\/\/pubmed.ncbi.nlm.nih.gov\/8662445).\n5. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#cite_ref-5)**\n   [\"Wave-particle duality of C60\"](https:\/\/web.archive.org\/web\/20120331115055\/http:\/\/www.quantum.at\/research\/molecule-interferometry-foundations\/wave-particle-duality-of-c60.html). 31 March 2012. Archived from the original on 31 March 2012.\n   \n   `{{cite web}}`: CS1 maint: bot: original URL status unknown ([link](https:\/\/en.wikipedia.org\/wiki\/Category:CS1_maint:_bot:_original_URL_status_unknown \"Category:CS1 maint: bot: original URL status unknown\"))\n6. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#cite_ref-6)**\n   Yaakov Y. Fein; Philipp Geyer; Patrick Zwick; Filip Kiałka; Sebastian Pedalino; Marcel Mayor; Stefan Gerlich; Markus Arndt (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).\n7. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#cite_ref-7)** Eibenberger, S., Gerlich, S., Arndt, M., Mayor, M., Tüxen, J. (2013). \"Matter-wave interference with particles selected from a molecular library with masses exceeding 10 000 amu\", *Physical Chemistry Chemical Physics*, **15**: 14696-14700 [arXiv](https:\/\/en.wikipedia.org\/wiki\/ArXiv_\\(identifier\\) \"ArXiv (identifier)\"):[1310\\.8343](https:\/\/arxiv.org\/abs\/1310.8343)\n8. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#cite_ref-8)**\n   Pedalino, Sebastian; Ramírez-Galindo, Bruno E.; Ferstl, Richard; Hornberger, Klaus; Arndt, Markus; Gerlich, Stefan (22 January 2026). [\"Probing quantum mechanics with nanoparticle matter-wave interferometry\"](https:\/\/www.nature.com\/articles\/s41586-025-09917-9). *Nature*. **649** (8098): 866–870\\. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1038\/s41586-025-09917-9](https:\/\/doi.org\/10.1038%2Fs41586-025-09917-9). [ISSN](https:\/\/en.wikipedia.org\/wiki\/ISSN_\\(identifier\\) \"ISSN (identifier)\") [0028-0836](https:\/\/search.worldcat.org\/issn\/0028-0836). [PMC](https:\/\/en.wikipedia.org\/wiki\/PMC_\\(identifier\\) \"PMC (identifier)\") [12823444](https:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC12823444). [PMID](https:\/\/en.wikipedia.org\/wiki\/PMID_\\(identifier\\) \"PMID (identifier)\") [41566007](https:\/\/pubmed.ncbi.nlm.nih.gov\/41566007).\n9. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#cite_ref-9)**\n   O’Connell, A. D.; Hofheinz, M.; Ansmann, M.; Bialczak, Radoslaw C.; Lenander, M.; Lucero, Erik; Neeley, M.; Sank, D.; Wang, H.; Weides, M.; Wenner, J.; Martinis, John M.; Cleland, A. N. (April 2010). [\"Quantum ground state and single-phonon control of a mechanical resonator\"](https:\/\/www.nature.com\/articles\/nature08967). *Nature*. **464** (7289): 697–703\\. [Bibcode](https:\/\/en.wikipedia.org\/wiki\/Bibcode_\\(identifier\\) \"Bibcode (identifier)\"):[2010Natur.464..697O](https:\/\/ui.adsabs.harvard.edu\/abs\/2010Natur.464..697O). [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1038\/nature08967](https:\/\/doi.org\/10.1038%2Fnature08967). [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)\") [20237473](https:\/\/pubmed.ncbi.nlm.nih.gov\/20237473).\n10. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#cite_ref-doi:10.1038\/nature08811_10-0)**\n    Scholes, Gregory; Elisabetta Collini; Cathy Y. Wong; Krystyna E. Wilk; Paul M. G. Curmi; Paul Brumer; Gregory D. Scholes (4 February 2010). \"Coherently wired light-harvesting in photosynthetic marine algae at ambient temperature\". *[Nature](https:\/\/en.wikipedia.org\/wiki\/Nature_\\(journal\\) \"Nature (journal)\")*. **463** (7281): 644–647\\. [Bibcode](https:\/\/en.wikipedia.org\/wiki\/Bibcode_\\(identifier\\) \"Bibcode (identifier)\"):[2010Natur.463..644C](https:\/\/ui.adsabs.harvard.edu\/abs\/2010Natur.463..644C). [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1038\/nature08811](https:\/\/doi.org\/10.1038%2Fnature08811). [PMID](https:\/\/en.wikipedia.org\/wiki\/PMID_\\(identifier\\) \"PMID (identifier)\") [20130647](https:\/\/pubmed.ncbi.nlm.nih.gov\/20130647). [S2CID](https:\/\/en.wikipedia.org\/wiki\/S2CID_\\(identifier\\) \"S2CID (identifier)\") [4369439](https:\/\/api.semanticscholar.org\/CorpusID:4369439).\n11. **[^](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#cite_ref-11)**\n    Moyer, Michael (September 2009). [\"Quantum Entanglement, Photosynthesis and Better Solar Cells\"](http:\/\/www.scientificamerican.com\/article.cfm?id=quantum-entanglement-and-photo). *Scientific American*. Retrieved 12 May 2010.\n12. ^ [***a***](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#cite_ref-Nielsen-Chuang_12-0) [***b***](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition#cite_ref-Nielsen-Chuang_12-1)\n    [Nielsen, Michael A.](https:\/\/en.wikipedia.org\/wiki\/Michael_Nielsen \"Michael Nielsen\"); [Chuang, Isaac](https:\/\/en.wikipedia.org\/wiki\/Isaac_Chuang \"Isaac Chuang\") (2010). [*Quantum Computation and Quantum Information*](https:\/\/www.cambridge.org\/9781107002173). Cambridge: [Cambridge University Press](https:\/\/en.wikipedia.org\/wiki\/Cambridge_University_Press \"Cambridge University Press\"). [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-1-10700-217-3](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-1-10700-217-3 \"Special:BookSources\/978-1-10700-217-3\")\n    \n    . [OCLC](https:\/\/en.wikipedia.org\/wiki\/OCLC_\\(identifier\\) \"OCLC (identifier)\") [43641333](https:\/\/search.worldcat.org\/oclc\/43641333).\n\n- [Bohr, N.](https:\/\/en.wikipedia.org\/wiki\/Niels_Bohr \"Niels Bohr\") (1927\/1928). The quantum postulate and the recent development of atomic theory, [*Nature* Supplement 14 April 1928, **121**: 580–590](http:\/\/www.nature.com\/nature\/journal\/v121\/n3050\/abs\/121580a0.html).\n- [Cohen-Tannoudji, C.](https:\/\/en.wikipedia.org\/wiki\/Claude_Cohen-Tannoudji \"Claude Cohen-Tannoudji\"), Diu, B., Laloë, F. (1973\/1977). *Quantum Mechanics*, translated from the French by S. R. Hemley, N. Ostrowsky, D. Ostrowsky, second edition, volume 1, Wiley, New York, [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  [0471164321](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/0471164321 \"Special:BookSources\/0471164321\")\n  .\n- [Einstein, A.](https:\/\/en.wikipedia.org\/wiki\/Albert_Einstein \"Albert Einstein\") (1949). Remarks concerning the essays brought together in this co-operative volume, translated from the original German by the editor, pp. 665–688 in [Schilpp, P. A.](https:\/\/en.wikipedia.org\/wiki\/Paul_Arthur_Schilpp \"Paul Arthur Schilpp\") editor (1949), [*Albert Einstein: Philosopher-Scientist*](https:\/\/www.worldcat.org\/oclc\/311439), volume II, Open Court, La Salle IL.\n- [Feynman, R. P.](https:\/\/en.wikipedia.org\/wiki\/Richard_Feynman \"Richard Feynman\"), Leighton, R.B., Sands, M. (1965). *The Feynman Lectures on Physics*, [volume 3](https:\/\/feynmanlectures.caltech.edu\/III_toc.html), Addison-Wesley, Reading, MA.\n- [Merzbacher, E.](https:\/\/en.wikipedia.org\/wiki\/Eugen_Merzbacher \"Eugen Merzbacher\") (1961\/1970). *Quantum Mechanics*, second edition, Wiley, New York.\n- [Messiah, A.](https:\/\/en.wikipedia.org\/wiki\/Albert_Messiah \"Albert Messiah\") (1961). *Quantum Mechanics*, volume 1, translated by G.M. Temmer from the French *Mécanique Quantique*, North-Holland, Amsterdam.\n- [Wheeler, J. A.](https:\/\/en.wikipedia.org\/wiki\/John_Archibald_Wheeler \"John Archibald Wheeler\"); [Zurek, W.H.](https:\/\/en.wikipedia.org\/wiki\/Wojciech_H._Zurek \"Wojciech H. Zurek\") (1983). *Quantum Theory and Measurement*. Princeton NJ: Princeton University Press.\n- [Nielsen, Michael A.](https:\/\/en.wikipedia.org\/wiki\/Michael_Nielsen \"Michael Nielsen\"); [Chuang, Isaac](https:\/\/en.wikipedia.org\/wiki\/Isaac_Chuang \"Isaac Chuang\") (2000). *Quantum Computation and Quantum Information*. Cambridge: [Cambridge University Press](https:\/\/en.wikipedia.org\/wiki\/Cambridge_University_Press \"Cambridge University Press\"). [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  \n  [0521632358](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/0521632358 \"Special:BookSources\/0521632358\")\n  \n  . [OCLC](https:\/\/en.wikipedia.org\/wiki\/OCLC_\\(identifier\\) \"OCLC (identifier)\") [43641333](https:\/\/search.worldcat.org\/oclc\/43641333).\n- Williams, Colin P. (2011). *Explorations in Quantum Computing*. [Springer](https:\/\/en.wikipedia.org\/wiki\/Springer_Science%2BBusiness_Media \"Springer Science+Business Media\"). [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  \n  [978-1-84628-887-6](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-1-84628-887-6 \"Special:BookSources\/978-1-84628-887-6\")\n  \n  .\n- Yanofsky, Noson S.; Mannucci, Mirco (2013). *Quantum computing for computer scientists*. [Cambridge University Press](https:\/\/en.wikipedia.org\/wiki\/Cambridge_University_Press \"Cambridge University Press\"). [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  \n  [978-0-521-87996-5](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-521-87996-5 \"Special:BookSources\/978-0-521-87996-5\")\n  \n  .","meta_canonical":null}

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Boilerpipe Text	From Wikipedia, the free encyclopedia Quantum superposition of states and decoherence Quantum superposition is a fundamental principle of quantum mechanics that states that linear combinations of solutions to the Schrödinger equation are also solutions of the Schrödinger equation. This follows from the fact that the Schrödinger equation is a linear differential equation in time and position. More precisely, the state of a system is given by a linear combination of all the eigenfunctions of the Schrödinger equation governing that system. An example is a qubit used in quantum information processing . A qubit state is most generally a superposition of the basis states and : where is the quantum state of the qubit, and , denote particular solutions to the Schrödinger equation in Dirac notation weighted by the two probability amplitudes and that both are complex numbers. Here corresponds to the classical 0 bit , and to the classical 1 bit. The probabilities of measuring the system in the or state are given by and respectively (see the Born rule ). Before the measurement occurs the qubit is in a superposition of both states. The interference fringes in the double-slit experiment provide another example of the superposition principle. The theory of quantum mechanics postulates that a wave equation completely determines the state of a quantum system at all times. Furthermore, this differential equation is restricted to be linear and homogeneous . These conditions mean that for any two solutions of the wave equation, and , a linear combination of those solutions also solve the wave equation: for arbitrary complex coefficients and . [ 1 ] : 61 If the wave equation has more than two solutions, combinations of all such solutions are again valid solutions. The quantum wave equation can be solved using functions of position, , or using functions of momentum, and consequently the superposition of momentum functions are also solutions: The position and momentum solutions are related by a linear transformation , a Fourier transformation . This transformation is itself a quantum superposition and every position wave function can be represented as a superposition of momentum wave functions and vice versa. These superpositions involve an infinite number of component waves. [ 1 ] : 244 Generalization to basis states [ edit ] Other transformations express a quantum solution as a superposition of eigenvectors , each corresponding to a possible result of a measurement on the quantum system. An eigenvector for a mathematical operator, , has the equation where is one possible measured quantum value for the observable . A superposition of these eigenvectors can represent any solution: The states like are called basis states. Compact notation for superpositions [ edit ] Important mathematical operations on quantum system solutions can be performed using only the coefficients of the superposition, suppressing the details of the superposed functions. This leads to quantum systems expressed in the Dirac bra-ket notation : [ 1 ] : 245 This approach is especially effective for systems like quantum spin with no classical coordinate analog. Such shorthand notation is very common in textbooks and papers on quantum mechanics, and superposition of basis states is a fundamental tool in quantum mechanics. Paul Dirac described the superposition principle as follows: The non-classical nature of the superposition process is brought out clearly if we consider the superposition of two states, A and B , such that there exists an observation which, when made on the system in state A , is certain to lead to one particular result, a say, and when made on the system in state B is certain to lead to some different result, b say. What will be the result of the observation when made on the system in the superposed state? The answer is that the result will be sometimes a and sometimes b , according to a probability law depending on the relative weights of A and B in the superposition process. It will never be different from both a and b [i.e., either a or b ]. The intermediate character of the state formed by superposition thus expresses itself through the probability of a particular result for an observation being intermediate between the corresponding probabilities for the original states, not through the result itself being intermediate between the corresponding results for the original states. [ 2 ] Anton Zeilinger , referring to the prototypical example of the double-slit experiment , has elaborated regarding the creation and destruction of quantum superposition: "[T]he superposition of amplitudes ... is only valid if there is no way to know, even in principle, which path the particle took. It is important to realize that this does not imply that an observer actually takes note of what happens. It is sufficient to destroy the interference pattern, if the path information is accessible in principle from the experiment or even if it is dispersed in the environment and beyond any technical possibility to be recovered, but in principle still ‘‘out there.’’ The absence of any such information is the essential criterion for quantum interference to appear. [ 3 ] Any quantum state can be expanded as a sum or superposition of the eigenstates of an Hermitian operator, like the Hamiltonian, because the eigenstates form a complete basis: where are the energy eigenstates of the Hamiltonian. For continuous variables like position eigenstates, : where is the projection of the state into the basis and is called the wave function of the particle. In both instances we notice that can be expanded as a superposition of an infinite number of basis states. Given the Schrödinger equation where indexes the set of eigenstates of the Hamiltonian with energy eigenvalues we see immediately that where is a solution of the Schrödinger equation but is not generally an eigenstate because and are not generally equal. We say that is made up of a superposition of energy eigenstates. Now consider the more concrete case of an electron that has either spin up or down. We now index the eigenstates with the spinors in the basis: where and denote spin-up and spin-down states respectively. As previously discussed, the magnitudes of the complex coefficients give the probability of finding the electron in either definite spin state: where the probability of finding the particle with either spin up or down is normalized to 1. Notice that and are complex numbers, so that is an example of an allowed state. We now get If we consider a qubit with both position and spin, the state is a superposition of all possibilities for both: where we have a general state is the sum of the tensor products of the position space wave functions and spinors. Successful experiments involving superpositions of relatively large (by the standards of quantum physics) objects have been performed. A beryllium ion has been trapped in a superposed state. [ 4 ] A double slit experiment has been performed with molecules as large as buckyballs and functionalized oligoporphyrins with up to 2000 atoms. [ 5 ] [ 6 ] Molecules with masses exceeding 10,000 and composed of over 810 atoms have successfully been superposed, [ 7 ] and metal clusters with masses over 170,000 Da and containing more than 7,000 atoms have also been demonstrated in quantum superposition. [ 8 ] Very sensitive magnetometers have been realized using superconducting quantum interference devices (SQUIDS) that operate using quantum interference effects in superconducting circuits. A piezoelectric " tuning fork " has been constructed, which can be placed into a superposition of vibrating and non-vibrating states. The resonator comprises about 10 trillion atoms. [ 9 ] Recent research indicates that chlorophyll within plants appears to exploit the feature of quantum superposition to achieve greater efficiency in transporting energy, allowing pigment proteins to be spaced further apart than would otherwise be possible. [ 10 ] [ 11 ] In quantum computers [ edit ] In quantum computers , a qubit is the analog of the classical information bit , but rather than having one of two distinct values, qubits are a superposition of two values. [ 12 ] : 13 Controlling this superposition qubits is a central challenge in quantum computation. The superposition needs to be robust to unintended interactions and yet interactions are needed for computing with qubits. Qubit systems like nuclear spins with small coupling strength are robust to outside disturbances but the same small coupling makes it difficult to readout results. [ 12 ] : 278 Eigenstate – Mathematical entity to describe the probability of each possible measurement on a system Mach–Zehnder interferometer – Device to determine relative phase shift Penrose interpretation – Interpretation of quantum mechanics Pure qubit state – Basic unit of quantum information Quantum computation – Computer hardware technology that uses quantum mechanics Schrödinger's cat – Thought experiment in quantum mechanics Superposition principle – Fundamental principle of physics Wave packet – Short "burst" or "envelope" of restricted wave action that travels as a unit ^ a b c Messiah, Albert (1976). Quantum mechanics. 1 (2 ed.). Amsterdam: North-Holland. ISBN 978-0-471-59766-7 . ^ P.A.M. Dirac (1947). The Principles of Quantum Mechanics (2nd ed.). Clarendon Press. p. 12. ^ Zeilinger A (1999). "Experiment and the foundations of quantum physics". Rev. Mod. Phys . 71 (2): S288– S297. Bibcode : 1999RvMPS..71..288Z . doi : 10.1103/revmodphys.71.s288 . ^ Monroe, C.; Meekhof, D. M.; King, B. E.; Wineland, D. J. (24 May 1996). "A "Schrödinger Cat" Superposition State of an Atom" . Science . 272 (5265): 1131– 1136. Bibcode : 1996Sci...272.1131M . doi : 10.1126/science.272.5265.1131 . ISSN 0036-8075 . PMID 8662445 . ^ "Wave-particle duality of C60" . 31 March 2012. Archived from the original on 31 March 2012. {{ cite web }} : CS1 maint: bot: original URL status unknown ( link ) ^ Yaakov Y. Fein; Philipp Geyer; Patrick Zwick; Filip Kiałka; Sebastian Pedalino; Marcel Mayor; Stefan Gerlich; Markus Arndt (September 2019). "Quantum superposition of molecules beyond 25 kDa". Nature Physics . 15 (12): 1242– 1245. Bibcode : 2019NatPh..15.1242F . doi : 10.1038/s41567-019-0663-9 . S2CID 203638258 . ^ Eibenberger, S., Gerlich, S., Arndt, M., Mayor, M., Tüxen, J. (2013). "Matter-wave interference with particles selected from a molecular library with masses exceeding 10 000 amu", Physical Chemistry Chemical Physics , 15 : 14696-14700 arXiv : 1310.8343 ^ Pedalino, Sebastian; Ramírez-Galindo, Bruno E.; Ferstl, Richard; Hornberger, Klaus; Arndt, Markus; Gerlich, Stefan (22 January 2026). "Probing quantum mechanics with nanoparticle matter-wave interferometry" . Nature . 649 (8098): 866– 870. doi : 10.1038/s41586-025-09917-9 . ISSN 0028-0836 . PMC 12823444 . PMID 41566007 . ^ O’Connell, A. D.; Hofheinz, M.; Ansmann, M.; Bialczak, Radoslaw C.; Lenander, M.; Lucero, Erik; Neeley, M.; Sank, D.; Wang, H.; Weides, M.; Wenner, J.; Martinis, John M.; Cleland, A. N. (April 2010). "Quantum ground state and single-phonon control of a mechanical resonator" . Nature . 464 (7289): 697– 703. Bibcode : 2010Natur.464..697O . doi : 10.1038/nature08967 . ISSN 0028-0836 . PMID 20237473 . ^ Scholes, Gregory; Elisabetta Collini; Cathy Y. Wong; Krystyna E. Wilk; Paul M. G. Curmi; Paul Brumer; Gregory D. Scholes (4 February 2010). "Coherently wired light-harvesting in photosynthetic marine algae at ambient temperature". Nature . 463 (7281): 644– 647. Bibcode : 2010Natur.463..644C . doi : 10.1038/nature08811 . PMID 20130647 . S2CID 4369439 . ^ Moyer, Michael (September 2009). "Quantum Entanglement, Photosynthesis and Better Solar Cells" . Scientific American . Retrieved 12 May 2010 . ^ a b Nielsen, Michael A. ; Chuang, Isaac (2010). Quantum Computation and Quantum Information . Cambridge: Cambridge University Press . ISBN 978-1-10700-217-3 . OCLC 43641333 . Bohr, N. (1927/1928). The quantum postulate and the recent development of atomic theory, Nature Supplement 14 April 1928, 121 : 580–590 . Cohen-Tannoudji, C. , Diu, B., Laloë, F. (1973/1977). Quantum Mechanics , translated from the French by S. R. Hemley, N. Ostrowsky, D. Ostrowsky, second edition, volume 1, Wiley, New York, ISBN 0471164321 . Einstein, A. (1949). Remarks concerning the essays brought together in this co-operative volume, translated from the original German by the editor, pp. 665–688 in Schilpp, P. A. editor (1949), Albert Einstein: Philosopher-Scientist , volume II , Open Court, La Salle IL. Feynman, R. P. , Leighton, R.B., Sands, M. (1965). The Feynman Lectures on Physics , volume 3 , Addison-Wesley, Reading, MA. Merzbacher, E. (1961/1970). Quantum Mechanics , second edition, Wiley, New York. Messiah, A. (1961). Quantum Mechanics , volume 1, translated by G.M. Temmer from the French Mécanique Quantique , North-Holland, Amsterdam. Wheeler, J. A. ; Zurek, W.H. (1983). Quantum Theory and Measurement . Princeton NJ: Princeton University Press. Nielsen, Michael A. ; Chuang, Isaac (2000). Quantum Computation and Quantum Information . Cambridge: Cambridge University Press . ISBN 0521632358 . OCLC 43641333 . Williams, Colin P. (2011). Explorations in Quantum Computing . Springer . ISBN 978-1-84628-887-6 . Yanofsky, Noson S.; Mannucci, Mirco (2013). Quantum computing for computer scientists . Cambridge University Press . ISBN 978-0-521-87996-5 .
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[o]") ## Contents move to sidebar hide - [(Top)](https://en.wikipedia.org/wiki/Quantum_superposition) - [1 Wave postulate](https://en.wikipedia.org/wiki/Quantum_superposition#Wave_postulate) - [2 Transformation](https://en.wikipedia.org/wiki/Quantum_superposition#Transformation) - [3 Generalization to basis states](https://en.wikipedia.org/wiki/Quantum_superposition#Generalization_to_basis_states) - [4 Compact notation for superpositions](https://en.wikipedia.org/wiki/Quantum_superposition#Compact_notation_for_superpositions) - [5 Consequences](https://en.wikipedia.org/wiki/Quantum_superposition#Consequences) - [6 Theory](https://en.wikipedia.org/wiki/Quantum_superposition#Theory) Toggle Theory subsection - [6\.1 General formalism](https://en.wikipedia.org/wiki/Quantum_superposition#General_formalism) - [6\.2 Example](https://en.wikipedia.org/wiki/Quantum_superposition#Example) - [7 Experiments](https://en.wikipedia.org/wiki/Quantum_superposition#Experiments) - [8 In quantum computers](https://en.wikipedia.org/wiki/Quantum_superposition#In_quantum_computers) - [9 See also](https://en.wikipedia.org/wiki/Quantum_superposition#See_also) - [10 References](https://en.wikipedia.org/wiki/Quantum_superposition#References) - [11 Further reading](https://en.wikipedia.org/wiki/Quantum_superposition#Further_reading) Toggle the table of contents # Quantum superposition 40 languages - [العربية](https://ar.wikipedia.org/wiki/%D8%AA%D8%B1%D8%A7%D9%83%D8%A8_%D9%83%D9%85%D9%8A "تراكب كمي – Arabic") - [Български](https://bg.