Wigner Quasiprobability Distribution - Definition and Meaning

Definition and Meaning

The Wigner distribution P(x, p) is defined as:

where ψ is the wavefunction and x and p are position and momentum but could be any conjugate variable pair. (i.e. real and imaginary parts of the electric field or frequency and time of a signal). Note it may have support in x even in regions where ψ has no support in x ("beats").

It is symmetric in x and p:

where φ is the Fourier transform of ψ.

In 3D,

In the general case, which includes mixed states, it is the Wigner transform of the density matrix:

This Wigner transformation (or map) is the inverse of the Weyl transform, which maps phase-space functions to Hilbert-space operators, in Weyl quantization. Thus, the Wigner function is the cornerstone of quantum mechanics in phase space.

In 1949, José Enrique Moyal elucidated how the Wigner function provides the integration measure (analogous to a probability density function) in phase space, to yield expectation values from phase-space c-number functions g(x,p) uniquely associated to suitably ordered operators through Weyl's transform (cf. Weyl quantization and property 7 below), in a manner evocative of classical probability theory.

Specifically, an operator's expectation value is a "phase-space average" of the Wigner transform of that operator,