# Born rule

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{{Short description|Calculation rule in quantum mechanics}}
{{Distinguish|Cauchy–Born rule|Born approximation}}
{{Quantum mechanics|cTopic=[Fundamentals](/source/Fundamentals)}}

The '''Born rule''' is a postulate of [quantum mechanics](/source/quantum_mechanics) that gives the [probability](/source/probability) that a [measurement of a quantum system](/source/measurement_in_quantum_mechanics) will yield a given result. In one commonly used application, it states that the [probability density](/source/probability_density_function) for finding a particle at a given position is proportional to the square of the amplitude of the system's [wavefunction](/source/wavefunction) at that position. It was formulated and published by German physicist [Max Born](/source/Max_Born) in July 1926.<ref>{{cite book | last=Hall | first=Brian C. | title=Graduate Texts in Mathematics | chapter=Quantum Theory for Mathematicians | publisher=Springer New York | publication-place=New York, NY | year=2013 | volume=267 | isbn=978-1-4614-7115-8 | issn=0072-5285 | doi=10.1007/978-1-4614-7116-5|pages=14–15, 58}}</ref>

== Details ==
The Born rule states that an [observable](/source/observable), measured in a system with normalized [wave function](/source/wave_function) <math>|\psi\rang</math> (see [Bra–ket notation](/source/Bra%E2%80%93ket_notation)), corresponds to a [self-adjoint operator](/source/self-adjoint_operator) <math>A</math> whose [spectrum](/source/spectrum_(functional_analysis)) is discrete if:
* the measured result will be one of the [eigenvalues](/source/Eigenvalues_and_eigenvectors) <math>\lambda</math> of <math>A</math>, and
* the probability of measuring a given eigenvalue <math>\lambda_i</math> will equal <math>\lang\psi|P_i|\psi\rang</math>, where <math>P_i</math> is the projection onto the [eigenspace](/source/eigenspace) of <math>A</math> corresponding to <math>\lambda_i</math>.
: (In the case where the eigenspace of <math>A</math> corresponding to <math>\lambda_i</math> is one-dimensional and spanned by the normalized eigenvector <math>|\lambda_i\rang</math>, <math>P_i</math> is equal to <math>|\lambda_i\rang\lang\lambda_i|</math>, so the probability <math>\lang\psi|P_i|\psi\rang</math> is equal to <math>\lang\psi|\lambda_i\rang\lang\lambda_i|\psi\rang</math>. Since the [complex number](/source/complex_number) <math>\lang\lambda_i|\psi\rang</math> is known as the ''[probability amplitude](/source/probability_amplitude)'' that the state vector <math>|\psi\rang</math> assigns to the eigenvector <math>|\lambda_i\rang</math>, it is common to describe the Born rule as saying that probability is equal to the amplitude-squared (really the amplitude times its own [complex conjugate](/source/complex_conjugate)). Equivalently, the probability can be written as <math>\big|\lang\lambda_i|\psi\rang\big|^2</math>.)

In the case where the spectrum of <math>A</math> is not wholly discrete, the [spectral theorem](/source/Spectral_theorem) proves the existence of a certain [projection-valued measure (PVM)](/source/projection-valued_measure) <math>Q</math>, the spectral measure of <math>A</math>. In this case:
* the probability that the result of the measurement lies in a measurable set <math>M</math> is given by <math>\lang\psi|Q(M)|\psi\rang</math>.

For example, a single structureless particle can be described by a wave function <math>\psi</math> that depends upon position coordinates <math>(x, y, z)</math> and a time coordinate <math>t</math>. The Born rule implies that the probability density function <math>p</math> for the result of a measurement of the particle's position at time <math>t_0</math> is:
<math display="block">p(x, y, z, t_0) = |\psi(x, y, z, t_0)|^2.</math>
The Born rule can also be employed to calculate probabilities (for measurements with discrete sets of outcomes) or probability densities (for continuous-valued measurements) for other observables, like momentum, energy, and angular momentum.

