
The Born rule is a postulate of quantum mechanics that gives the probability that a measurement of a quantum system will yield a given result. In one commonly used application, it states that the probability density for finding a particle at a given position is proportional to the square of the amplitude of the system's wavefunction at that position. It was formulated and published by German physicist Max Born in July, 1926.
Details
The Born rule states that an observable, measured in a system with normalized wave function (see Bra–ket notation), corresponds to a self-adjoint operator
whose spectrum is discrete if:
- the measured result will be one of the eigenvalues
of
, and
- the probability of measuring a given eigenvalue
will equal
, where
is the projection onto the eigenspace of
corresponding to
.
- (In the case where the eigenspace of
corresponding to
is one-dimensional and spanned by the normalized eigenvector
,
is equal to
, so the probability
is equal to
. Since the complex number
is known as the probability amplitude that the state vector
assigns to the eigenvector
, 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). Equivalently, the probability can be written as
.)
In the case where the spectrum of is not wholly discrete, the spectral theorem proves the existence of a certain projection-valued measure (PVM)
, the spectral measure of
. In this case:
- the probability that the result of the measurement lies in a measurable set
is given by
.
For example, a single structureless particle can be described by a wave function that depends upon position coordinates
and a time coordinate
. The Born rule implies that the probability density function
for the result of a measurement of the particle's position at time
is:
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). A POVM is a measure whose values are positive semi-definite operators on a 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 is to a pure state. Mixed states are needed to specify the state of a subsystem of a larger system (see 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. They are extensively used in the field of quantum information.
In the simplest case, of a POVM with a finite number of elements acting on a finite-dimensional Hilbert space, a POVM is a set of positive semi-definite matrices on a Hilbert space
that sum to the identity matrix,:: 90
The POVM element is associated with the measurement outcome
, such that the probability of obtaining it when making a measurement on the quantum state
is given by:
where is the trace operator. This is the POVM version of the Born rule. When the quantum state being measured is a pure state
this formula reduces to:
The Born rule, together with the unitarity of the time evolution operator (or, equivalently, the Hamiltonian
being Hermitian), implies the unitarity 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, proper normalization is ensured by requiring that the time evolution is a trace-preserving, completely positive map.)
History
The Born rule was formulated by Born in a 1926 paper. In this paper, Born solves the Schrödinger equation for a scattering problem and, inspired by Albert Einstein and Einstein's probabilistic rule for the photoelectric effect, 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, Born was awarded the Nobel Prize in Physics for this and other work.John von Neumann discussed the application of spectral theory to Born's rule in his 1932 book.
Derivation from more basic principles
Gleason's theorem shows that the Born rule can be derived from the usual mathematical representation of measurements in quantum physics together with the assumption of non-contextuality. Andrew M. Gleason first proved the theorem in 1957, prompted by a question posed by George W. Mackey. This theorem was historically significant for the role it played in showing that wide classes of hidden-variable theories are inconsistent with quantum physics.
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. These include the decision-theory approach pioneered by David Deutsch and later developed by Hilary Greaves and David Wallace; and an "envariance" approach by Wojciech H. Zurek. These proofs have, however, been criticized as circular. In 2018, an approach based on self-locating uncertainty was suggested by Charles Sebens and Sean M. Carroll; this has also been criticized.Simon Saunders, in 2021, 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. 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.
In 2019, Lluís Masanes, Thomas Galley, and Markus Müller proposed a derivation based on postulates including the possibility of state estimation.
It has also been claimed that pilot-wave theory can be used to statistically derive the Born rule, though this remains controversial.
Within the QBist interpretation of quantum theory, the Born rule is seen as an extension of the normative principle of coherence, which ensures self-consistency of probability assessments across a whole set of such assessments. It can be shown that an agent who thinks they are gambling on the outcomes of measurements on a sufficiently quantum-like system but refuses to use the Born rule when placing their bets is vulnerable to a Dutch book.
References
- Hall, Brian C. (2013). "Quantum Theory for Mathematicians". Graduate Texts in Mathematics. New York, NY: Springer New York. pp. 14–15, 58. doi:10.1007/978-1-4614-7116-5. ISBN 978-1-4614-7115-8. ISSN 0072-5285.
