Born rule

The Born rule (also called Born's rule) is a key postulate of quantum mechanics which gives the probability that a measurement of a quantum system will yield a given result.[1] In its simplest form, it states that the probability density of finding a particle at a given point, when measured, is proportional to the square of the magnitude of the particle's wavefunction at that point. It was formulated by German physicist Max Born in 1926.

Details

The Born rule states that if an observable corresponding to a self-adjoint operator ${\textstyle A}$ with discrete spectrum is measured in a system with normalized wave function ${\textstyle |\psi \rangle }$ (see Bra–ket notation), then

• the measured result will be one of the eigenvalues ${\displaystyle \lambda }$ of ${\displaystyle A}$, and
• the probability of measuring a given eigenvalue ${\displaystyle \lambda _{i}}$ will equal ${\displaystyle \langle \psi |P_{i}|\psi \rangle }$, where ${\displaystyle P_{i}}$ is the projection onto the eigenspace of ${\displaystyle A}$ corresponding to ${\displaystyle \lambda _{i}}$.

(In the case where the eigenspace of ${\displaystyle A}$ corresponding to ${\displaystyle \lambda _{i}}$ is one-dimensional and spanned by the normalized eigenvector ${\displaystyle |\lambda _{i}\rangle }$, ${\displaystyle P_{i}}$ is equal to ${\displaystyle |\lambda _{i}\rangle \langle \lambda _{i}|}$, so the probability ${\displaystyle \langle \psi |P_{i}|\psi \rangle }$ is equal to ${\displaystyle \langle \psi |\lambda _{i}\rangle \langle \lambda _{i}|\psi \rangle }$. Since the complex number ${\displaystyle \langle \lambda _{i}|\psi \rangle }$ is known as the probability amplitude that the state vector ${\displaystyle |\psi \rangle }$ assigns to the eigenvector ${\displaystyle |\lambda _{i}\rangle }$, 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 ${\displaystyle |\langle \lambda _{i}|\psi \rangle |^{2}}$.)

In the case where the spectrum of ${\displaystyle A}$ is not wholly discrete, the spectral theorem proves the existence of a certain projection-valued measure ${\displaystyle Q}$, the spectral measure of ${\displaystyle A}$. In this case,

• the probability that the result of the measurement lies in a measurable set ${\displaystyle M}$ is given by ${\displaystyle \langle \psi |Q(M)|\psi \rangle }$.

Given a wave function ${\displaystyle \psi }$ for a single structureless particle in position space, implies that the probability density function ${\displaystyle p(x,y,z)}$ for a measurement of the position at time ${\displaystyle t_{0}}$ is

${\displaystyle p(x,y,z)=|\psi (x,y,z,t_{0})|^{2}}$.

In some applications, this treatment of the Born rule is generalized using positive-operator-valued measures. A POVM is a measure whose values are positive semi-definite operators on a Hilbert space. POVMs are a generalisation of von Neumann measurements and, correspondingly, quantum measurements described by POVMs are a generalisation of quantum measurement 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.[2] 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 ${\displaystyle \{F_{i}\}}$ on a Hilbert space ${\displaystyle {\mathcal {H}}}$ that sum to the identity matrix,[3]^:90

${\displaystyle \sum _{i=1}^{n}F_{i}=\operatorname {I} .}$

The POVM element ${\displaystyle F_{i}}$ is associated with the measurement outcome ${\displaystyle i}$, such that the probability of obtaining it when making a measurement on the quantum state ${\displaystyle \rho }$ is given by

${\displaystyle p(i)=\operatorname {tr} (\rho F_{i})}$,

where ${\displaystyle \operatorname {tr} }$ is the trace operator. This is the POVM version of the Born rule. When the quantum state being measured is a pure state ${\displaystyle |\psi \rangle }$ this formula reduces to

${\displaystyle p(i)=\operatorname {tr} (|\psi \rangle \langle \psi |F_{i})=\langle \psi |F_{i}|\psi \rangle }$.

History

The Born rule was formulated by Born in a 1926 paper.[4] In this paper, Born solves the Schrödinger equation for a scattering problem and, inspired by Einstein's work on the photoelectric effect,[5] concludes, in a footnote, that the Born rule gives the only possible interpretation of the solution. In 1954, together with Walther Bothe, Born was awarded the Nobel Prize in Physics for this and other work.[5] John von Neumann discussed the application of spectral theory to Born's rule in his 1932 book.[6]

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,[7] prompted by a question posed by George W. Mackey.[8][9] This theorem was historically significant for the role it played in showing that wide classes of hidden-variable theories are inconsistent with quantum physics.[10]

Interpretations

The Born rule, together with the unitarity of the time evolution operator ${\displaystyle e^{-i{\hat {H}}t}}$ (or, equivalently, the Hamiltonian ${\displaystyle {\hat {H}}}$ being Hermitian), implies the unitarity of the theory, which is considered required for consistency. For example, unitarity ensures that the probabilities of all possible outcomes sum to 1 (though it is not the only option to get this particular requirement).

Within the QBist interpretation of quantum theory, the Born rule is seen as a modification of the standard Law of Total Probability, which takes into account the Hilbert space dimension of the physical system involved.[11] It has been claimed that Pilot wave theory can also statistically derive the Born rule, though this remains controversial.[12] While it has been claimed that the Born rule can be derived from the many-worlds interpretation, the existing proofs have been criticized as circular.[13] Kastner claims that the transactional interpretation is unique in giving a physical explanation for the Born rule.[14]

See also

• Einstein and the quantum
• Gleason's theorem
• Unitarity

References

1. The time evolution of a quantum system is entirely deterministic according to the Schrödinger equation. It is through the Born Rule that probability enters into the theory.
2. 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.
3. 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.
4. Born, Max (1926). "I.2". In Wheeler, J. A.; Zurek, W. H. (eds.). Zur Quantenmechanik der Stoßvorgänge [On the quantum mechanics of collisions]. Zeitschrift für Physik. 37. Princeton University Press (published 1983). pp. 863–867. Bibcode:1926ZPhy...37..863B. doi:10.1007/BF01397477. ISBN 978-0-691-08316-2.
5. 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).
6. 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.
7. 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.
8. 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.
9. Chernoff, Paul R. "Andy Gleason and Quantum Mechanics" (PDF). Notices of the AMS. 56 (10): 1253–1259.
10. 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.
11. Healey, Richard (2016). "Quantum-Bayesian and Pragmatist Views of Quantum Theory". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University.
12. Goldstein, Sheldon (2017). "Bohmian Mechanics". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University.
13. 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 3-540-70622-4. 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
14. Kastner, R. E. (2013). The Transactional Interpretation of Quantum Mechanics. Cambridge University Press. p. 35. ISBN 978-0-521-76415-5.

External links

• Quantum Mechanics Not in Jeopardy: Physicists Confirm a Decades-Old Key Principle Experimentally ScienceDaily (July 23, 2010)