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D0%BE%D0%B2%D0%B0_%D1%81%D1%83%D0%BF%D0%B5%D1%80%D0%BF%D0%BE%D0%B7%D0%B8%D1%86%D0%B8%D1%8F "Квантова суперпозиция – Bulgarian") - [Català](https://ca.wikipedia.org/wiki/Superposici%C3%B3_qu%C3%A0ntica "Superposició quàntica – Catalan") - [Čeština](https://cs.wikipedia.org/wiki/Kvantov%C3%A1_superpozice "Kvantová superpozice – Czech") - [Чӑвашла](https://cv.wikipedia.org/wiki/%D0%A1%D1%83%D0%BF%D0%B5%D1%80%D0%BF%D0%BE%D0%B7%D0%B8%D1%86%D0%B8_%D0%BF%D1%80%D0%B8%D0%BD%D1%86%D0%B8%D0%BF%C4%95_\(%D0%BA%D0%B2%D0%B0%D0%BD%D1%82%D0%BB%D0%B0_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0\) "Суперпозици принципĕ (квантла механика) – Chuvash") - [Dansk](https://da.wikipedia.org/wiki/Kvantemekanisk_superposition "Kvantemekanisk superposition – Danish") - [Deutsch](https://de.wikipedia.org/wiki/Superposition_\(Quantenmechanik\) "Superposition (Quantenmechanik) – German") - [Ελληνικά](https://el.wikipedia.org/wiki/%CE%9A%CE%B2%CE%B1%CE%BD%CF%84%CE%B9%CE%BA%CE%AE_%CF%85%CF%80%CE%AD%CF%81%CE%B8%CE%B5%CF%83%CE%B7 "Κβαντική υπέρθεση – Greek") - [Español](https://es.wikipedia.org/wiki/Superposici%C3%B3n_cu%C3%A1ntica "Superposición cuántica – Spanish") - [Eesti](https://et.wikipedia.org/wiki/Superpositsioon "Superpositsioon – Estonian") - [فارسی](https://fa.wikipedia.org/wiki/%D8%A8%D8%B1%D9%87%D9%85%E2%80%8C%D9%86%D9%87%DB%8C_%DA%A9%D9%88%D8%A7%D9%86%D8%AA%D9%88%D9%85%DB%8C "برهم‌نهی کوانتومی – Persian") - [Suomi](https://fi.wikipedia.org/wiki/Kvanttisuperpositio "Kvanttisuperpositio – Finnish") - [Français](https://fr.wikipedia.org/wiki/Principe_de_superposition_quantique "Principe de superposition quantique – French") - [עברית](https://he.wikipedia.org/wiki/%D7%A1%D7%95%D7%A4%D7%A8%D7%A4%D7%95%D7%96%D7%99%D7%A6%D7%99%D7%94_%D7%A7%D7%95%D7%95%D7%A0%D7%98%D7%99%D7%AA "סופרפוזיציה קוונטית – Hebrew") - 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[한국어](https://ko.wikipedia.org/wiki/%EC%96%91%EC%9E%90_%EC%A4%91%EC%B2%A9 "양자 중첩 – Korean") - [Lietuvių](https://lt.wikipedia.org/wiki/Kvantin%C4%97_superpozicija "Kvantinė superpozicija – Lithuanian") - [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%B5%E0%B4%BF%E0%B4%B6%E0%B4%BF%E0%B4%B7%E0%B5%8D%E0%B4%9F%E0%B4%B8%E0%B5%8D%E0%B4%A5%E0%B4%BF%E0%B4%A4%E0%B4%BF "ക്വാണ്ടം വിശിഷ്ടസ്ഥിതി – Malayalam") - [Occitan](https://oc.wikipedia.org/wiki/Estats_superpausats_\(quantics\) "Estats superpausats (quantics) – 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%B8%E0%A9%81%E0%A8%AA%E0%A8%B0%E0%A8%AA%E0%A9%81%E0%A8%9C%E0%A9%80%E0%A8%B8%E0%A8%BC%E0%A8%A8 "ਕੁਆਂਟਮ ਸੁਪਰਪੁਜੀਸ਼ਨ – Punjabi") - [Português](https://pt.wikipedia.org/wiki/Sobreposi%C3%A7%C3%A3o_qu%C3%A2ntica "Sobreposição quântica – Portuguese") - [Русский](https://ru.wikipedia.org/wiki/%D0%9F%D1%80%D0%B8%D0%BD%D1%86%D0%B8%D0%BF_%D1%81%D1%83%D0%BF%D0%B5%D1%80%D0%BF%D0%BE%D0%B7%D0%B8%D1%86%D0%B8%D0%B8_\(%D0%BA%D0%B2%D0%B0%D0%BD%D1%82%D0%BE%D0%B2%D0%B0%D1%8F_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0\) "Принцип суперпозиции (квантовая механика) – Russian") - [Simple English](https://simple.wikipedia.org/wiki/Quantum_superposition "Quantum superposition – Simple English") - [Slovenščina](https://sl.wikipedia.org/wiki/Kvantna_superpozicija "Kvantna superpozicija – Slovenian") - [Shqip](https://sq.wikipedia.org/wiki/Mbivendosja_kuantike "Mbivendosja kuantike – Albanian") - [Српски / srpski](https://sr.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D0%BD%D0%B0_%D1%81%D1%83%D0%BF%D0%B5%D1%80%D0%BF%D0%BE%D0%B7%D0%B8%D1%86%D0%B8%D1%98%D0%B0 "Квантна суперпозиција – Serbian") - [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%A8%E0%AF%87%E0%AE%B0%E0%AE%9F%E0%AF%81%E0%AE%95%E0%AF%8D%E0%AE%95%E0%AF%81%E0%AE%AE%E0%AF%88 "குவாண்டம் நேரடுக்குமை – Tamil") - [ไทย](https://th.wikipedia.org/wiki/%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B8%8B%E0%B9%89%E0%B8%AD%E0%B8%99%E0%B8%97%E0%B8%B1%E0%B8%9A%E0%B8%84%E0%B8%A7%E0%B8%AD%E0%B8%99%E0%B8%95%E0%B8%B1%E0%B8%A1 "การซ้อนทับควอนตัม – Thai") - [Tagalog](https://tl.wikipedia.org/wiki/Superposisyong_quantum "Superposisyong quantum – Tagalog") - [Татарча / tatarça](https://tt.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82_%D0%BA%D1%83%D1%88%D1%83%D1%8B "Квант кушуы – Tatar") - [Українська](https://uk.wikipedia.org/wiki/%D0%9F%D1%80%D0%B8%D0%BD%D1%86%D0%B8%D0%BF_%D1%81%D1%83%D0%BF%D0%B5%D1%80%D0%BF%D0%BE%D0%B7%D0%B8%D1%86%D1%96%D1%97_\(%D0%BA%D0%B2%D0%B0%D0%BD%D1%82%D0%BE%D0%B2%D0%B0_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D1%96%D0%BA%D0%B0\) "Принцип суперпозиції (квантова механіка) – Ukrainian") - [Tiếng Việt](https://vi.wikipedia.org/wiki/Ch%E1%BB%93ng_ch%E1%BA%ADp_l%C6%B0%E1%BB%A3ng_t%E1%BB%AD "Chồng chập lượng tử – Vietnamese") - [閩南語 / Bân-lâm-gí](https://zh-min-nan.wikipedia.org/wiki/Li%C5%8Dng-ch%C3%AD_tia%CC%8Dp-ka "Liōng-chí tia̍p-ka – Minnan") - [中文](https://zh.wikipedia.org/wiki/%E6%80%81%E5%8F%A0%E5%8A%A0%E5%8E%9F%E7%90%86 "态叠加原理 – Chinese") [Edit links](https://www.wikidata.org/wiki/Special:EntityPage/Q830791#sitelinks-wikipedia "Edit interlanguage links") - [Article](https://en.wikipedia.org/wiki/Quantum_superposition "View the content page [c]") - [Talk](https://en.wikipedia.org/wiki/Talk:Quantum_superposition "Discuss improvements to the content page [t]") English - [Read](https://en.wikipedia.org/wiki/Quantum_superposition) - [Edit](https://en.wikipedia.org/w/index.php?title=Quantum_superposition&action=edit "Edit this page [e]") - [View history](https://en.wikipedia.org/w/index.php?title=Quantum_superposition&action=history "Past revisions of this page [h]") Tools Tools move to sidebar hide Actions - [Read](https://en.wikipedia.org/wiki/Quantum_superposition) - [Edit](https://en.wikipedia.org/w/index.php?title=Quantum_superposition&action=edit "Edit this page [e]") - [View history](https://en.wikipedia.org/w/index.php?title=Quantum_superposition&action=history) General - [What links here](https://en.wikipedia.org/wiki/Special:WhatLinksHere/Quantum_superposition "List of all English Wikipedia pages containing links to this page [j]") - 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[Printable version](https://en.wikipedia.org/w/index.php?title=Quantum_superposition&printable=yes "Printable version of this page [p]") In other projects - [Wikidata item](https://www.wikidata.org/wiki/Special:EntityPage/Q830791 "Structured data on this page hosted by Wikidata [g]") Appearance move to sidebar hide From Wikipedia, the free encyclopedia Principle of quantum mechanics For broader coverage of this topic, see [Superposition principle](https://en.wikipedia.org/wiki/Superposition_principle "Superposition principle"). Quantum superposition of states and decoherence \| \| \|---\| \| Part of a series of articles about \| \| [Quantum mechanics](https://en.wikipedia.org/wiki/Quantum_mechanics "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 superposition is a fundamental principle of [quantum mechanics](https://en.wikipedia.org/wiki/Quantum_mechanics "Quantum mechanics") that states that [linear combinations](https://en.wikipedia.org/wiki/Linear_combination "Linear combination") of [solutions](https://en.wikipedia.org/wiki/Solution_\(mathematics\) "Solution (mathematics)") to the [Schrödinger equation](https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation "Schrödinger equation") are also solutions of the Schrödinger equation. This follows from the fact that the Schrödinger equation is a [linear differential equation](https://en.wikipedia.org/wiki/Linear_differential_equation "Linear differential equation") in time and position. More precisely, the state of a system is given by a [linear combination](https://en.wikipedia.org/wiki/Linear_combination "Linear combination") of all the [eigenfunctions](https://en.wikipedia.org/wiki/Eigenfunction "Eigenfunction") of the Schrödinger equation governing that system. An example is a [qubit](https://en.wikipedia.org/wiki/Qubit "Qubit") used in [quantum information processing](https://en.wikipedia.org/wiki/Quantum_information_processing "Quantum information processing"). A qubit state is most generally a superposition of the basis states \\| 0 ⟩ {\\displaystyle \\|0\\rangle } ![{\\displaystyle \\|0\\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/ed066a3ad158da0ad6d6a421a606b1c8a35eb95b) and \\| 1 ⟩ {\\displaystyle \\|1\\rangle } ![{\\displaystyle \\|1\\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f53021ca18e77477ee5bd3c1523e5830189ec5c): \\| Ψ ⟩ \= c 0 \\| 0 ⟩ \+ c 1 \\| 1 ⟩ , {\\displaystyle \\|\\Psi \\rangle =c\_{0}\\|0\\rangle +c\_{1}\\|1\\rangle ,} ![{\\displaystyle \\|\\Psi \\rangle =c\_{0}\\|0\\rangle +c\_{1}\\|1\\rangle ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f10213c27e86b15bff8015bd81b1d5983d14c5a4) where \\| Ψ ⟩ {\\displaystyle \\|\\Psi \\rangle } ![{\\displaystyle \\|\\Psi \\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e77f6b1e903837c5765c9683da41dd93199621c) is the [quantum state](https://en.wikipedia.org/wiki/Quantum_state "Quantum state") of the qubit, and \\| 0 ⟩ {\\displaystyle \\|0\\rangle } ![{\\displaystyle \\|0\\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/ed066a3ad158da0ad6d6a421a606b1c8a35eb95b), \\| 1 ⟩ {\\displaystyle \\|1\\rangle } ![{\\displaystyle \\|1\\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f53021ca18e77477ee5bd3c1523e5830189ec5c) denote particular solutions to the Schrödinger equation in [Dirac notation](https://en.wikipedia.org/wiki/Bra%E2%80%93ket_notation "Bra–ket notation") weighted by the two [probability amplitudes](https://en.wikipedia.org/wiki/Probability_amplitude "Probability amplitude") c 0 {\\displaystyle c\_{0}} ![{\\displaystyle c\_{0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1882ba8f1dc60f0c68a642abb5af093c73910921) and c 1 {\\displaystyle c\_{1}} ![{\\displaystyle c\_{1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77b7dc6d279091d354e0b90889b463bfa7eb7247) that both are complex numbers. Here \\| 0 ⟩ {\\displaystyle \\|0\\rangle } ![{\\displaystyle \\|0\\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/ed066a3ad158da0ad6d6a421a606b1c8a35eb95b) corresponds to the classical 0 [bit](https://en.wikipedia.org/wiki/Bit "Bit"), and \\| 1 ⟩ {\\displaystyle \\|1\\rangle } ![{\\displaystyle \\|1\\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f53021ca18e77477ee5bd3c1523e5830189ec5c) to the classical 1 bit. The probabilities of measuring the system in the \\| 0 ⟩ {\\displaystyle \\|0\\rangle } ![{\\displaystyle \\|0\\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/ed066a3ad158da0ad6d6a421a606b1c8a35eb95b) or \\| 1 ⟩ {\\displaystyle \\|1\\rangle } ![{\\displaystyle \\|1\\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f53021ca18e77477ee5bd3c1523e5830189ec5c) state are given by \\| c 0 \\| 2 {\\displaystyle \\|c\_{0}\\|^{2}} ![{\\displaystyle \\|c\_{0}\\|^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8529f071ccb861751297a04659e8dbddca09bad8) and \\| c 1 \\| 2 {\\displaystyle \\|c\_{1}\\|^{2}} ![{\\displaystyle \\|c\_{1}\\|^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a9c67c5c8bf07a77edd704f880a64a35b8d9f1c5) respectively (see the [Born rule](https://en.wikipedia.org/wiki/Born_rule "Born rule")). Before the measurement occurs the qubit is in a superposition of both states. The interference fringes in the [double-slit experiment](https://en.wikipedia.org/wiki/Double-slit_experiment "Double-slit experiment") provide another example of the superposition principle. ## Wave postulate \[[edit](https://en.wikipedia.org/w/index.php?title=Quantum_superposition&action=edit&section=1 "Edit section: Wave postulate")\] The theory of quantum mechanics postulates that a [wave equation](https://en.wikipedia.org/wiki/Wave_equation "Wave equation") completely determines the state of a quantum system at all times. Furthermore, this differential equation is restricted to be [linear](https://en.wikipedia.org/wiki/Linear_differential_equation "Linear differential equation") and [homogeneous](https://en.wikipedia.org/wiki/Homogeneous_differential_equation "Homogeneous differential equation"). These conditions mean that for any two solutions of the wave equation, Ψ 1 {\\displaystyle \\Psi \_{1}} ![{\\displaystyle \\Psi \_{1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ddcd58cf9c1c4dd1eb32a29c1e3abf425ec72600) and Ψ 2 {\\displaystyle \\Psi \_{2}} ![{\\displaystyle \\Psi \_{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1125553c0a026635c63ab2ad542b43174aca9bbe), a linear combination of those solutions also solve the wave equation: Ψ \= c 1 Ψ 1 \+ c 2 Ψ 2 {\\displaystyle \\Psi =c\_{1}\\Psi \_{1}+c\_{2}\\Psi \_{2}} ![{\\displaystyle \\Psi =c\_{1}\\Psi \_{1}+c\_{2}\\Psi \_{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6823dfb8e3163bbbfd61bd2d925ce8d6fdac949) for arbitrary complex coefficients c 1 {\\displaystyle c\_{1}} ![{\\displaystyle c\_{1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77b7dc6d279091d354e0b90889b463bfa7eb7247) and c 2 {\\displaystyle c\_{2}} ![{\\displaystyle c\_{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b30ba1b247fb8d334580cec68561e749d24aff2).[\[1\]](https://en.wikipedia.org/wiki/Quantum_superposition#cite_note-Messiah-1): 61 If the wave equation has more than two solutions, combinations of all such solutions are again valid solutions. ## Transformation \[[edit](https://en.wikipedia.org/w/index.php?title=Quantum_superposition&action=edit&section=2 "Edit section: Transformation")\] The quantum wave equation can be solved using functions of position, Ψ ( r → ) {\\displaystyle \\Psi ({\\vec {r}})} ![{\\displaystyle \\Psi ({\\vec {r}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/860ed8dcca03bb55dda0fe92c92a079038be797b), or using functions of momentum, Φ ( p → ) {\\displaystyle \\Phi ({\\vec {p}})} ![{\\displaystyle \\Phi ({\\vec {p}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/96584b5a2ed608930bfa349e822c564a5298250d) and consequently the superposition of momentum functions are also solutions: Φ ( p → ) \= d 1 Φ 1 ( p → ) \+ d 2 Φ 2 ( p → ) {\\displaystyle \\Phi ({\\vec {p}})=d\_{1}\\Phi \_{1}({\\vec {p}})+d\_{2}\\Phi \_{2}({\\vec {p}})} ![{\\displaystyle \\Phi ({\\vec {p}})=d\_{1}\\Phi \_{1}({\\vec {p}})+d\_{2}\\Phi \_{2}({\\vec {p}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f72ca07aa31123cf18829faed1ebb4abeb3a5120) The position and momentum solutions are related by a [linear transformation](https://en.wikipedia.org/wiki/Linear_transformation "Linear transformation"), a [Fourier transformation](https://en.wikipedia.org/wiki/Fourier_transformation "Fourier transformation"). This transformation is itself a quantum superposition and every position wave function can be represented as a superposition of momentum wave functions and vice versa. These superpositions involve an infinite number of component waves.