In some applications, this treatment of the Born rule is generalized using [positive-operator-valued measures (POVM)](/source/POVM). A POVM is a [measure](/source/Measure_(mathematics)) whose values are [positive semi-definite operators](/source/Definiteness_of_a_matrix) on a [Hilbert space](/source/Hilbert_space). POVMs are a generalization of von Neumann measurements and, correspondingly, quantum measurements described by POVMs are a generalization of quantum measurements described by self-adjoint observables. In rough analogy, a POVM is to a PVM what a [mixed state](/source/Quantum_state) is to a [pure state](/source/Quantum_state). Mixed states are needed to specify the state of a subsystem of a larger system (see [purification of quantum state](/source/purification_of_quantum_state)); analogously, POVMs are necessary to describe the effect on a subsystem of a projective measurement performed on a larger system. POVMs are the most general kind of measurement in quantum mechanics and can also be used in [quantum field theory](/source/quantum_field_theory).<ref>{{cite journal |last1=Peres |first1=Asher |author-link1=Asher Peres |last2=Terno |first2=Daniel R. |title=Quantum information and relativity theory |journal=[Reviews of Modern Physics](/source/Reviews_of_Modern_Physics) |volume=76 |number=1 |year=2004 |pages=93–123 |arxiv=quant-ph/0212023 |doi=10.1103/RevModPhys.76.93 |bibcode=2004RvMP...76...93P |s2cid=7481797 }}</ref> They are extensively used in the field of [quantum information](/source/quantum_information).

In the simplest case of a POVM with a finite number of elements acting on a finite-dimensional [Hilbert space](/source/Hilbert_space), a POVM is a set of [positive semi-definite](/source/Definiteness_of_a_matrix) [matrices](/source/Matrix_(mathematics)) <math>\{F_i\}</math> on a Hilbert space <math>\mathcal{H}</math> that sum to the [identity matrix](/source/identity_matrix),:<ref name="mike_ike">{{Cite book |last1=Nielsen |first1=Michael A. |author-link1=Michael Nielsen |last2=Chuang |first2=Isaac L. |author-link2=Isaac Chuang |title=Quantum Computation and Quantum Information|title-link=Quantum Computation and Quantum Information |publisher=[Cambridge University Press](/source/Cambridge_University_Press) |location=Cambridge |year=2000 |edition=1st |oclc=634735192 |isbn=978-0-521-63503-5}}</ref>{{rp|90}}
<math display="block">\sum_{i=1}^n F_i = I.</math>

The POVM element <math>F_i</math> is associated with the measurement outcome <math>i</math>, such that the probability of obtaining it when making a measurement on the quantum state <math>\rho</math> is given by:

<math display="block">p(i) = \operatorname{tr}(\rho F_i),</math>

where <math>\operatorname{tr}</math> is the [trace](/source/Trace_(linear_algebra)) operator. This is the POVM version of the Born rule. When the quantum state being measured is a pure state <math>|\psi\rangle</math> this formula reduces to:
<math display="block">p(i) = \operatorname{tr}\big(|\psi\rangle\langle\psi| F_i\big) = \langle\psi|F_i|\psi\rangle.</math>

The Born rule, together with the [unitarity](/source/Unitary_operator) of the time evolution operator <math>e^{-i\hat{H}t}</math> (or, equivalently, the [Hamiltonian](/source/Hamiltonian_(quantum_mechanics)) <math>\hat{H}</math> being [Hermitian](/source/Hermitian_matrix)), implies the [unitarity](/source/Unitarity_(physics)) of the theory: a wave function that is time-evolved by a unitary operator will remain properly normalized. (In the more general case where one considers the time evolution of a [density matrix](/source/density_matrix), proper normalization is ensured by requiring that the time evolution is a [trace-preserving, completely positive map](/source/quantum_channel).)

