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. 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.
(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
, 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 position space implies that the probability density function
for a measurement of the position at time
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. They are extensively used in the field of quantum information.
on a Hilbert space
that sum to the identity matrix,: 90
The POVM element
is associated with the measurement outcome
, such that the probability of obtaining it when making a measurement on the quantum state
is given by
is the trace operator. This is the POVM version of the Born rule. When the quantum state being measured is a pure state
this formula reduces to
The Born rule, together with the unitarity of the time evolution operator
(or, equivalently, the Hamiltonian
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).