[\[1\]](https://en.wikipedia.org/wiki/Quantum_superposition#cite_note-Messiah-1): 244 ## Generalization to basis states \[[edit](https://en.wikipedia.org/w/index.php?title=Quantum_superposition&action=edit&section=3 "Edit section: Generalization to basis states")\] Other transformations express a quantum solution as a superposition of [eigenvectors](https://en.wikipedia.org/wiki/Eigenvectors "Eigenvectors"), each corresponding to a possible result of a measurement on the quantum system. An eigenvector ψ i {\\displaystyle \\psi \_{i}} ![{\\displaystyle \\psi \_{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e709a809cfd3cfbb647fe22c8d1f81573f4c7ae) for a mathematical operator, A ^ {\\displaystyle {\\hat {A}}} ![{\\displaystyle {\\hat {A}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f595a6c73d1183d6a1b2ac21fe47ac28c1483821), has the equation A ^ ψ i \= λ i ψ i {\\displaystyle {\\hat {A}}\\psi \_{i}=\\lambda \_{i}\\psi \_{i}} ![{\\displaystyle {\\hat {A}}\\psi \_{i}=\\lambda \_{i}\\psi \_{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f7c7cfad33ac61dd3bcde805172bb9897ac0dfe) where λ i {\\displaystyle \\lambda \_{i}} ![{\\displaystyle \\lambda \_{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/72fde940918edf84caf3d406cc7d31949166820f) is one possible measured quantum value for the observable A {\\displaystyle A} ![{\\displaystyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3). A superposition of these eigenvectors can represent any solution: Ψ \= ∑ n a i ψ i . {\\displaystyle \\Psi =\\sum \_{n}a\_{i}\\psi \_{i}.} ![{\\displaystyle \\Psi =\\sum \_{n}a\_{i}\\psi \_{i}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/da5aeb99c104b07a8842ea74cc61a45a859aa2a3) The states like ψ i {\\displaystyle \\psi \_{i}} ![{\\displaystyle \\psi \_{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e709a809cfd3cfbb647fe22c8d1f81573f4c7ae) are called basis states. ## Compact notation for superpositions \[[edit](https://en.wikipedia.org/w/index.php?title=Quantum_superposition&action=edit&section=4 "Edit section: Compact notation for superpositions")\] Important mathematical operations on quantum system solutions can be performed using only the coefficients of the superposition, suppressing the details of the superposed functions. This leads to quantum systems expressed in the [Dirac bra-ket notation](https://en.wikipedia.org/wiki/Bra-ket_notation "Bra-ket notation"):[\[1\]](https://en.wikipedia.org/wiki/Quantum_superposition#cite_note-Messiah-1): 245 \\| v ⟩ \= d 1 \\| 1 ⟩ \+ d 2 \\| 2 ⟩ {\\displaystyle \\|v\\rangle =d\_{1}\\|1\\rangle +d\_{2}\\|2\\rangle } ![{\\displaystyle \\|v\\rangle =d\_{1}\\|1\\rangle +d\_{2}\\|2\\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/acc7ac146085e844fd0f2988bed9f23c30af497a) This approach is especially effective for systems like quantum spin with no classical coordinate analog. Such shorthand notation is very common in textbooks and papers on quantum mechanics, and superposition of basis states is a fundamental tool in quantum mechanics. ## Consequences \[[edit](https://en.wikipedia.org/w/index.php?title=Quantum_superposition&action=edit&section=5 "Edit section: Consequences")\] [Paul Dirac](https://en.wikipedia.org/wiki/Paul_Dirac "Paul Dirac") described the superposition principle as follows: > The non-classical nature of the superposition process is brought out clearly if we consider the superposition of two states, A and B, such that there exists an observation which, when made on the system in state A, is certain to lead to one particular result, a say, and when made on the system in state B is certain to lead to some different result, b say. What will be the result of the observation when made on the system in the superposed state? The answer is that the result will be sometimes a and sometimes b, according to a probability law depending on the relative weights of A and B in the superposition process. It will never be different from both a and b \[i.e., either a or b\]. The intermediate character of the state formed by superposition thus expresses itself through the probability of a particular result for an observation being intermediate between the corresponding probabilities for the original states, not through the result itself being intermediate between the corresponding results for the original states.[\[2\]](https://en.wikipedia.org/wiki/Quantum_superposition#cite_note-Dirac1947-2) [Anton Zeilinger](https://en.wikipedia.org/wiki/Anton_Zeilinger "Anton Zeilinger"), referring to the prototypical example of the [double-slit experiment](https://en.wikipedia.org/wiki/Double-slit_experiment "Double-slit experiment"), has elaborated regarding the creation and destruction of quantum superposition: > "\[T\]he superposition of amplitudes ... is only valid if there is no way to know, even in principle, which path the particle took. It is important to realize that this does not imply that an observer actually takes note of what happens. It is sufficient to destroy the interference pattern, if the path information is accessible in principle from the experiment or even if it is dispersed in the environment and beyond any technical possibility to be recovered, but in principle still ‘‘out there.’’ The absence of any such information is the essential criterion for quantum interference to appear.[\[3\]](https://en.wikipedia.org/wiki/Quantum_superposition#cite_note-Zeilinger-3) ## Theory \[[edit](https://en.wikipedia.org/w/index.php?title=Quantum_superposition&action=edit&section=6 "Edit section: Theory")\] ### General formalism \[[edit](https://en.wikipedia.org/w/index.php?title=Quantum_superposition&action=edit&section=7 "Edit section: General formalism")\] Any quantum state can be expanded as a sum or superposition of the eigenstates of an Hermitian operator, like the Hamiltonian, because the eigenstates form a complete basis: \\| α ⟩ \= ∑ n c n \\| n ⟩ , {\\displaystyle \\|\\alpha \\rangle =\\sum \_{n}c\_{n}\\|n\\rangle ,} ![{\\displaystyle \\|\\alpha \\rangle =\\sum \_{n}c\_{n}\\|n\\rangle ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/df393c7be36c417cc59f5a1a605aea04beef2e7b) where \\| n ⟩ {\\displaystyle \\|n\\rangle } ![{\\displaystyle \\|n\\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/89c45bda6ac852da785c06476f2d3e3b6cba812a) are the energy eigenstates of the Hamiltonian. For continuous variables like position eigenstates, \\| x ⟩ {\\displaystyle \\|x\\rangle } ![{\\displaystyle \\|x\\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/48004887d8f9dfc489bd2bc793780b7f1d8039ad): \\| α ⟩ \= ∫ d x ′ \\| x ′ ⟩ ⟨ x ′ \\| α ⟩ , {\\displaystyle \\|\\alpha \\rangle =\\int dx'\\|x'\\rangle \\langle x'\\|\\alpha \\rangle ,} ![{\\displaystyle \\|\\alpha \\rangle =\\int dx'\\|x'\\rangle \\langle x'\\|\\alpha \\rangle ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9495e664ba18698edce2bfcc82c345de5ceaf6c6) where ϕ α ( x ) \= ⟨ x \\| α ⟩ {\\displaystyle \\phi \_{\\alpha }(x)=\\langle x\\|\\alpha \\rangle } ![{\\displaystyle \\phi \_{\\alpha }(x)=\\langle x\\|\\alpha \\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/2ef30d5158d3fe8417e7fe37540e5370e46d9887) is the projection of the state into the \\| x ⟩ {\\displaystyle \\|x\\rangle } ![{\\displaystyle \\|x\\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/48004887d8f9dfc489bd2bc793780b7f1d8039ad) basis and is called the wave function of the particle. In both instances we notice that \\| α ⟩ {\\displaystyle \\|\\alpha \\rangle } ![{\\displaystyle \\|\\alpha \\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f42032e642ee1c9d27adb318d34c7cc85f7a95d5) can be expanded as a superposition of an infinite number of basis states. ### Example \[[edit](https://en.wikipedia.org/w/index.php?title=Quantum_superposition&action=edit&section=8 "Edit section: Example")\] Given the Schrödinger equation H ^ \\| n ⟩ \= E n \\| n ⟩ , {\\displaystyle {\\hat {H}}\\|n\\rangle =E\_{n}\\|n\\rangle ,} ![{\\displaystyle {\\hat {H}}\\|n\\rangle =E\_{n}\\|n\\rangle ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a6afb6d354132f7d244ee60cf28e4608aa6e922) where \\| n ⟩ {\\displaystyle \\|n\\rangle } ![{\\displaystyle \\|n\\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/89c45bda6ac852da785c06476f2d3e3b6cba812a) indexes the set of eigenstates of the Hamiltonian with energy eigenvalues E n , {\\displaystyle E\_{n},} ![{\\displaystyle E\_{n},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/578d4bb07de7b29269be95abb6a214848c09aea7) we see immediately that H ^ ( \\| n ⟩ \+ \\| n ′ ⟩ ) \= E n \\| n ⟩ \+ E n ′ \\| n ′ ⟩ , {\\displaystyle {\\hat {H}}{\\big (}\\|n\\rangle +\\|n'\\rangle {\\big )}=E\_{n}\\|n\\rangle +E\_{n'}\\|n'\\rangle ,} ![{\\displaystyle {\\hat {H}}{\\big (}\\|n\\rangle +\\|n'\\rangle {\\big )}=E\_{n}\\|n\\rangle +E\_{n'}\\|n'\\rangle ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/806bff6ca40452045baac80fd266eb3c453e7340) where \\| Ψ ⟩ \= \\| n ⟩ \+ \\| n ′ ⟩ {\\displaystyle \\|\\Psi \\rangle =\\|n\\rangle +\\|n'\\rangle } ![{\\displaystyle \\|\\Psi \\rangle =\\|n\\rangle +\\|n'\\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/99dd060b4c8a46870132855acbe5a64eefdf0ea3) is a solution of the Schrödinger equation but is not generally an eigenstate because E n {\\displaystyle E\_{n}} ![{\\displaystyle E\_{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ad6b82f2a00af6c9efd4c16d4e99329605645c0c) and E n ′ {\\displaystyle E\_{n'}} ![{\\displaystyle E\_{n'}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d618f0a54e5627c8baa3cfb77b3e2dfac8f31bc) are not generally equal. We say that \\| Ψ ⟩ {\\displaystyle \\|\\Psi \\rangle } ![{\\displaystyle \\|\\Psi \\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e77f6b1e903837c5765c9683da41dd93199621c) is made up of a superposition of energy eigenstates. Now consider the more concrete case of an [electron](https://en.wikipedia.org/wiki/Electron "Electron") that has either [spin](https://en.wikipedia.org/wiki/Spin_\(physics\) "Spin (physics)") up or down. We now index the eigenstates with the [spinors](https://en.wikipedia.org/wiki/Spinor "Spinor") in the z ^ {\\displaystyle {\\hat {z}}} ![{\\displaystyle {\\hat {z}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/722665b45e05afe79f4395a3de0237d8ce856273) basis: \\| Ψ ⟩ \= c 1 \\| ↑ ⟩ \+ c 2 \\| ↓ ⟩ , {\\displaystyle \\|\\Psi \\rangle =c\_{1}\\|{\\uparrow }\\rangle +c\_{2}\\|{\\downarrow }\\rangle ,} ![{\\displaystyle \\|\\Psi \\rangle =c\_{1}\\|{\\uparrow }\\rangle +c\_{2}\\|{\\downarrow }\\rangle ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/42f17d6ba89011bea1b2f5f2747b88a7482c681e) where \\| ↑ ⟩ {\\displaystyle \\|{\\uparrow }\\rangle } ![{\\displaystyle \\|{\\uparrow }\\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e81fba5ac4e54a40f5b9d63e66548328b7ffb6e) and \\| ↓ ⟩ {\\displaystyle \\|{\\downarrow }\\rangle } ![{\\displaystyle \\|{\\downarrow }\\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/622f3f29fa4b88926ffa22e3da1daa5f2893d17c) denote spin-up and spin-down states respectively. As previously discussed, the magnitudes of the complex coefficients give the probability of finding the electron in either definite spin state: P ( \\| ↑ ⟩ ) \= \\| c 1 \\| 2 , {\\displaystyle P{\\big (}\\|{\\uparrow }\\rangle {\\big )}=\\|c\_{1}\\|^{2},} ![{\\displaystyle P{\\big (}\\|{\\uparrow }\\rangle {\\big )}=\\|c\_{1}\\|^{2},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e31d0a1610fddb7d86d8548a0ae43a9dff80529) P ( \\| ↓ ⟩ ) \= \\| c 2 \\| 2 , {\\displaystyle P{\\big (}\\|{\\downarrow }\\rangle {\\big )}=\\|c\_{2}\\|^{2},} ![{\\displaystyle P{\\big (}\\|{\\downarrow }\\rangle {\\big )}=\\|c\_{2}\\|^{2},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ad7f70c0f22f0405876a0e5c22cfdf96bcf05a9) P total \= P ( \\| ↑ ⟩ ) \+ P ( \\| ↓ ⟩ ) \= \\| c 1 \\| 2 \+ \\| c 2 \\| 2 \= 1 , {\\displaystyle P\_{\\text{total}}=P{\\big (}\\|{\\uparrow }\\rangle {\\big )}+P{\\big (}\\|{\\downarrow }\\rangle {\\big )}=\\|c\_{1}\\|^{2}+\\|c\_{2}\\|^{2}=1,} ![{\\displaystyle P\_{\\text{total}}=P{\\big (}\\|{\\uparrow }\\rangle {\\big )}+P{\\big (}\\|{\\downarrow }\\rangle {\\big )}=\\|c\_{1}\\|^{2}+\\|c\_{2}\\|^{2}=1,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5fd7a45939635ea1e22247c9c7293c938999cefe) where the probability of finding the particle with either spin up or down is normalized to 1. Notice that c 1 {\\displaystyle c\_{1}} ![{\\displaystyle c\_{1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77b7dc6d279091d354e0b90889b463bfa7eb7247) and c 2 {\\displaystyle c\_{2}} ![{\\displaystyle c\_{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b30ba1b247fb8d334580cec68561e749d24aff2) are complex numbers, so that \\| Ψ ⟩ \= 3 5 i \\| ↑ ⟩ \+ 4 5 \\| ↓ ⟩ . {\\displaystyle \\|\\Psi \\rangle ={\\frac {3}{5}}i\\|{\\uparrow }\\rangle +{\\frac {4}{5}}\\|{\\downarrow }\\rangle .