Wigner Function and Phase Space: ISW

The ordinary position-space quantum wave function can be transformed into the momentum-space wave function using the Fourier transform:

eq17

A similar-looking construct, but with a different interpretation, was conceived by Wigner over seventy years ago

eq18

This construct continues to interest and be useful physicists today.  The function introduced by Wigner can be interpreted as a quasi- or pseudo-probability distribution corresponding to a general quantum state.  The Wigner distribution allows us to study quantum correlations to classical mechanics by giving a phase-space description of quantum mechanics with a quasi-probability distribution that is joint in both x and p even though the simultaneous measurement of position and momentum violates the uncertainty principle. The Wigner function is real and integral over x (p) yields the correct momentum-space (position-space) probability density.  It can be negative, however, which points to the quantum-mechanical aspects of the system.  Some authors have suggested that the negative part of the Wigner function can be used to numerically describe the non-classicality of a quantum-mechanical system.

The Wigner Function program was developed by Wolfgang Christian (Davidson College) using the Open Source Physics Java code library and published by Mario Belloni (Davidson College) and Wolfgang Christian in a package for the teaching of the time evolution and visualization of energy eigenstates and their superpositions in quantum mechanics https://www.compadre.org/osp/items/detail.cfm?ID=7302. It was converted from Java to JavaScript by Wolfgang Christian and Robert Hanson (St. Olaf College) using the SwingJS system developed at St. Olaf College.