- Peres, Asher; Terno, Daniel R. (2004). "Quantum information and relativity theory". Reviews of Modern Physics. 76 (1): 93–123. arXiv:quant-ph/0212023. Bibcode:2004RvMP...76...93P. doi:10.1103/RevModPhys.76.93. S2CID 7481797.
- Nielsen, Michael A.; Chuang, Isaac L. (2000). Quantum Computation and Quantum Information (1st ed.). Cambridge: Cambridge University Press. ISBN 978-0-521-63503-5. OCLC 634735192.
- Born, Max (1926). "Zur Quantenmechanik der Stoßvorgänge" [On the quantum mechanics of collisions]. Zeitschrift für Physik. 37 (12): 863–867. Bibcode:1926ZPhy...37..863B. doi:10.1007/BF01397477. S2CID 119896026. Reprinted as Born, Max (1983). "On the quantum mechanics of collisions". In Wheeler, J. A.; Zurek, W. H. (eds.). Quantum Theory and Measurement. Princeton University Press. pp. 52–55. ISBN 978-0-691-08316-2.
- Born, Max (11 December 1954). "The statistical interpretation of quantum mechanics" (PDF). www.nobelprize.org. nobelprize.org. Retrieved 7 November 2018.
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: |psi|2 ought to represent the probability density for electrons (or other particles).
- Neumann (von), John (1932). Mathematische Grundlagen der Quantenmechanik [Mathematical Foundations of Quantum Mechanics]. Translated by Beyer, Robert T. Princeton University Press (published 1996). ISBN 978-0691028934.
- Gleason, Andrew M. (1957). "Measures on the closed subspaces of a Hilbert space". Indiana University Mathematics Journal. 6 (4): 885–893. doi:10.1512/iumj.1957.6.56050. MR 0096113.
- Mackey, George W. (1957). "Quantum Mechanics and Hilbert Space". The American Mathematical Monthly. 64 (8P2): 45–57. doi:10.1080/00029890.1957.11989120. JSTOR 2308516.
- Chernoff, Paul R. (November 2009). "Andy Gleason and Quantum Mechanics" (PDF). Notices of the AMS. 56 (10): 1253–1259.
- Mermin, N. David (1993-07-01). "Hidden variables and the two theorems of John Bell". Reviews of Modern Physics. 65 (3): 803–815. arXiv:1802.10119. Bibcode:1993RvMP...65..803M. doi:10.1103/RevModPhys.65.803. S2CID 119546199.
- Deutsch, David (8 August 1999). "Quantum Theory of Probability and Decisions". Proceedings of the Royal Society A. 455 (1988): 3129–3137. arXiv:quant-ph/9906015. Bibcode:1999RSPSA.455.3129D. doi:10.1098/rspa.1999.0443. S2CID 5217034. Retrieved December 5, 2022.
- Greaves, Hilary (21 December 2006). "Probability in the Everett Interpretation" (PDF). Philosophy Compass. 2 (1): 109–128. doi:10.1111/j.1747-9991.2006.00054.x. Retrieved 6 December 2022.
- Wallace, David (2010). "How to Prove the Born Rule". In Kent, Adrian; Wallace, David; Barrett, Jonathan; Saunders, Simon (eds.). Many Worlds? Everett, Quantum Theory, & Reality. Oxford University Press. pp. 227–263. arXiv:0906.2718. ISBN 978-0-191-61411-8.
- Zurek, Wojciech H. (25 May 2005). "Probabilities from entanglement, Born's rule from envariance". Physical Review A. 71: 052105. arXiv:quant-ph/0405161. doi:10.1103/PhysRevA.71.052105. Retrieved 6 December 2022.
- Landsman, N. P. (2008). "The Born rule and its interpretation" (PDF). In Weinert, F.; Hentschel, K.; Greenberger, D.; Falkenburg, B. (eds.). Compendium of Quantum Physics. Springer. ISBN 978-3-540-70622-9.
The conclusion seems to be that no generally accepted derivation of the Born rule has been given to date, but this does not imply that such a derivation is impossible in principle
- Sebens, Charles T.; Carroll, Sean M. (March 2018). "Self-Locating Uncertainty and the Origin of Probability in Everettian Quantum Mechanics". The British Journal for the Philosophy of Science. 69 (1): 25–74. arXiv:1405.7577. doi:10.1093/bjps/axw004.