} ![{\\displaystyle \\|\\Psi \\rangle ={\\frac {3}{5}}i\\|{\\uparrow }\\rangle +{\\frac {4}{5}}\\|{\\downarrow }\\rangle .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1751456a557a43b4c870d3dd5351854eb5556233) is an example of an allowed state. We now get P ( \\| ↑ ⟩ ) \= \\| 3 i 5 \\| 2 \= 9 25 , {\\displaystyle P{\\big (}\\|{\\uparrow }\\rangle {\\big )}=\\left\\|{\\frac {3i}{5}}\\right\\|^{2}={\\frac {9}{25}},} ![{\\displaystyle P{\\big (}\\|{\\uparrow }\\rangle {\\big )}=\\left\\|{\\frac {3i}{5}}\\right\\|^{2}={\\frac {9}{25}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eeca39a54f3fc393e54a5fdc7feeb5aeb0b527ac) P ( \\| ↓ ⟩ ) \= \\| 4 5 \\| 2 \= 16 25 , {\\displaystyle P{\\big (}\\|{\\downarrow }\\rangle {\\big )}=\\left\\|{\\frac {4}{5}}\\right\\|^{2}={\\frac {16}{25}},} ![{\\displaystyle P{\\big (}\\|{\\downarrow }\\rangle {\\big )}=\\left\\|{\\frac {4}{5}}\\right\\|^{2}={\\frac {16}{25}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/863ea6a63d777554353bd1f448c5d4d989136162) P total \= P ( \\| ↑ ⟩ ) \+ P ( \\| ↓ ⟩ ) \= 9 25 \+ 16 25 \= 1\. {\\displaystyle P\_{\\text{total}}=P{\\big (}\\|{\\uparrow }\\rangle {\\big )}+P{\\big (}\\|{\\downarrow }\\rangle {\\big )}={\\frac {9}{25}}+{\\frac {16}{25}}=1.} ![{\\displaystyle P\_{\\text{total}}=P{\\big (}\\|{\\uparrow }\\rangle {\\big )}+P{\\big (}\\|{\\downarrow }\\rangle {\\big )}={\\frac {9}{25}}+{\\frac {16}{25}}=1.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/845feda8d1aa925d760f2a5e4b88e350100ccfa1) If we consider a qubit with both position and spin, the state is a superposition of all possibilities for both: Ψ \= ψ \+ ( x ) ⊗ \\| ↑ ⟩ \+ ψ − ( x ) ⊗ \\| ↓ ⟩ , {\\displaystyle \\Psi =\\psi \_{+}(x)\\otimes \\|{\\uparrow }\\rangle +\\psi \_{-}(x)\\otimes \\|{\\downarrow }\\rangle ,} ![{\\displaystyle \\Psi =\\psi \_{+}(x)\\otimes \\|{\\uparrow }\\rangle +\\psi \_{-}(x)\\otimes \\|{\\downarrow }\\rangle ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4ece787cabf4ceba45438338ca75ca96b78c13f4) where we have a general state Ψ {\\displaystyle \\Psi } ![{\\displaystyle \\Psi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5471531a3fe80741a839bc98d49fae862a6439a) is the sum of the [tensor products](https://en.wikipedia.org/wiki/Tensor_products "Tensor products") of the position space wave functions and spinors. ## Experiments \[[edit](https://en.wikipedia.org/w/index.php?title=Quantum_superposition&action=edit&section=9 "Edit section: Experiments")\] Successful experiments involving superpositions of [relatively large](https://en.wikipedia.org/wiki/Mesoscopic "Mesoscopic") (by the standards of quantum physics) objects have been performed. - A [beryllium](https://en.wikipedia.org/wiki/Beryllium "Beryllium") [ion](https://en.wikipedia.org/wiki/Ion "Ion") has been trapped in a superposed state.[\[4\]](https://en.wikipedia.org/wiki/Quantum_superposition#cite_note-4) - A [double slit experiment](https://en.wikipedia.org/wiki/Double_slit_experiment "Double slit experiment") has been performed with molecules as large as [buckyballs](https://en.wikipedia.org/wiki/Buckminsterfullerene "Buckminsterfullerene") and functionalized oligoporphyrins with up to 2000 atoms.[\[5\]](https://en.wikipedia.org/wiki/Quantum_superposition#cite_note-5)[\[6\]](https://en.wikipedia.org/wiki/Quantum_superposition#cite_note-6) - Molecules with masses exceeding 10,000 and composed of over 810 atoms have successfully been superposed,[\[7\]](https://en.wikipedia.org/wiki/Quantum_superposition#cite_note-7) and metal clusters with masses over 170,000 [Da](https://en.wikipedia.org/wiki/Dalton_\(unit\) "Dalton (unit)") and containing more than 7,000 atoms have also been demonstrated in quantum superposition.[\[8\]](https://en.wikipedia.org/wiki/Quantum_superposition#cite_note-8) - Very sensitive magnetometers have been realized using [superconducting quantum interference devices](https://en.wikipedia.org/wiki/SQUID "SQUID") (SQUIDS) that operate using quantum interference effects in superconducting circuits. - A [piezoelectric](https://en.wikipedia.org/wiki/Piezoelectric "Piezoelectric") "[tuning fork](https://en.wikipedia.org/wiki/Tuning_fork "Tuning fork")" has been constructed, which can be placed into a superposition of vibrating and non-vibrating states. The resonator comprises about 10 trillion atoms.[\[9\]](https://en.wikipedia.org/wiki/Quantum_superposition#cite_note-9) - Recent research indicates that [chlorophyll](https://en.wikipedia.org/wiki/Chlorophyll "Chlorophyll") within [plants](https://en.wikipedia.org/wiki/Plants "Plants") appears to exploit the feature of quantum superposition to achieve greater efficiency in transporting energy, allowing pigment proteins to be spaced further apart than would otherwise be possible.[\[10\]](https://en.wikipedia.org/wiki/Quantum_superposition#cite_note-doi:10.1038/nature08811-10)[\[11\]](https://en.wikipedia.org/wiki/Quantum_superposition#cite_note-11) ## In quantum computers \[[edit](https://en.wikipedia.org/w/index.php?title=Quantum_superposition&action=edit&section=10 "Edit section: In quantum computers")\] In [quantum computers](https://en.wikipedia.org/wiki/Quantum_computers "Quantum computers"), a [qubit](https://en.wikipedia.org/wiki/Qubit "Qubit") is the analog of the classical information [bit](https://en.wikipedia.org/wiki/Bit "Bit"), but rather than having one of two distinct values, qubits are a superposition of two values.[\[12\]](https://en.wikipedia.org/wiki/Quantum_superposition#cite_note-Nielsen-Chuang-12): 13 Controlling this superposition qubits is a central challenge in quantum computation. The superposition needs to be robust to unintended interactions and yet interactions are needed for computing with qubits. Qubit systems like [nuclear spins](https://en.wikipedia.org/wiki/Nuclear_spin "Nuclear spin") with small coupling strength are robust to outside disturbances but the same small coupling makes it difficult to readout results.[\[12\]](https://en.wikipedia.org/wiki/Quantum_superposition#cite_note-Nielsen-Chuang-12): 278 ## See also \[[edit](https://en.wikipedia.org/w/index.php?title=Quantum_superposition&action=edit&section=11 "Edit section: See also")\] - [Eigenstate](https://en.wikipedia.org/wiki/Eigenstate "Eigenstate") – Mathematical entity to describe the probability of each possible measurement on a systemPages displaying short descriptions of redirect targets - [Mach–Zehnder interferometer](https://en.wikipedia.org/wiki/Mach%E2%80%93Zehnder_interferometer "Mach–Zehnder interferometer") – Device to determine relative phase shift - [Penrose interpretation](https://en.wikipedia.org/wiki/Penrose_interpretation "Penrose interpretation") – Interpretation of quantum mechanics - [Pure qubit state](https://en.wikipedia.org/wiki/Pure_qubit_state "Pure qubit state") – Basic unit of quantum informationPages displaying short descriptions of redirect targets - [Quantum computation](https://en.wikipedia.org/wiki/Quantum_computation "Quantum computation") – Computer hardware technology that uses quantum mechanicsPages displaying short descriptions of redirect targets - [Schrödinger's cat](https://en.wikipedia.org/wiki/Schr%C3%B6dinger%27s_cat "Schrödinger's cat") – Thought experiment in quantum mechanics - [Superposition principle](https://en.wikipedia.org/wiki/Superposition_principle "Superposition principle") – Fundamental principle of physics - [Wave packet](https://en.wikipedia.org/wiki/Wave_packet "Wave packet") – Short "burst" or "envelope" of restricted wave action that travels as a unit ## References \[[edit](https://en.wikipedia.org/w/index.php?title=Quantum_superposition&action=edit&section=12 "Edit section: References")\] 1. ^ [*a](https://en.wikipedia.org/wiki/Quantum_superposition#cite_ref-Messiah_1-0) [b](https://en.wikipedia.org/wiki/Quantum_superposition#cite_ref-Messiah_1-1) [c](https://en.wikipedia.org/wiki/Quantum_superposition#cite_ref-Messiah_1-2) Messiah, Albert (1976). 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[^](https://en.wikipedia.org/wiki/Quantum_superposition#cite_ref-4) Monroe, C.; Meekhof, D. M.; King, B. E.; Wineland, D. J. (24 May 1996). ["A "Schrödinger Cat" Superposition State of an Atom"](https://www.science.org/doi/10.1126/science.272.5265.1131). Science. 272 (5265): 1131–1136\. [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[1996Sci...272.1131M](https://ui.adsabs.harvard.edu/abs/1996Sci...272.1131M). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1126/science.272.5265.1131](https://doi.org/10.1126%2Fscience.272.5265.1131). [ISSN](https://en.wikipedia.org/wiki/ISSN_\(identifier\) "ISSN (identifier)") [0036-8075](https://search.worldcat.org/issn/0036-8075). [PMID](https://en.wikipedia.org/wiki/PMID_\(identifier\) "PMID (identifier)") [8662445](https://pubmed.ncbi.nlm.nih.gov/8662445). 5. [^](https://en.wikipedia.org/wiki/Quantum_superposition#cite_ref-5) ["Wave-particle duality of C60"](https://web.archive.org/web/20120331115055/http://www.quantum.at/research/molecule-interferometry-foundations/wave-particle-duality-of-c60.html). 31 March 2012. Archived from the original on 31 March 2012. `{{cite web}}`: CS1 maint: bot: original URL status unknown ([link](https://en.wikipedia.org/wiki/Category:CS1_maint:_bot:_original_URL_status_unknown "Category:CS1 maint: bot: original URL status unknown")) 6. [^](https://en.wikipedia.org/wiki/Quantum_superposition#cite_ref-6) Yaakov Y. Fein; Philipp Geyer; Patrick Zwick; Filip Kiałka; Sebastian Pedalino; Marcel Mayor; Stefan Gerlich; Markus Arndt (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). 7. [^](https://en.wikipedia.org/wiki/Quantum_superposition#cite_ref-7) Eibenberger, S., Gerlich, S., Arndt, M., Mayor, M., Tüxen, J. (2013). "Matter-wave interference with particles selected from a molecular library with masses exceeding 10 000 amu", Physical Chemistry Chemical Physics, 15: 14696-14700 [arXiv](https://en.wikipedia.org/wiki/ArXiv_\(identifier\) "ArXiv (identifier)"):[1310\.8343](https://arxiv.org/abs/1310.8343) 8. [^](https://en.wikipedia.org/wiki/Quantum_superposition#cite_ref-8) Pedalino, Sebastian; Ramírez-Galindo, Bruno E.; Ferstl, Richard; Hornberger, Klaus; Arndt, Markus; Gerlich, Stefan (22 January 2026). ["Probing quantum mechanics with nanoparticle matter-wave interferometry"](https://www.nature.com/articles/s41586-025-09917-9). Nature. 649 (8098): 866–870\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1038/s41586-025-09917-9](https://doi.org/10.1038%2Fs41586-025-09917-9). [ISSN](https://en.wikipedia.org/wiki/ISSN_\(identifier\) "ISSN (identifier)") [0028-0836](https://search.worldcat.org/issn/0028-0836). [PMC](https://en.wikipedia.org/wiki/PMC_\(identifier\) "PMC (identifier)") [12823444](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12823444). [PMID](https://en.wikipedia.org/wiki/PMID_\(identifier\) "PMID (identifier)") [41566007](https://pubmed.ncbi.nlm.nih.gov/41566007). 9. [^](https://en.wikipedia.org/wiki/Quantum_superposition#cite_ref-9) O’Connell, A. D.; Hofheinz, M.; Ansmann, M.; Bialczak, Radoslaw C.; Lenander, M.; Lucero, Erik; Neeley, M.; Sank, D.; Wang, H.; Weides, M.; Wenner, J.; Martinis, John M.; Cleland, A. N. (April 2010). ["Quantum ground state and single-phonon control of a mechanical resonator"](https://www.nature.com/articles/nature08967). Nature. 464 (7289): 697–703\. [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[2010Natur.464..697O](https://ui.adsabs.harvard.edu/abs/2010Natur.464..697O). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1038/nature08967](https://doi.org/10.1038%2Fnature08967). [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)") [20237473](https://pubmed.ncbi.nlm.nih.gov/20237473). 10. [^](https://en.wikipedia.org/wiki/Quantum_superposition#cite_ref-doi:10.1038/nature08811_10-0) Scholes, Gregory; Elisabetta Collini; Cathy Y. Wong; Krystyna E. Wilk; Paul M. G. Curmi; Paul Brumer; Gregory D. Scholes (4 February 2010). "Coherently wired light-harvesting in photosynthetic marine algae at ambient temperature". [Nature](https://en.wikipedia.org/wiki/Nature_\(journal\) "Nature (journal)"). 463 (7281): 644–647\. [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[2010Natur.463..644C](https://ui.adsabs.harvard.edu/abs/2010Natur.463..644C). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1038/nature08811](https://doi.org/10.1038%2Fnature08811). [PMID](https://en.wikipedia.org/wiki/PMID_\(identifier\) "PMID (identifier)") [20130647](https://pubmed.ncbi.nlm.nih.gov/20130647). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [4369439](https://api.semanticscholar.org/CorpusID:4369439). 