Wigner 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.
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).
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.
A classical particle has a definite position and momentum, and hence it is represented by a point in phase space.
For instance, the Wigner distribution can and normally does take on negative values for states which have no classical model—and is a convenient indicator of quantum-mechanical interference.
Smoothing the Wigner distribution through a filter of size larger than ħ (e.g., convolving with a phase-space Gaussian, a Weierstrass transform, to yield the Husimi representation, below), results in a positive-semidefinite function, i.e., it may be thought to have been coarsened to a semi-classical one.
They are shielded by the uncertainty principle, which does not allow precise location within phase-space regions smaller than ħ, and thus renders such "negative probabilities" less paradoxical.
where ψ is the wavefunction, and x and p are position and momentum, but could be any conjugate variable pair (e.g. real and imaginary parts of the electric field or frequency and time of a signal).
It is symmetric in x and p: where φ is the normalized momentum-space wave function, proportional to the Fourier transform of ψ.
In 3D, In the general case, which includes mixed states, it is the Wigner transform of the density matrix:
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 Ĝ through Weyl's transform (see Wigner–Weyl transform and property 7 below), in a manner evocative of classical probability theory.
Thus the trace of an operator with the density matrix Wigner-transforms to the equivalent phase-space integral overlap of g(x, p) with the Wigner function.
This method transfers to quantum systems, where the characteristics' "trajectories" now determine the evolution of Wigner functions.
The solution of the Moyal evolution equation for the Wigner function is represented formally as where
-products, successive operations by them have been adapted to a phase-space path integral, to solve the evolution equation for the Wigner function[9] (see also [10][11][12]).
This non-local feature of Moyal time evolution[13] is illustrated in the gallery below, for Hamiltonians more complex than the harmonic oscillator.
In the classical limit, the trajectory nature of the time evolution of Wigner functions becomes more and more distinct.
This same time evolution occurs with quantum states of light modes, which are harmonic oscillators.
[14][15] It has been suggested that the Wigner function approach can be viewed as a quantum analogy to the operatorial formulation of classical mechanics introduced in 1932 by Bernard Koopman and John von Neumann: the time evolution of the Wigner function approaches, in the limit ħ → 0, the time evolution of the Koopman–von Neumann wavefunction of a classical particle.
The Wigner representation is thus very well suited for making semi-classical approximations in quantum optics[17] and field theory of Bose-Einstein condensates where high mode occupation approaches a semiclassical limit.
[18] As already noted, the Wigner function of quantum state typically takes some negative values.
Thus, pure states with non-negative Wigner functions are not necessarily minimum-uncertainty states in the sense of the Heisenberg uncertainty formula; rather, they give equality in the Schrödinger uncertainty formula, which includes an anticommutator term in addition to the commutator term.
(With careful definition of the respective variances, all pure-state Wigner functions lead to Heisenberg's inequality all the same.)
Thus, we obtain an explicit form of the Segal–Bargmann transform of any pure state whose Wigner function is non-negative.
We can then invert the Segal–Bargmann transform to obtain the claimed form of the position wave function.
[21] Basil Hiley has shown that the quasi-probability distribution may be understood as the density matrix re-expressed in terms of a mean position and momentum of a "cell" in phase space, and the de Broglie–Bohm interpretation allows one to describe the dynamics of the centers of such "cells".
As in the case of coordinate systems, on account of varying properties, several such have with various advantages for specific applications: Nevertheless, in some sense, the Wigner distribution holds a privileged position among all these distributions, since it is the only one whose requisite star-product drops out (integrates out by parts to effective unity) in the evaluation of expectation values, as illustrated above, and so can be visualized as a quasiprobability measure analogous to the classical ones.
As indicated, the formula for the Wigner function was independently derived several times in different contexts.
In fact, apparently, Wigner was unaware that even within the context of quantum theory, it had been introduced previously by Heisenberg and Dirac,[26][27] albeit purely formally: these two missed its significance, and that of its negative values, as they merely considered it as an approximation to the full quantum description of a system such as the atom.
