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Concept
Born rule
Summary
The Born rule (also called Born's rule) is a postulate of quantum mechanics which gives the probability that a measurement of a quantum system will yield a given result. In its simplest form, it states that the probability density of finding a system in a given state, when measured, is proportional to the square of the amplitude of the system's wavefunction at that state. It was formulated by German physicist Max Born in 1926.
The Born rule states that if an observable corresponding to a self-adjoint operator with discrete spectrum is measured in a system with normalized wave function (see Bra–ket notation), then:
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 , the spectral measure of . In this case:
the probability that the result of the measurement lies in a measurable set is given by .
A wave function for a single structureless particle in space position implies that the probability density function for a measurement of the particles's position at time is:
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.
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