11. [^](https://en.wikipedia.org/wiki/Quantum_superposition#cite_ref-11) Moyer, Michael (September 2009). ["Quantum Entanglement, Photosynthesis and Better Solar Cells"](http://www.scientificamerican.com/article.cfm?id=quantum-entanglement-and-photo). Scientific American. Retrieved 12 May 2010. 12. ^ [*a](https://en.wikipedia.org/wiki/Quantum_superposition#cite_ref-Nielsen-Chuang_12-0) [b](https://en.wikipedia.org/wiki/Quantum_superposition#cite_ref-Nielsen-Chuang_12-1) [Nielsen, Michael A.](https://en.wikipedia.org/wiki/Michael_Nielsen "Michael Nielsen"); [Chuang, Isaac](https://en.wikipedia.org/wiki/Isaac_Chuang "Isaac Chuang") (2010). [Quantum Computation and Quantum Information](https://www.cambridge.org/9781107002173). Cambridge: [Cambridge University Press](https://en.wikipedia.org/wiki/Cambridge_University_Press "Cambridge University Press"). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-1-10700-217-3](https://en.wikipedia.org/wiki/Special:BookSources/978-1-10700-217-3 "Special:BookSources/978-1-10700-217-3") . [OCLC](https://en.wikipedia.org/wiki/OCLC_\(identifier\) "OCLC (identifier)") [43641333](https://search.worldcat.org/oclc/43641333). ## Further reading \[[edit](https://en.wikipedia.org/w/index.php?title=Quantum_superposition&action=edit&section=13 "Edit section: Further reading")\] - [Bohr, N.](https://en.wikipedia.org/wiki/Niels_Bohr "Niels Bohr") (1927/1928). The quantum postulate and the recent development of atomic theory, [Nature* Supplement 14 April 1928, 121: 580–590](http://www.nature.com/nature/journal/v121/n3050/abs/121580a0.html). - [Cohen-Tannoudji, C.](https://en.wikipedia.org/wiki/Claude_Cohen-Tannoudji "Claude Cohen-Tannoudji"), Diu, B., Laloë, F. (1973/1977). Quantum Mechanics, translated from the French by S. R. Hemley, N. Ostrowsky, D. Ostrowsky, second edition, volume 1, Wiley, New York, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [0471164321](https://en.wikipedia.org/wiki/Special:BookSources/0471164321 "Special:BookSources/0471164321") . - [Einstein, A.](https://en.wikipedia.org/wiki/Albert_Einstein "Albert Einstein") (1949). Remarks concerning the essays brought together in this co-operative volume, translated from the original German by the editor, pp. 665–688 in [Schilpp, P. A.](https://en.wikipedia.org/wiki/Paul_Arthur_Schilpp "Paul Arthur Schilpp") editor (1949), [Albert Einstein: Philosopher-Scientist](https://www.worldcat.org/oclc/311439), volume II, Open Court, La Salle IL. - [Feynman, R. P.](https://en.wikipedia.org/wiki/Richard_Feynman "Richard Feynman"), Leighton, R.B., Sands, M. (1965). The Feynman Lectures on Physics, [volume 3](https://feynmanlectures.caltech.edu/III_toc.html), Addison-Wesley, Reading, MA. - [Merzbacher, E.](https://en.wikipedia.org/wiki/Eugen_Merzbacher "Eugen Merzbacher") (1961/1970). Quantum Mechanics, second edition, Wiley, New York. - [Messiah, A.](https://en.wikipedia.org/wiki/Albert_Messiah "Albert Messiah") (1961). Quantum Mechanics, volume 1, translated by G.M. Temmer from the French Mécanique Quantique, North-Holland, Amsterdam. - [Wheeler, J. A.](https://en.wikipedia.org/wiki/John_Archibald_Wheeler "John Archibald Wheeler"); [Zurek, W.H.](https://en.wikipedia.org/wiki/Wojciech_H._Zurek "Wojciech H. Zurek") (1983). Quantum Theory and Measurement. Princeton NJ: Princeton University Press. - [Nielsen, Michael A.](https://en.wikipedia.org/wiki/Michael_Nielsen "Michael Nielsen"); [Chuang, Isaac](https://en.wikipedia.org/wiki/Isaac_Chuang "Isaac Chuang") (2000). Quantum Computation and Quantum Information. Cambridge: [Cambridge University Press](https://en.wikipedia.org/wiki/Cambridge_University_Press "Cambridge University Press"). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [0521632358](https://en.wikipedia.org/wiki/Special:BookSources/0521632358 "Special:BookSources/0521632358") . [OCLC](https://en.wikipedia.org/wiki/OCLC_\(identifier\) "OCLC (identifier)") [43641333](https://search.worldcat.org/oclc/43641333). - Williams, Colin P. (2011). Explorations in Quantum Computing. [Springer](https://en.wikipedia.org/wiki/Springer_Science%2BBusiness_Media "Springer Science+Business Media"). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-1-84628-887-6](https://en.wikipedia.org/wiki/Special:BookSources/978-1-84628-887-6 "Special:BookSources/978-1-84628-887-6") . - Yanofsky, Noson S.; Mannucci, Mirco (2013). Quantum computing for computer scientists. [Cambridge University Press](https://en.wikipedia.org/wiki/Cambridge_University_Press "Cambridge University Press"). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-521-87996-5](https://en.wikipedia.org/wiki/Special:BookSources/978-0-521-87996-5 "Special:BookSources/978-0-521-87996-5") . \| [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](https://en.wikipedia.org/wiki/Quantum_mechanics "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]() [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") \| \| \| \| \| \|---\|---\| \| [Authority control databases](https://en.wikipedia.org/wiki/Help:Authority_control "Help:Authority control") [![Edit this at Wikidata](https://upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/20px-OOjs_UI_icon_edit-ltr-progressive.svg.png)](https://www.wikidata.org/wiki/Q830791#identifiers "Edit this at Wikidata") \| [GND](https://d-nb.info/gnd/4346012-4) \| 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Readable Markdown	From Wikipedia, the free encyclopedia Quantum superposition of states and decoherence Quantum superposition is a fundamental principle of [quantum mechanics](https://en.wikipedia.org/wiki/Quantum_mechanics "Quantum mechanics") that states that [linear combinations](https://en.wikipedia.org/wiki/Linear_combination "Linear combination") of [solutions](https://en.wikipedia.org/wiki/Solution_\(mathematics\) "Solution (mathematics)") to the [Schrödinger equation](https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation "Schrödinger equation") are also solutions of the Schrödinger equation. This follows from the fact that the Schrödinger equation is a [linear differential equation](https://en.wikipedia.org/wiki/Linear_differential_equation "Linear differential equation") in time and position. More precisely, the state of a system is given by a [linear combination](https://en.wikipedia.org/wiki/Linear_combination "Linear combination") of all the [eigenfunctions](https://en.wikipedia.org/wiki/Eigenfunction "Eigenfunction") of the Schrödinger equation governing that system. An example is a [qubit](https://en.wikipedia.org/wiki/Qubit "Qubit") used in [quantum information processing](https://en.wikipedia.org/wiki/Quantum_information_processing "Quantum information processing"). A qubit state is most generally a superposition of the basis states ![{\\displaystyle \\|0\\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/ed066a3ad158da0ad6d6a421a606b1c8a35eb95b) and ![{\\displaystyle \\|1\\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f53021ca18e77477ee5bd3c1523e5830189ec5c): ![{\\displaystyle \\|\\Psi \\rangle =c\_{0}\\|0\\rangle +c\_{1}\\|1\\rangle ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f10213c27e86b15bff8015bd81b1d5983d14c5a4) where ![{\\displaystyle \\|\\Psi \\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e77f6b1e903837c5765c9683da41dd93199621c) is the [quantum state](https://en.wikipedia.org/wiki/Quantum_state "Quantum state") of the qubit, and ![{\\displaystyle \\|0\\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/ed066a3ad158da0ad6d6a421a606b1c8a35eb95b), ![{\\displaystyle \\|1\\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f53021ca18e77477ee5bd3c1523e5830189ec5c) denote particular solutions to the Schrödinger equation in [Dirac notation](https://en.wikipedia.org/wiki/Bra%E2%80%93ket_notation "Bra–ket notation") weighted by the two [probability amplitudes](https://en.wikipedia.org/wiki/Probability_amplitude "Probability amplitude") ![{\\displaystyle c\_{0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1882ba8f1dc60f0c68a642abb5af093c73910921) and ![{\\displaystyle c\_{1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77b7dc6d279091d354e0b90889b463bfa7eb7247) that both are complex numbers. Here ![{\\displaystyle \\|0\\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/ed066a3ad158da0ad6d6a421a606b1c8a35eb95b) corresponds to the classical 0 [bit](https://en.wikipedia.org/wiki/Bit "Bit"), and ![{\\displaystyle \\|1\\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f53021ca18e77477ee5bd3c1523e5830189ec5c) to the classical 1 bit. The probabilities of measuring the system in the ![{\\displaystyle \\|0\\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/ed066a3ad158da0ad6d6a421a606b1c8a35eb95b) or ![{\\displaystyle \\|1\\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f53021ca18e77477ee5bd3c1523e5830189ec5c) state are given by ![{\\displaystyle \\|c\_{0}\\|^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8529f071ccb861751297a04659e8dbddca09bad8) and ![{\\displaystyle \\|c\_{1}\\|^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a9c67c5c8bf07a77edd704f880a64a35b8d9f1c5) respectively (see the [Born rule](https://en.wikipedia.org/wiki/Born_rule "Born rule")). Before the measurement occurs the qubit is in a superposition of both states. The interference fringes in the [double-slit experiment](https://en.wikipedia.org/wiki/Double-slit_experiment "Double-slit experiment") provide another example of the superposition principle. The theory of quantum mechanics postulates that a [wave equation](https://en.wikipedia.org/wiki/Wave_equation "Wave equation") completely determines the state of a quantum system at all times. Furthermore, this differential equation is restricted to be [linear](https://en.wikipedia.org/wiki/Linear_differential_equation "Linear differential equation") and [homogeneous](https://en.wikipedia.org/wiki/Homogeneous_differential_equation "Homogeneous differential equation"). These conditions mean that for any two solutions of the wave equation, ![{\\displaystyle \\Psi \_{1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ddcd58cf9c1c4dd1eb32a29c1e3abf425ec72600) and ![{\\displaystyle \\Psi \_{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1125553c0a026635c63ab2ad542b43174aca9bbe), a linear combination of those solutions also solve the wave equation: ![{\\displaystyle \\Psi =c\_{1}\\Psi \_{1}+c\_{2}\\Psi \_{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6823dfb8e3163bbbfd61bd2d925ce8d6fdac949) for arbitrary complex coefficients ![{\\displaystyle c\_{1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77b7dc6d279091d354e0b90889b463bfa7eb7247) and ![{\\displaystyle c\_{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b30ba1b247fb8d334580cec68561e749d24aff2).[\[1\]](https://en.wikipedia.org/wiki/Quantum_superposition#cite_note-Messiah-1): 61 If the wave equation has more than two solutions, combinations of all such solutions are again valid solutions. The quantum wave equation can be solved using functions of position, ![{\\displaystyle \\Psi ({\\vec {r}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/860ed8dcca03bb55dda0fe92c92a079038be797b), or using functions of momentum, ![{\\displaystyle \\Phi ({\\vec {p}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/96584b5a2ed608930bfa349e822c564a5298250d) and consequently the superposition of momentum functions are also solutions: ![{\\displaystyle \\Phi ({\\vec {p}})=d\_{1}\\Phi \_{1}({\\vec {p}})+d\_{2}\\Phi \_{2}({\\vec {p}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f72ca07aa31123cf18829faed1ebb4abeb3a5120) The position and momentum solutions are related by a [linear transformation](https://en.wikipedia.org/wiki/Linear_transformation "Linear transformation"), a [Fourier transformation](https://en.wikipedia.org/wiki/Fourier_transformation "Fourier transformation"). This transformation is itself a quantum superposition and every position wave function can be represented as a superposition of momentum wave functions and vice versa. These superpositions involve an infinite number of component waves.