== History ==

The Born rule was formulated by Born in a 1926 paper.<ref name=Zeitschrift>
{{cite journal
|last=Born
|first=Max
|author-link=Max Born
|title=Zur Quantenmechanik der Stoßvorgänge
|journal=Zeitschrift für Physik
|volume=37
|issue=12
|trans-title=On the quantum mechanics of collisions
|date=1926
|bibcode = 1926ZPhy...37..863B
|doi=10.1007/BF01397477
|pages=863–867
|s2cid=119896026
}}
Reprinted as {{cite book 
|last=Born
|first=Max
|author-link=Max Born
|title=Quantum Theory and Measurement
|editor1-last=Wheeler 
|editor1-first=J. A.
|editor1-link=John Archibald Wheeler
|editor2-last=Zurek
|editor2-first=W. H.
|editor2-link=Wojciech H. Zurek
|publisher=Princeton University Press
|publication-date=1983
|isbn=978-0-691-08316-2
|chapter=On the quantum mechanics of collisions
|pages=52–55
}}
</ref> In this paper, Born solves the [Schrödinger equation](/source/Schr%C3%B6dinger_equation) for a scattering problem and, inspired by [Albert Einstein](/source/Albert_Einstein) and [Einstein's probabilistic rule](/source/Einstein's_thought_experiments) for the [photoelectric effect](/source/photoelectric_effect),<ref name=Nobel>
{{cite web
|url=https://www.nobelprize.org/uploads/2018/06/born-lecture.pdf
|title=The statistical interpretation of quantum mechanics
|last1=Born
|first1=Max
|author-link=Max Born
|date=11 December 1954
|website=www.nobelprize.org
|publisher=nobelprize.org
|access-date=7 November 2018
|quote=Again an idea of Einstein's gave me the lead. He had tried to make the duality of particles - light quanta or photons - and waves comprehensible by interpreting the square of the optical wave amplitudes as probability density for the occurrence of photons. This concept could at once be carried over to the psi-function: <math>||\psi>|</math><sup>2</sup> ought to represent the probability density for electrons (or other particles).
}}</ref> concludes, in a footnote, that the Born rule gives the only possible interpretation of the solution. (The main body of the article says that the amplitude "gives the probability" [''bestimmt die Wahrscheinlichkeit''], while the footnote added in proof says that the probability is proportional to the square of its magnitude.) In 1954, together with [Walther Bothe](/source/Walther_Bothe), Born was awarded the [Nobel Prize in Physics](/source/Nobel_Prize_in_Physics) for this and other work.<ref name=Nobel/> [John von Neumann](/source/John_von_Neumann) discussed the application of [spectral theory](/source/spectral_theory) to Born's rule in his 1932 book.<ref name=Grundlagen>
{{cite book 
|last=Neumann (von)
|first=John
|author-link=John von Neumann
|title=Mathematische Grundlagen der Quantenmechanik
|translator-last=Beyer
|translator-first=Robert T.
|trans-title=Mathematical Foundations of Quantum Mechanics
|date=1932
|publisher=Princeton University Press
|publication-date=1996
|isbn=978-0-691-02893-4
|title-link=Mathematische Grundlagen der Quantenmechanik
}}</ref>

===Derivation from more basic principles===
[Gleason's theorem](/source/Gleason's_theorem) shows that the Born rule can be derived from the usual mathematical representation of measurements in quantum physics. [Andrew M. Gleason](/source/Andrew_M._Gleason) first proved the theorem in 1957,<ref name="gleason1957">{{cite journal |first=Andrew M. |author-link=Andrew M. Gleason |year = 1957 |title = Measures on the closed subspaces of a Hilbert space |url = http://www.iumj.indiana.edu/IUMJ/FULLTEXT/1957/6/56050 |journal = [Indiana University Mathematics Journal](/source/Indiana_University_Mathematics_Journal) |volume = 6 |issue=4 |pages = 885–893 |doi=10.1512/iumj.1957.6.56050 |mr=0096113 |last = Gleason |doi-access = free|url-access = subscription }}</ref> prompted by a question posed by [George W. Mackey](/source/George_Mackey).<ref>{{Cite journal |last=Mackey |first=George W. |author-link=George Mackey |title=Quantum Mechanics and Hilbert Space |journal=[The American Mathematical Monthly](/source/The_American_Mathematical_Monthly) |year=1957 |volume=64 |number=8P2 |pages=45–57 |doi=10.1080/00029890.1957.11989120 |jstor=2308516}}</ref><ref name="chernoff2009">{{Cite journal|last=Chernoff |first=Paul R. |author-link=Paul Chernoff |title=Andy Gleason and Quantum Mechanics |date=November 2009 |journal=[Notices of the AMS](/source/Notices_of_the_AMS) |volume=56 |number=10 |pages=1253–1259 |url=https://www.ams.org/notices/200910/rtx091001236p.pdf}}</ref> This theorem was historically significant for the role it played in showing that wide classes of [hidden-variable theories](/source/hidden-variable_theory) are inconsistent with quantum physics.<ref name="mermin1993">{{Cite journal |last=Mermin |first=N. David |author-link=David Mermin |date=1993-07-01 |title=Hidden variables and the two theorems of John Bell |journal=[Reviews of Modern Physics](/source/Reviews_of_Modern_Physics) |volume=65 |issue=3 |pages=803–815 |doi=10.1103/RevModPhys.65.803 |bibcode=1993RvMP...65..803M |arxiv=1802.10119 |s2cid=119546199}}</ref>

