The Rule
The Born rule states that if an observable corresponding to a Hermitian 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 telling us 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 will be given by .
If we are given a wave function for a single structureless particle in position space, this reduces to saying that the probability density function for a measurement of the position at time will be given by
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