[\[1\]](https://en.wikipedia.org/wiki/Quantum_superposition#cite_note-Messiah-1): 244 ## Generalization to basis states \[[edit](https://en.wikipedia.org/w/index.php?title=Quantum_superposition&action=edit&section=3 "Edit section: Generalization to basis states")\] Other transformations express a quantum solution as a superposition of [eigenvectors](https://en.wikipedia.org/wiki/Eigenvectors "Eigenvectors"), each corresponding to a possible result of a measurement on the quantum system. An eigenvector ![{\\displaystyle \\psi \_{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e709a809cfd3cfbb647fe22c8d1f81573f4c7ae) for a mathematical operator, ![{\\displaystyle {\\hat {A}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f595a6c73d1183d6a1b2ac21fe47ac28c1483821), has the equation ![{\\displaystyle {\\hat {A}}\\psi \_{i}=\\lambda \_{i}\\psi \_{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f7c7cfad33ac61dd3bcde805172bb9897ac0dfe) where ![{\\displaystyle \\lambda \_{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/72fde940918edf84caf3d406cc7d31949166820f) is one possible measured quantum value for the observable ![{\\displaystyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3). A superposition of these eigenvectors can represent any solution: ![{\\displaystyle \\Psi =\\sum \_{n}a\_{i}\\psi \_{i}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/da5aeb99c104b07a8842ea74cc61a45a859aa2a3) The states like ![{\\displaystyle \\psi \_{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e709a809cfd3cfbb647fe22c8d1f81573f4c7ae) are called basis states. ## Compact notation for superpositions \[[edit](https://en.wikipedia.org/w/index.php?title=Quantum_superposition&action=edit&section=4 "Edit section: Compact notation for superpositions")\] Important mathematical operations on quantum system solutions can be performed using only the coefficients of the superposition, suppressing the details of the superposed functions. This leads to quantum systems expressed in the [Dirac bra-ket notation](https://en.wikipedia.org/wiki/Bra-ket_notation "Bra-ket notation"):[\[1\]](https://en.wikipedia.org/wiki/Quantum_superposition#cite_note-Messiah-1): 245 ![{\\displaystyle \\|v\\rangle =d\_{1}\\|1\\rangle +d\_{2}\\|2\\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/acc7ac146085e844fd0f2988bed9f23c30af497a) This approach is especially effective for systems like quantum spin with no classical coordinate analog. Such shorthand notation is very common in textbooks and papers on quantum mechanics, and superposition of basis states is a fundamental tool in quantum mechanics. [Paul Dirac](https://en.wikipedia.org/wiki/Paul_Dirac "Paul Dirac") described the superposition principle as follows: > The non-classical nature of the superposition process is brought out clearly if we consider the superposition of two states, A and B, such that there exists an observation which, when made on the system in state A, is certain to lead to one particular result, a say, and when made on the system in state B is certain to lead to some different result, b say. What will be the result of the observation when made on the system in the superposed state? The answer is that the result will be sometimes a and sometimes b, according to a probability law depending on the relative weights of A and B in the superposition process. It will never be different from both a and b \[i.e., either a or b\]. The intermediate character of the state formed by superposition thus expresses itself through the probability of a particular result for an observation being intermediate between the corresponding probabilities for the original states, not through the result itself being intermediate between the corresponding results for the original states.[\[2\]](https://en.wikipedia.org/wiki/Quantum_superposition#cite_note-Dirac1947-2) [Anton Zeilinger](https://en.wikipedia.org/wiki/Anton_Zeilinger "Anton Zeilinger"), referring to the prototypical example of the [double-slit experiment](https://en.wikipedia.org/wiki/Double-slit_experiment "Double-slit experiment"), has elaborated regarding the creation and destruction of quantum superposition: > "\[T\]he superposition of amplitudes ... is only valid if there is no way to know, even in principle, which path the particle took. It is important to realize that this does not imply that an observer actually takes note of what happens. It is sufficient to destroy the interference pattern, if the path information is accessible in principle from the experiment or even if it is dispersed in the environment and beyond any technical possibility to be recovered, but in principle still ‘‘out there.’’ The absence of any such information is the essential criterion for quantum interference to appear.[\[3\]](https://en.wikipedia.org/wiki/Quantum_superposition#cite_note-Zeilinger-3) Any quantum state can be expanded as a sum or superposition of the eigenstates of an Hermitian operator, like the Hamiltonian, because the eigenstates form a complete basis: ![{\\displaystyle \\|\\alpha \\rangle =\\sum \_{n}c\_{n}\\|n\\rangle ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/df393c7be36c417cc59f5a1a605aea04beef2e7b) where ![{\\displaystyle \\|n\\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/89c45bda6ac852da785c06476f2d3e3b6cba812a) are the energy eigenstates of the Hamiltonian. For continuous variables like position eigenstates, ![{\\displaystyle \\|x\\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/48004887d8f9dfc489bd2bc793780b7f1d8039ad): ![{\\displaystyle \\|\\alpha \\rangle =\\int dx'\\|x'\\rangle \\langle x'\\|\\alpha \\rangle ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9495e664ba18698edce2bfcc82c345de5ceaf6c6) where ![{\\displaystyle \\phi \_{\\alpha }(x)=\\langle x\\|\\alpha \\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/2ef30d5158d3fe8417e7fe37540e5370e46d9887) is the projection of the state into the ![{\\displaystyle \\|x\\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/48004887d8f9dfc489bd2bc793780b7f1d8039ad) basis and is called the wave function of the particle. In both instances we notice that ![{\\displaystyle \\|\\alpha \\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f42032e642ee1c9d27adb318d34c7cc85f7a95d5) can be expanded as a superposition of an infinite number of basis states. Given the Schrödinger equation ![{\\displaystyle {\\hat {H}}\\|n\\rangle =E\_{n}\\|n\\rangle ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a6afb6d354132f7d244ee60cf28e4608aa6e922) where ![{\\displaystyle \\|n\\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/89c45bda6ac852da785c06476f2d3e3b6cba812a) indexes the set of eigenstates of the Hamiltonian with energy eigenvalues ![{\\displaystyle E\_{n},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/578d4bb07de7b29269be95abb6a214848c09aea7) we see immediately that ![{\\displaystyle {\\hat {H}}{\\big (}\\|n\\rangle +\\|n'\\rangle {\\big )}=E\_{n}\\|n\\rangle +E\_{n'}\\|n'\\rangle ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/806bff6ca40452045baac80fd266eb3c453e7340) where ![{\\displaystyle \\|\\Psi \\rangle =\\|n\\rangle +\\|n'\\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/99dd060b4c8a46870132855acbe5a64eefdf0ea3) is a solution of the Schrödinger equation but is not generally an eigenstate because ![{\\displaystyle E\_{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ad6b82f2a00af6c9efd4c16d4e99329605645c0c) and ![{\\displaystyle E\_{n'}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d618f0a54e5627c8baa3cfb77b3e2dfac8f31bc) are not generally equal. We say that ![{\\displaystyle \\|\\Psi \\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e77f6b1e903837c5765c9683da41dd93199621c) is made up of a superposition of energy eigenstates. Now consider the more concrete case of an [electron](https://en.wikipedia.org/wiki/Electron "Electron") that has either [spin](https://en.wikipedia.org/wiki/Spin_\(physics\) "Spin (physics)") up or down. We now index the eigenstates with the [spinors](https://en.wikipedia.org/wiki/Spinor "Spinor") in the ![{\\displaystyle {\\hat {z}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/722665b45e05afe79f4395a3de0237d8ce856273) basis: ![{\\displaystyle \\|\\Psi \\rangle =c\_{1}\\|{\\uparrow }\\rangle +c\_{2}\\|{\\downarrow }\\rangle ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/42f17d6ba89011bea1b2f5f2747b88a7482c681e) where ![{\\displaystyle \\|{\\uparrow }\\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e81fba5ac4e54a40f5b9d63e66548328b7ffb6e) and ![{\\displaystyle \\|{\\downarrow }\\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/622f3f29fa4b88926ffa22e3da1daa5f2893d17c) denote spin-up and spin-down states respectively. As previously discussed, the magnitudes of the complex coefficients give the probability of finding the electron in either definite spin state: ![{\\displaystyle P{\\big (}\\|{\\uparrow }\\rangle {\\big )}=\\|c\_{1}\\|^{2},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e31d0a1610fddb7d86d8548a0ae43a9dff80529) ![{\\displaystyle P{\\big (}\\|{\\downarrow }\\rangle {\\big )}=\\|c\_{2}\\|^{2},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ad7f70c0f22f0405876a0e5c22cfdf96bcf05a9) ![{\\displaystyle P\_{\\text{total}}=P{\\big (}\\|{\\uparrow }\\rangle {\\big )}+P{\\big (}\\|{\\downarrow }\\rangle {\\big )}=\\|c\_{1}\\|^{2}+\\|c\_{2}\\|^{2}=1,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5fd7a45939635ea1e22247c9c7293c938999cefe) where the probability of finding the particle with either spin up or down is normalized to 1. Notice that ![{\\displaystyle c\_{1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77b7dc6d279091d354e0b90889b463bfa7eb7247) and ![{\\displaystyle c\_{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b30ba1b247fb8d334580cec68561e749d24aff2) are complex numbers, so that ![{\\displaystyle \\|\\Psi \\rangle ={\\frac {3}{5}}i\\|{\\uparrow }\\rangle +{\\frac {4}{5}}\\|{\\downarrow }\\rangle .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1751456a557a43b4c870d3dd5351854eb5556233) is an example of an allowed state. We now get ![{\\displaystyle P{\\big (}\\|{\\uparrow }\\rangle {\\big )}=\\left\\|{\\frac {3i}{5}}\\right\\|^{2}={\\frac {9}{25}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eeca39a54f3fc393e54a5fdc7feeb5aeb0b527ac) ![{\\displaystyle P{\\big (}\\|{\\downarrow }\\rangle {\\big )}=\\left\\|{\\frac {4}{5}}\\right\\|^{2}={\\frac {16}{25}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/863ea6a63d777554353bd1f448c5d4d989136162) ![{\\displaystyle P\_{\\text{total}}=P{\\big (}\\|{\\uparrow }\\rangle {\\big )}+P{\\big (}\\|{\\downarrow }\\rangle {\\big )}={\\frac {9}{25}}+{\\frac {16}{25}}=1.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/845feda8d1aa925d760f2a5e4b88e350100ccfa1) If we consider a qubit with both position and spin, the state is a superposition of all possibilities for both: ![{\\displaystyle \\Psi =\\psi \_{+}(x)\\otimes \\|{\\uparrow }\\rangle +\\psi \_{-}(x)\\otimes \\|{\\downarrow }\\rangle ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4ece787cabf4ceba45438338ca75ca96b78c13f4) where we have a general state ![{\\displaystyle \\Psi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5471531a3fe80741a839bc98d49fae862a6439a) is the sum of the [tensor products](https://en.wikipedia.org/wiki/Tensor_products "Tensor products") of the position space wave functions and spinors. Successful experiments involving superpositions of [relatively large](https://en.wikipedia.org/wiki/Mesoscopic "Mesoscopic") (by the standards of quantum physics) objects have been performed. - A [beryllium](https://en.wikipedia.org/wiki/Beryllium "Beryllium") [ion](https://en.wikipedia.org/wiki/Ion "Ion") has been trapped in a superposed state.[\[4\]](https://en.wikipedia.org/wiki/Quantum_superposition#cite_note-4) - A [double slit experiment](https://en.wikipedia.org/wiki/Double_slit_experiment "Double slit experiment") has been performed with molecules as large as [buckyballs](https://en.wikipedia.org/wiki/Buckminsterfullerene "Buckminsterfullerene") and functionalized oligoporphyrins with up to 2000 atoms.[\[5\]](https://en.wikipedia.org/wiki/Quantum_superposition#cite_note-5)[\[6\]](https://en.wikipedia.org/wiki/Quantum_superposition#cite_note-6) - Molecules with masses exceeding 10,000 and composed of over 810 atoms have successfully been superposed,[\[7\]](https://en.wikipedia.org/wiki/Quantum_superposition#cite_note-7) and metal clusters with masses over 170,000 [Da](https://en.wikipedia.org/wiki/Dalton_\(unit\) "Dalton (unit)") and containing more than 7,000 atoms have also been demonstrated in quantum superposition.[\[8\]](https://en.wikipedia.org/wiki/Quantum_superposition#cite_note-8) - Very sensitive magnetometers have been realized using [superconducting quantum interference devices](https://en.wikipedia.org/wiki/SQUID "SQUID") (SQUIDS) that operate using quantum interference effects in superconducting circuits. - A [piezoelectric](https://en.wikipedia.org/wiki/Piezoelectric "Piezoelectric") "[tuning fork](https://en.wikipedia.org/wiki/Tuning_fork "Tuning fork")" has been constructed, which can be placed into a superposition of vibrating and non-vibrating states. The resonator comprises about 10 trillion atoms.[\[9\]](https://en.wikipedia.org/wiki/Quantum_superposition#cite_note-9) - Recent research indicates that [chlorophyll](https://en.wikipedia.org/wiki/Chlorophyll "Chlorophyll") within [plants](https://en.wikipedia.org/wiki/Plants "Plants") appears to exploit the feature of quantum superposition to achieve greater efficiency in transporting energy, allowing pigment proteins to be spaced further apart than would otherwise be possible.