Several other researchers have also tried to derive the Born rule from more basic principles. A number of derivations have been proposed in the context of the [many-worlds interpretation](/source/many-worlds_interpretation). These include the decision-theory approach pioneered by [David Deutsch](/source/David_Deutsch)<ref>{{cite journal |last1=Deutsch |first1=David |author-link=David Deutsch |title=Quantum Theory of Probability and Decisions |journal=Proceedings of the Royal Society A |date=8 August 1999 |volume=455 |issue=1988 |pages=3129–3137 |doi=10.1098/rspa.1999.0443 |arxiv=quant-ph/9906015 |bibcode=1999RSPSA.455.3129D |s2cid=5217034 |url=https://royalsocietypublishing.org/doi/10.1098/rspa.1999.0443 |access-date=December 5, 2022 |ref=deutsch}}</ref> and later developed by [Hilary Greaves](/source/Hilary_Greaves)<ref>{{cite journal |last1=Greaves |first1=Hilary |title=Probability in the Everett Interpretation |journal=Philosophy Compass |date=21 December 2006 |volume=2 |issue=1 |pages=109–128 |doi=10.1111/j.1747-9991.2006.00054.x |url=http://philsci-archive.pitt.edu/3103/2/pitei.pdf |access-date=6 December 2022 |ref=greaves }}</ref> and David Wallace;<ref>{{Cite book|last=Wallace |first=David |chapter=How to Prove the Born Rule |title=Many Worlds? Everett, Quantum Theory, & Reality |pages=227–263 |publisher=Oxford University Press |year=2010 |isbn=978-0-191-61411-8 |editor-first1=Adrian |editor-last1=Kent |editor-first2=David |editor-last2=Wallace |editor-first3=Jonathan |editor-last3=Barrett |editor-first4=Simon |editor-last4=Saunders |arxiv=0906.2718|ref=wallace}}</ref> and an "envariance" approach by [Wojciech H. Zurek](/source/Wojciech_H._Zurek).<ref>{{cite journal |last1=Zurek |first1=Wojciech H. |title=Probabilities from entanglement, Born's rule from envariance |journal=Physical Review A |date=25 May 2005 |volume=71 |article-number=052105 |doi=10.1103/PhysRevA.71.052105 |arxiv=quant-ph/0405161 |url=https://journals.aps.org/pra/abstract/10.1103/PhysRevA.71.052105 |access-date=6 December 2022 |ref=zurek}}</ref> These proofs have, however, been criticized as circular.<ref>{{cite book |first=N. P. |last=Landsman |chapter-url=https://www.math.ru.nl/~landsman/Born.pdf |quote=The conclusion seems to be that no generally accepted derivation of the Born rule has been given to date, but this does not imply that such a derivation is impossible in principle |chapter=The Born rule and its interpretation |title=Compendium of Quantum Physics |editor-first=F. |editor-last=Weinert |editor2-first=K. |editor2-last=Hentschel |editor3-first=D. |editor3-last=Greenberger |editor4-first=B. |editor4-last=Falkenburg |publisher=Springer |year=2008 |isbn=978-3-540-70622-9 }}</ref> In 2018, an approach based on self-locating uncertainty was suggested by Charles Sebens and [Sean M. Carroll](/source/Sean_M._Carroll);<ref>{{cite journal |last1=Sebens |first1=Charles T. |last2=Carroll |first2=Sean M. |date=March 2018 |title=Self-Locating Uncertainty and the Origin of Probability in Everettian Quantum Mechanics |journal=The British Journal for the Philosophy of Science |volume=69 |issue=1 |pages=25–74 |doi=10.1093/bjps/axw004 |ref=sebens-carroll|doi-access=free |arxiv=1405.7577 }}</ref> this has also been criticized.<ref>{{cite book|first=Lev |last=Vaidman |author-link=Lev Vaidman |chapter=Derivations of the Born Rule |title=Quantum, Probability, Logic |series=Jerusalem Studies in Philosophy and History of Science |publisher=Springer |year=2020 |pages=567–584 |isbn=978-3-030-34315-6 |doi=10.1007/978-3-030-34316-3_26 |s2cid=156046920 |chapter-url=https://philsci-archive.pitt.edu/15943/1/BornRule24-4-19.pdf}}</ref> In 2019, Lluís Masanes, Thomas Galley, and Markus Müller proposed a derivation based on postulates including the possibility of state estimation.<ref>{{cite journal |last1=Masanes |first1=Lluís |last2=Galley |first2=Thomas |last3=Müller |first3=Markus |url= |title=The measurement postulates of quantum mechanics are operationally redundant |journal=[Nature Communications](/source/Nature_Communications) |volume=10 |year=2019 |issue=1 |page=1361 |doi=10.1038/s41467-019-09348-x |pmid=30911009 |pmc=6434053 |arxiv=1811.11060 |bibcode=2019NatCo..10.1361M }}</ref><ref>{{cite web |last=Ball |first=Philip |author-link=Philip Ball |date=February 13, 2019 |title=Mysterious Quantum Rule Reconstructed From Scratch |url=https://www.quantamagazine.org/the-born-rule-has-been-derived-from-simple-physical-principles-20190213/ |url-status=live |website=[Quanta Magazine](/source/Quanta_Magazine)|archive-url=https://web.archive.org/web/20190213180115/https://www.quantamagazine.org/the-born-rule-has-been-derived-from-simple-physical-principles-20190213/ |archive-date=2019-02-13 }}</ref> In 2021, [Simon Saunders](/source/Simon_Saunders) produced a branch counting derivation of the Born rule. The crucial feature of this approach is to define the branches so that they all have the same magnitude or [2-norm](/source/norm_(mathematics)). The ratios of the numbers of branches thus defined give the probabilities of the various outcomes of a measurement, in accordance with the Born rule.<ref name=BranchCounting>{{Cite journal|last=Saunders|first=Simon|author-link=Simon Saunders|date=24 November 2021|title=Branch-counting in the Everett interpretation of quantum mechanics.|journal=Proceedings of the Royal Society A|language=en|volume=477|issue=2255|pages=1–22 |article-number=20210600 |doi=10.1098/rspa.2021.0600|issn=|arxiv=2201.06087|bibcode=2021RSPSA.47710600S |s2cid=244491576 }}</ref>

