Wigner Function for Harmonic Oscillator and The Classical Limit

TL;DR

This paper demonstrates how the Wigner function for a quantum harmonic oscillator approaches the classical microcanonical ensemble in the high-energy limit, providing insights into quantum-classical correspondence.

Contribution

It explicitly derives the Wigner function for all states and shows its classical limit using semi-classical wavefunctions for highly excited states.

Findings

Wigner function is exactly found for all states

Classical limit corresponds to the microcanonical ensemble

Semi-classical wavefunctions effectively describe the transition to classical behavior

Abstract

The Wigner function is a quantum analogue of the classical joined distribution of position and momentum. As such is should be a good tool to study quantum-classical correspondence. In this paper, the classical limit of the Wigner function is shown using the quantum harmonic oscillator as an example. The Wigner function is found exactly for all states. The semi-classical wavefunctions for highly excited states are used as the approach to the classical limit. Therefore, one can found the classical limit of the Wigner function for highly excited states and shown that it gives the classical microcanonical ensemble.