[\[10\]](https://en.wikipedia.org/wiki/Quantum_superposition#cite_note-doi:10.1038/nature08811-10)[\[11\]](https://en.wikipedia.org/wiki/Quantum_superposition#cite_note-11) ## In quantum computers \[[edit](https://en.wikipedia.org/w/index.php?title=Quantum_superposition&action=edit&section=10 "Edit section: In quantum computers")\] In [quantum computers](https://en.wikipedia.org/wiki/Quantum_computers "Quantum computers"), a [qubit](https://en.wikipedia.org/wiki/Qubit "Qubit") is the analog of the classical information [bit](https://en.wikipedia.org/wiki/Bit "Bit"), but rather than having one of two distinct values, qubits are a superposition of two values.[\[12\]](https://en.wikipedia.org/wiki/Quantum_superposition#cite_note-Nielsen-Chuang-12): 13 Controlling this superposition qubits is a central challenge in quantum computation. The superposition needs to be robust to unintended interactions and yet interactions are needed for computing with qubits. Qubit systems like [nuclear spins](https://en.wikipedia.org/wiki/Nuclear_spin "Nuclear spin") with small coupling strength are robust to outside disturbances but the same small coupling makes it difficult to readout results.[\[12\]](https://en.wikipedia.org/wiki/Quantum_superposition#cite_note-Nielsen-Chuang-12): 278 - [Eigenstate](https://en.wikipedia.org/wiki/Eigenstate "Eigenstate") – Mathematical entity to describe the probability of each possible measurement on a system - [Mach–Zehnder interferometer](https://en.wikipedia.org/wiki/Mach%E2%80%93Zehnder_interferometer "Mach–Zehnder interferometer") – Device to determine relative phase shift - [Penrose interpretation](https://en.wikipedia.org/wiki/Penrose_interpretation "Penrose interpretation") – Interpretation of quantum mechanics - [Pure qubit state](https://en.wikipedia.org/wiki/Pure_qubit_state "Pure qubit state") – Basic unit of quantum information - [Quantum computation](https://en.wikipedia.org/wiki/Quantum_computation "Quantum computation") – Computer hardware technology that uses quantum mechanics - [Schrödinger's cat](https://en.wikipedia.org/wiki/Schr%C3%B6dinger%27s_cat "Schrödinger's cat") – Thought experiment in quantum mechanics - [Superposition principle](https://en.wikipedia.org/wiki/Superposition_principle "Superposition principle") – Fundamental principle of physics - [Wave packet](https://en.wikipedia.org/wiki/Wave_packet "Wave packet") – Short "burst" or "envelope" of restricted wave action that travels as a unit 1. ^ [*a](https://en.wikipedia.org/wiki/Quantum_superposition#cite_ref-Messiah_1-0) [b](https://en.wikipedia.org/wiki/Quantum_superposition#cite_ref-Messiah_1-1) [c](https://en.wikipedia.org/wiki/Quantum_superposition#cite_ref-Messiah_1-2) Messiah, Albert (1976). Quantum mechanics. 1* (2 ed.). Amsterdam: North-Holland. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-471-59766-7](https://en.wikipedia.org/wiki/Special:BookSources/978-0-471-59766-7 "Special:BookSources/978-0-471-59766-7") . 2. [^](https://en.wikipedia.org/wiki/Quantum_superposition#cite_ref-Dirac1947_2-0) P.A.M. Dirac (1947). The Principles of Quantum Mechanics (2nd ed.). Clarendon Press. p. 12. 3. [^](https://en.wikipedia.org/wiki/Quantum_superposition#cite_ref-Zeilinger_3-0) Zeilinger A (1999). "Experiment and the foundations of quantum physics". Rev. Mod. Phys. 71 (2): S288–S297. [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[1999RvMPS..71..288Z](https://ui.adsabs.harvard.edu/abs/1999RvMPS..71..288Z). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1103/revmodphys.71.s288](https://doi.org/10.1103%2Frevmodphys.71.s288). 4. [^](https://en.wikipedia.org/wiki/Quantum_superposition#cite_ref-4) Monroe, C.; Meekhof, D. M.; King, B. E.; Wineland, D. J. (24 May 1996). ["A "Schrödinger Cat" Superposition State of an Atom"](https://www.science.org/doi/10.1126/science.272.5265.1131). Science. 272 (5265): 1131–1136\. [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[1996Sci...272.1131M](https://ui.adsabs.harvard.edu/abs/1996Sci...272.1131M). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1126/science.272.5265.1131](https://doi.org/10.1126%2Fscience.272.5265.1131). [ISSN](https://en.wikipedia.org/wiki/ISSN_\(identifier\) "ISSN (identifier)") [0036-8075](https://search.worldcat.org/issn/0036-8075). [PMID](https://en.wikipedia.org/wiki/PMID_\(identifier\) "PMID (identifier)") [8662445](https://pubmed.ncbi.nlm.nih.gov/8662445). 5. [^](https://en.wikipedia.org/wiki/Quantum_superposition#cite_ref-5) ["Wave-particle duality of C60"](https://web.archive.org/web/20120331115055/http://www.quantum.at/research/molecule-interferometry-foundations/wave-particle-duality-of-c60.html). 31 March 2012. Archived from the original on 31 March 2012. `{{cite web}}`: CS1 maint: bot: original URL status unknown ([link](https://en.wikipedia.org/wiki/Category:CS1_maint:_bot:_original_URL_status_unknown "Category:CS1 maint: bot: original URL status unknown")) 6. [^](https://en.wikipedia.org/wiki/Quantum_superposition#cite_ref-6) Yaakov Y. Fein; Philipp Geyer; Patrick Zwick; Filip Kiałka; Sebastian Pedalino; Marcel Mayor; Stefan Gerlich; Markus Arndt (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). 7. [^](https://en.wikipedia.org/wiki/Quantum_superposition#cite_ref-7) Eibenberger, S., Gerlich, S., Arndt, M., Mayor, M., Tüxen, J. (2013). "Matter-wave interference with particles selected from a molecular library with masses exceeding 10 000 amu", Physical Chemistry Chemical Physics, 15: 14696-14700 [arXiv](https://en.wikipedia.org/wiki/ArXiv_\(identifier\) "ArXiv (identifier)"):[1310\.8343](https://arxiv.org/abs/1310.8343) 8. [^](https://en.wikipedia.org/wiki/Quantum_superposition#cite_ref-8) Pedalino, Sebastian; Ramírez-Galindo, Bruno E.; Ferstl, Richard; Hornberger, Klaus; Arndt, Markus; Gerlich, Stefan (22 January 2026). ["Probing quantum mechanics with nanoparticle matter-wave interferometry"](https://www.nature.com/articles/s41586-025-09917-9). Nature. 649 (8098): 866–870\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1038/s41586-025-09917-9](https://doi.org/10.1038%2Fs41586-025-09917-9). [ISSN](https://en.wikipedia.org/wiki/ISSN_\(identifier\) "ISSN (identifier)") [0028-0836](https://search.worldcat.org/issn/0028-0836). [PMC](https://en.wikipedia.org/wiki/PMC_\(identifier\) "PMC (identifier)") [12823444](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12823444). [PMID](https://en.wikipedia.org/wiki/PMID_\(identifier\) "PMID (identifier)") [41566007](https://pubmed.ncbi.nlm.nih.gov/41566007). 9. [^](https://en.wikipedia.org/wiki/Quantum_superposition#cite_ref-9) O’Connell, A. D.; Hofheinz, M.; Ansmann, M.; Bialczak, Radoslaw C.; Lenander, M.; Lucero, Erik; Neeley, M.; Sank, D.; Wang, H.; Weides, M.; Wenner, J.; Martinis, John M.; Cleland, A. N. (April 2010). ["Quantum ground state and single-phonon control of a mechanical resonator"](https://www.nature.com/articles/nature08967). Nature. 464 (7289): 697–703\. [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[2010Natur.464..697O](https://ui.adsabs.harvard.edu/abs/2010Natur.464..697O). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1038/nature08967](https://doi.org/10.1038%2Fnature08967). [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)") [20237473](https://pubmed.ncbi.nlm.nih.gov/20237473). 10. [^](https://en.wikipedia.org/wiki/Quantum_superposition#cite_ref-doi:10.1038/nature08811_10-0) Scholes, Gregory; Elisabetta Collini; Cathy Y. Wong; Krystyna E. Wilk; Paul M. G. Curmi; Paul Brumer; Gregory D. Scholes (4 February 2010). "Coherently wired light-harvesting in photosynthetic marine algae at ambient temperature". [Nature](https://en.wikipedia.org/wiki/Nature_\(journal\) "Nature (journal)"). 463 (7281): 644–647\. [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[2010Natur.463..644C](https://ui.adsabs.harvard.edu/abs/2010Natur.463..644C). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1038/nature08811](https://doi.org/10.1038%2Fnature08811). [PMID](https://en.wikipedia.org/wiki/PMID_\(identifier\) "PMID (identifier)") [20130647](https://pubmed.ncbi.nlm.nih.gov/20130647). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [4369439](https://api.semanticscholar.org/CorpusID:4369439). 11. [^](https://en.wikipedia.org/wiki/Quantum_superposition#cite_ref-11) Moyer, Michael (September 2009). ["Quantum Entanglement, Photosynthesis and Better Solar Cells"](http://www.scientificamerican.com/article.cfm?id=quantum-entanglement-and-photo). Scientific American. Retrieved 12 May 2010. 12. ^ [*a](https://en.wikipedia.org/wiki/Quantum_superposition#cite_ref-Nielsen-Chuang_12-0) [b](https://en.wikipedia.org/wiki/Quantum_superposition#cite_ref-Nielsen-Chuang_12-1) [Nielsen, Michael A.](https://en.wikipedia.org/wiki/Michael_Nielsen "Michael Nielsen"); [Chuang, Isaac](https://en.wikipedia.org/wiki/Isaac_Chuang "Isaac Chuang") (2010). [Quantum Computation and Quantum Information](https://www.cambridge.org/9781107002173). Cambridge: [Cambridge University Press](https://en.wikipedia.org/wiki/Cambridge_University_Press "Cambridge University Press"). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-1-10700-217-3](https://en.wikipedia.org/wiki/Special:BookSources/978-1-10700-217-3 "Special:BookSources/978-1-10700-217-3") . [OCLC](https://en.wikipedia.org/wiki/OCLC_\(identifier\) "OCLC (identifier)") [43641333](https://search.worldcat.org/oclc/43641333). - [Bohr, N.](https://en.wikipedia.org/wiki/Niels_Bohr "Niels Bohr") (1927/1928). The quantum postulate and the recent development of atomic theory, [Nature* Supplement 14 April 1928, 121: 580–590](http://www.nature.com/nature/journal/v121/n3050/abs/121580a0.html). - [Cohen-Tannoudji, C.](https://en.wikipedia.org/wiki/Claude_Cohen-Tannoudji "Claude Cohen-Tannoudji"), Diu, B., Laloë, F. (1973/1977). Quantum Mechanics, translated from the French by S. R. Hemley, N. Ostrowsky, D. Ostrowsky, second edition, volume 1, Wiley, New York, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [0471164321](https://en.wikipedia.org/wiki/Special:BookSources/0471164321 "Special:BookSources/0471164321") . - [Einstein, A.](https://en.wikipedia.org/wiki/Albert_Einstein "Albert Einstein") (1949). Remarks concerning the essays brought together in this co-operative volume, translated from the original German by the editor, pp. 665–688 in [Schilpp, P. A.](https://en.wikipedia.org/wiki/Paul_Arthur_Schilpp "Paul Arthur Schilpp") editor (1949), [Albert Einstein: Philosopher-Scientist](https://www.worldcat.org/oclc/311439), volume II, Open Court, La Salle IL. - [Feynman, R. P.](https://en.wikipedia.org/wiki/Richard_Feynman "Richard Feynman"), Leighton, R.B., Sands, M. (1965). The Feynman Lectures on Physics, [volume 3](https://feynmanlectures.caltech.edu/III_toc.html), Addison-Wesley, Reading, MA. - [Merzbacher, E.](https://en.wikipedia.org/wiki/Eugen_Merzbacher "Eugen Merzbacher") (1961/1970). Quantum Mechanics, second edition, Wiley, New York. - [Messiah, A.](https://en.wikipedia.org/wiki/Albert_Messiah "Albert Messiah") (1961). Quantum Mechanics, volume 1, translated by G.M. Temmer from the French Mécanique Quantique, North-Holland, Amsterdam. - [Wheeler, J. A.](https://en.wikipedia.org/wiki/John_Archibald_Wheeler "John Archibald Wheeler"); [Zurek, W.H.](https://en.wikipedia.org/wiki/Wojciech_H._Zurek "Wojciech H. Zurek") (1983). Quantum Theory and Measurement. Princeton NJ: Princeton University Press. - [Nielsen, Michael A.](https://en.wikipedia.org/wiki/Michael_Nielsen "Michael Nielsen"); [Chuang, Isaac](https://en.wikipedia.org/wiki/Isaac_Chuang "Isaac Chuang") (2000). Quantum Computation and Quantum Information. Cambridge: [Cambridge University Press](https://en.wikipedia.org/wiki/Cambridge_University_Press "Cambridge University Press"). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [0521632358](https://en.wikipedia.org/wiki/Special:BookSources/0521632358 "Special:BookSources/0521632358") . [OCLC](https://en.wikipedia.org/wiki/OCLC_\(identifier\) "OCLC (identifier)") [43641333](https://search.worldcat.org/oclc/43641333). - Williams, Colin P. (2011). Explorations in Quantum Computing. [Springer](https://en.wikipedia.org/wiki/Springer_Science%2BBusiness_Media "Springer Science+Business Media"). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-1-84628-887-6](https://en.wikipedia.org/wiki/Special:BookSources/978-1-84628-887-6 "Special:BookSources/978-1-84628-887-6") . - Yanofsky, Noson S.; Mannucci, Mirco (2013). Quantum computing for computer scientists. [Cambridge University Press](https://en.wikipedia.org/wiki/Cambridge_University_Press "Cambridge University Press"). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-521-87996-5](https://en.wikipedia.org/wiki/Special:BookSources/978-0-521-87996-5 "Special:BookSources/978-0-521-87996-5") .
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