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Born rule

January 14, 2022
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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.

Calculation rule in quantum mechanics
Not to be confused with Cauchy–Born rule or Born approximation.
Part of a series of articles about
Quantum mechanics

iℏ∂∂t|ψ(t)⟩=H^|ψ(t)⟩{displaystyle ihbar {frac {partial }{partial t}}|psi (t)rangle ={hat {H}}|psi (t)rangle }

. . . Born rule . . .

The Born rule states that if an observable corresponding to a self-adjoint operator

A{displaystyle A}

with discrete spectrum is measured in a system with normalized wave function

|ψ⟩{displaystyle |psi rangle }

(see Bra–ket notation), then

  • the measured result will be one of the eigenvalues
    λ{displaystyle lambda }

    of

    A{displaystyle A}

    , and

  • the probability of measuring a given eigenvalue
    λi{displaystyle lambda _{i}}

    will equal

    ⟨ψ|Pi|ψ⟩{displaystyle langle psi |P_{i}|psi rangle }

    , where

    Pi{displaystyle P_{i}}

    is the projection onto the eigenspace of

    A{displaystyle A}

    corresponding to

    λi{displaystyle lambda _{i}}

    .

(In the case where the eigenspace of

A{displaystyle A}

corresponding to

λi{displaystyle lambda _{i}}

is one-dimensional and spanned by the normalized eigenvector

|λi⟩{displaystyle |lambda _{i}rangle }

,

Pi{displaystyle P_{i}}

is equal to

|λi⟩⟨λi|{displaystyle |lambda _{i}rangle langle lambda _{i}|}

, so the probability

⟨ψ|Pi|ψ⟩{displaystyle langle psi |P_{i}|psi rangle }

is equal to

⟨ψ|λi⟩⟨λi|ψ⟩{displaystyle langle psi |lambda _{i}rangle langle lambda _{i}|psi rangle }

. Since the complex number

⟨λi|ψ⟩{displaystyle langle lambda _{i}|psi rangle }

is known as the probability amplitude that the state vector

|ψ⟩{displaystyle |psi rangle }

assigns to the eigenvector

|λi⟩{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

|⟨λi|ψ⟩|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

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

    ⟨ψ|Q(M)|ψ⟩{displaystyle langle psi |Q(M)|psi rangle }

    .

A wave function

ψ{displaystyle psi }

for a single structureless particle in position space implies that the probability density function

p(x,y,z){displaystyle p(x,y,z)}

for a measurement of the position at time

t0{displaystyle t_{0}}

is

p(x,y,z)=|ψ(x,y,z,t0)|2.{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-definitematrices

{Fi}{displaystyle {F_{i}}}

on a Hilbert space

H{displaystyle {mathcal {H}}}

that sum to the identity matrix,[3]: 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

ρ{displaystyle rho }

is given by

p(i)=tr⁡(ρFi),{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

|ψ⟩{displaystyle |psi rangle }

this formula reduces to

p(i)=tr⁡(|ψ⟩⟨ψ|Fi)=⟨ψ|Fi|ψ⟩.{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, 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).

. . . Born rule . . .

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. . . Born rule . . .