- Vaidman, Lev (2020). "Derivations of the Born Rule" (PDF). Quantum, Probability, Logic. Jerusalem Studies in Philosophy and History of Science. Springer. pp. 567–584. doi:10.1007/978-3-030-34316-3_26. ISBN 978-3-030-34315-6. S2CID 156046920.
- Saunders, Simon (24 November 2021). "Branch-counting in the Everett interpretation of quantum mechanics". Proceedings of the Royal Society A. 477 (2255): 1–22. arXiv:2201.06087. Bibcode:2021RSPSA.47710600S. doi:10.1098/rspa.2021.0600. S2CID 244491576.
- Masanes, Lluís; Galley, Thomas; Müller, Markus (2019). "The measurement postulates of quantum mechanics are operationally redundant". Nature Communications. 10 (1): 1361. arXiv:1811.11060. Bibcode:2019NatCo..10.1361M. doi:10.1038/s41467-019-09348-x. PMC 6434053. PMID 30911009.
- Ball, Philip (February 13, 2019). "Mysterious Quantum Rule Reconstructed From Scratch". Quanta Magazine. Archived from the original on 2019-02-13.
- Goldstein, Sheldon (2017). "Bohmian Mechanics". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University.
- DeBrota, John B.; Fuchs, Christopher A.; Pienaar, Jacques L.; Stacey, Blake C. (2021). "Born's rule as a quantum extension of Bayesian coherence". Phys. Rev. A. 104 (2). 022207. arXiv:2012.14397. Bibcode:2021PhRvA.104b2207D. doi:10.1103/PhysRevA.104.022207.
External links
The Born rule is a postulate of quantum mechanics that gives the probability that a measurement of a quantum system will yield a given result In one commonly used application it states that the probability density for finding a particle at a given position is proportional to the square of the amplitude of the system s wavefunction at that position It was formulated and published by German physicist Max Born in July 1926 DetailsThe Born rule states that an observable measured in a system with normalized wave function ps displaystyle psi rangle see Bra ket notation corresponds to a self adjoint operator A displaystyle A whose spectrum is discrete if the measured result will be one of the eigenvalues l displaystyle lambda of A displaystyle A and the probability of measuring a given eigenvalue li displaystyle lambda i will equal ps Pi ps displaystyle langle psi P i psi rangle where Pi displaystyle P i is the projection onto the eigenspace of A displaystyle A corresponding to li displaystyle lambda i In the case where the eigenspace of A displaystyle A corresponding to li displaystyle lambda i is one dimensional and spanned by the normalized eigenvector li displaystyle lambda i rangle Pi displaystyle P i is equal to li li displaystyle lambda i rangle langle lambda i so the probability ps Pi ps displaystyle langle psi P i psi rangle is equal to ps li li ps displaystyle langle psi lambda i rangle langle lambda i psi rangle Since the complex number li ps displaystyle langle lambda i psi rangle is known as the probability amplitude that the state vector ps displaystyle psi rangle assigns to the eigenvector li displaystyle lambda i rangle 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 Equivalently the probability can be written as li ps 2 displaystyle big langle lambda i psi rangle big 2 In the case where the spectrum of A displaystyle A is not wholly discrete the spectral theorem proves the existence of a certain projection valued measure PVM Q displaystyle Q the spectral measure of A displaystyle A In this case the probability that the result of the measurement lies in a measurable set M displaystyle M is given by ps Q M ps displaystyle langle psi Q M psi rangle For example a single structureless particle can be described by a wave function ps displaystyle psi that depends upon position coordinates x y z displaystyle x y z and a time coordinate t displaystyle t The Born rule implies that the probability density function p displaystyle p for the result of a measurement of the particle s position at time t0 displaystyle t 0 is p x y z t0 ps x y z t0 2 displaystyle p x y z t 0 psi x y z t 0 2 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 A POVM is a measure whose values are positive semi definite operators on a 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 is to a pure state Mixed states are needed to specify