nLab Born rule



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Probability theory



The Born rule is the core statement of coordination in the foundations of quantum physics:

In its version for pure states the Born rule says that

then the probability of observing the given eigenstate in a quantum system which is in state ψ\psi equals (in bra-ket notation):

ψ|P|ψψ|ψ. \frac{ \langle \psi \vert P \vert \psi \rangle }{ \langle \psi \vert \psi \rangle } \,.

In particular, if WW is an orthonormal basis for the Hilbert space of states associated with a quantum measurement-procedure, then the probability of measuring the result ww on a system in state |ψ\left\vert \psi \right\rangle is

ψ|ww|ψψ|ψ=|w|ψ| 2ψ|ψ \frac{ \langle \psi \vert w \rangle \langle w \vert \psi \rangle }{ \langle \psi \vert \psi \rangle } \;\;=\;\; \frac{ \big\vert \langle w \vert \psi \rangle \big\vert^2 }{ \langle \psi \vert \psi \rangle }

and so if the state is normalized to begin with (ψ|ψ\langle \psi \vert \psi \rangle = 1 ), then the probability is

P ψ(w)=|w|ψ| 2. P_\psi(w) \;=\; \big\vert \langle w \vert \psi \rangle \big\vert^2 \,.

Essentially in this form the rule was first formulated by Born 1926b p 805.

quantum probability theoryobservables and states


Historical origins

The Born rule is named in honor of

where it appears (though disregarding the norm symbol) as a brief footnote-added-in-proof:

and its expanded version

which provides the mathematical details (p. 805):

and finally in the followup

which makes the interpretation more explicit (p. 171):

The generalization of this idea to probability densities for continuous observables is due to Wolfgang Pauli, as first recounted in:

and then by Pauli himself, again in a footnote:

Early review of the Born-Pauli rule:

where it is once again a footnote:

and in

The full recognition and amplification of the Born rule as a pillar of quantum physics making it a probabilistic theory (cf. quantum probability) is (maybe besides Jordan 1927) due to:

Historical survey:

  • Jagdish Mehra, Helmut Rechenberg, The Probability Interpretation and the Statistical Transformation Theory, the Physical Interpretation, and the Empirical and Mathematical Foundations of Quantum Mechanics 1926-1932, Part 1 in: The Historical Development of Quantum Theory. Volume 6: The Completion of Quantum Mechanics, 1926-1941, Springer (2001) [ISBN:978-0-387-98971-6]

Further discussion


See also:

Last revised on September 3, 2023 at 20:05:44. See the history of this page for a list of all contributions to it.