It has also been claimed that [pilot-wave theory](/source/pilot-wave_theory) can be used to statistically derive the Born rule, though this remains controversial.<ref>{{cite book |chapter=Bohmian Mechanics |title=[Stanford Encyclopedia of Philosophy](/source/Stanford_Encyclopedia_of_Philosophy) |chapter-url=https://plato.stanford.edu/entries/qm-bohm/ |year=2017 |publisher=Metaphysics Research Lab, Stanford University |editor-first=Edward N. |editor-last=Zalta |first=Sheldon |last=Goldstein}}</ref>

Within the [QBist](/source/Quantum_Bayesianism) interpretation of quantum theory, the Born rule is seen as an extension of the normative principle of [coherence](/source/Coherence_(philosophical_gambling_strategy)), which ensures self-consistency of probability assessments across a whole set of such assessments. It can be shown that an agent gambling on the outcomes of measurements on a sufficiently quantum-like system, refusing to use the Born rule when placing bets, is vulnerable to a [Dutch book](/source/Dutch_book).<ref>{{cite journal |first1=John B. |last1=DeBrota |first2=Christopher A. |last2=Fuchs |author-link2=Christopher A. Fuchs |first3=Jacques L. |last3=Pienaar |first4=Blake C. |last4=Stacey |title=Born's rule as a quantum extension of Bayesian coherence |journal=Phys. Rev. A |volume=104 |issue= 2|at=022207 |year=2021 |doi=10.1103/PhysRevA.104.022207 |arxiv=2012.14397 |bibcode=2021PhRvA.104b2207D }}</ref>

==References==
{{reflist}}

==External links==
{{Wikiquote}}
{{Quantum mechanics topics}}

{{DEFAULTSORT:Born Rule}}
Category:Quantum measurement
Category:Max Born

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Adapted from the Wikipedia article [Born rule](https://en.wikipedia.org/wiki/Born_rule) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Born_rule?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.