the state of a subsystem of a larger system see 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 They are extensively used in the field of quantum information In the simplest case of a POVM with a finite number of elements acting on a finite dimensional Hilbert space a POVM is a set of positive semi definite matrices Fi displaystyle F i on a Hilbert space H displaystyle mathcal H that sum to the identity matrix 90 i 1nFi I displaystyle sum i 1 n F i I The POVM element Fi displaystyle F i is associated with the measurement outcome i displaystyle i such that the probability of obtaining it when making a measurement on the quantum state r displaystyle rho is given by p i tr rFi displaystyle p i operatorname tr rho F i where tr displaystyle operatorname tr is the trace operator This is the POVM version of the Born rule When the quantum state being measured is a pure state ps displaystyle psi rangle this formula reduces to p i tr ps ps Fi ps Fi ps displaystyle p i operatorname tr big psi rangle langle psi F i big langle psi F i psi rangle The Born rule together with the unitarity of the time evolution operator e iH t displaystyle e i hat H t or equivalently the Hamiltonian H displaystyle hat H being Hermitian implies the unitarity 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 proper normalization is ensured by requiring that the time evolution is a trace preserving completely positive map HistoryThe Born rule was formulated by Born in a 1926 paper In this paper Born solves the Schrodinger equation for a scattering problem and inspired by Albert Einstein and Einstein s probabilistic rule for the photoelectric effect 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 Born was awarded the Nobel Prize in Physics for this and other work John von Neumann discussed the application of spectral theory to Born s rule in his 1932 book Derivation from more basic principles Gleason s theorem shows that the Born rule can be derived from the usual mathematical representation of measurements in quantum physics together with the assumption of non contextuality Andrew M Gleason first proved the theorem in 1957 prompted by a question posed by George W Mackey This theorem was historically significant for the role it played in showing that wide classes of hidden variable theories are inconsistent with quantum physics 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 These include the decision theory approach pioneered by David Deutsch and later developed by Hilary Greaves and David Wallace and an envariance approach by Wojciech H Zurek These proofs have however been criticized as circular In 2018 an approach based on self locating uncertainty was suggested by Charles Sebens and Sean M Carroll this has also been criticized Simon Saunders in 2021 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 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 In 2019 Lluis Masanes Thomas Galley and Markus Muller proposed a derivation based on postulates including the possibility of state estimation It has also been claimed that pilot wave theory can be used to statistically derive the Born rule though this remains controversial Within the QBist interpretation of quantum theory the Born rule is seen as an extension of the normative principle of coherence which ensures self consistency of probability assessments across a whole set of such assessments It can be shown that an agent who thinks they are gambling on the outcomes of measurements on a sufficiently quantum like system but refuses to use the Born rule when placing their bets is vulnerable to a Dutch book ReferencesHall Brian C 2013 Quantum Theory for Mathematicians Graduate Texts in Mathematics New York NY Springer New York pp 14 15 58 doi 10 1007 978 1 4614 7116 5 ISBN 978 1 4614 7115 8 ISSN 0072 5285 Peres Asher Terno Daniel R 2004 Quantum information and relativity theory Reviews of Modern Physics 76 1 93 123 arXiv quant ph 0212023 Bibcode 2004RvMP 76 93P doi 10 1103 RevModPhys 76 93 S2CID 7481797 Nielsen Michael A Chuang Isaac L 2000 Quantum Computation and Quantum Information 1st ed Cambridge Cambridge University Press ISBN 978 0 521 63503 5 OCLC 634735192 Born Max 1926 