Wigner quasiprobability distribution

The Wigner quasiprobability distribution (also called the Wigner function or the Wigner–Ville distribution, after Eugene Wigner and Jean-André Ville) is a quasiprobability distribution. It was introduced by Eugene Wigner in 1932^[1] to study quantum corrections to classical statistical mechanics. The goal was to link the wavefunction that appears in Schrödinger's equation to a probability distribution in phase space.

It is a generating function for all spatial autocorrelation functions of a given quantum-mechanical wavefunction ψ(x). Thus, it maps^[2] on the quantum density matrix in the map between real phase-space functions and Hermitian operators introduced by Hermann Weyl in 1927,^[3] in a context related to representation theory in mathematics (see Weyl quantization). In effect, it is the Wigner–Weyl transform of the density matrix, so the realization of that operator in phase space. It was later rederived by Jean Ville in 1948 as a quadratic (in signal) representation of the local time-frequency energy of a signal,^[4] effectively a spectrogram.

In 1949, José Enrique Moyal, who had derived it independently, recognized it as the quantum moment-generating functional,^[5] and thus as the basis of an elegant encoding of all quantum expectation values, and hence quantum mechanics, in phase space (see Phase-space formulation). It has applications in statistical mechanics, quantum chemistry, quantum optics, classical optics and signal analysis in diverse fields, such as electrical engineering, seismology, time–frequency analysis for music signals, spectrograms in biology and speech processing, and engine design.