Zur Quantenmechanik der Stossvorgange On the quantum mechanics of collisions Zeitschrift fur Physik 37 12 863 867 Bibcode 1926ZPhy 37 863B doi 10 1007 BF01397477 S2CID 119896026 Reprinted as Born Max 1983 On the quantum mechanics of collisions In Wheeler J A Zurek W H eds Quantum Theory and Measurement Princeton University Press pp 52 55 ISBN 978 0 691 08316 2 Born Max 11 December 1954 The statistical interpretation of quantum mechanics PDF www nobelprize org nobelprize org Retrieved 7 November 2018 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 psi 2 ought to represent the probability density for electrons or other particles Neumann von John 1932 Mathematische Grundlagen der Quantenmechanik Mathematical Foundations of Quantum Mechanics Translated by Beyer Robert T Princeton University Press published 1996 ISBN 978 0691028934 Gleason Andrew M 1957 Measures on the closed subspaces of a Hilbert space Indiana University Mathematics Journal 6 4 885 893 doi 10 1512 iumj 1957 6 56050 MR 0096113 Mackey George W 1957 Quantum Mechanics and Hilbert Space The American Mathematical Monthly 64 8P2 45 57 doi 10 1080 00029890 1957 11989120 JSTOR 2308516 Chernoff Paul R November 2009 Andy Gleason and Quantum Mechanics PDF Notices of the AMS 56 10 1253 1259 Mermin N David 1993 07 01 Hidden variables and the two theorems of John Bell Reviews of Modern Physics 65 3 803 815 arXiv 1802 10119 Bibcode 1993RvMP 65 803M doi 10 1103 RevModPhys 65 803 S2CID 119546199 Deutsch David 8 August 1999 Quantum Theory of Probability and Decisions Proceedings of the Royal Society A 455 1988 3129 3137 arXiv quant ph 9906015 Bibcode 1999RSPSA 455 3129D doi 10 1098 rspa 1999 0443 S2CID 5217034 Retrieved December 5 2022 Greaves Hilary 21 December 2006 Probability in the Everett Interpretation PDF Philosophy Compass 2 1 109 128 doi 10 1111 j 1747 9991 2006 00054 x Retrieved 6 December 2022 Wallace David 2010 How to Prove the Born Rule In Kent Adrian Wallace David Barrett Jonathan Saunders Simon eds Many Worlds Everett Quantum Theory amp Reality Oxford University Press pp 227 263 arXiv 0906 2718 ISBN 978 0 191 61411 8 Zurek Wojciech H 25 May 2005 Probabilities from entanglement Born s rule from envariance Physical Review A 71 052105 arXiv quant ph 0405161 doi 10 1103 PhysRevA 71 052105 Retrieved 6 December 2022 Landsman N P 2008 The Born rule and its interpretation PDF In Weinert F Hentschel K Greenberger D Falkenburg B eds Compendium of Quantum Physics Springer ISBN 978 3 540 70622 9 The conclusion seems to be that no generally accepted derivation of the Born rule has been given to date but this does not imply that such a derivation is impossible in principle Sebens Charles T Carroll Sean M March 2018 Self Locating Uncertainty and the Origin of Probability in Everettian Quantum Mechanics The British Journal for the Philosophy of Science 69 1 25 74 arXiv 1405 7577 doi 10 1093 bjps axw004 Vaidman Lev 2020 Derivations of the Born Rule PDF Quantum Probability Logic Jerusalem Studies in Philosophy and History of Science Springer pp 567 584 doi 10 1007 978 3 030 34316 3 26 ISBN 978 3 030 34315 6 S2CID 156046920 Saunders Simon 24 November 2021 Branch counting in the Everett interpretation of quantum mechanics Proceedings of the Royal Society A 477 2255 1 22 arXiv 2201 06087 Bibcode 2021RSPSA 47710600S doi 10 1098 rspa 2021 0600 S2CID 244491576 Masanes Lluis Galley Thomas Muller Markus 2019 The measurement postulates of quantum mechanics are operationally redundant Nature Communications 10 1 1361 arXiv 1811 11060 Bibcode 2019NatCo 10 1361M doi 10 1038 s41467 019 09348 x PMC 6434053 PMID 30911009 Ball Philip February 13 2019 Mysterious Quantum Rule Reconstructed From Scratch Quanta Magazine Archived from the original on 2019 02 13 Goldstein Sheldon 2017 Bohmian Mechanics In Zalta Edward N ed Stanford Encyclopedia of Philosophy Metaphysics Research Lab Stanford University DeBrota John B Fuchs Christopher A Pienaar Jacques L Stacey Blake C 2021 Born s rule as a quantum extension of Bayesian coherence Phys Rev A 104 2 022207 arXiv 2012 14397 Bibcode 2021PhRvA 104b2207D doi 10 1103 PhysRevA 104 022207 External linksWikiquote has quotations related to Born rule