The Wentzel - Kramers - Brillouin approximation method applied to the Wigner function

TL;DR

This paper adapts the WKB approximation within the deformation quantization framework to solve the eigenvalue problem in phase space quantum mechanics, deriving formulas and analyzing properties of the Wigner function.

Contribution

It introduces a WKB-based method for phase space quantum mechanics, including formulas for Wigner functions of products and superpositions, and applies it to specific potentials.

Findings

Derived formulas for Wigner functions of superpositions and products

Analyzed properties of Wigner functions for interfering states

Obtained Wigner functions for unbound states in Pöschl-Teller potential

Abstract

An adaptation of the WKB method in the deformation quantization formalism is presented with the aim to obtain an approximate technique of solving the eigenvalue problem for energy in the phase space quantum approach. A relationship between the phase $\sigma(\vec{r})$ of a wave function $\exp \left(\frac{i}{\hbar} \sigma(\vec{r}) \right)$ and its respective Wigner function is derived. Formulas to calculate the Wigner function of a product and of a superposition of wave functions are proposed. Properties of a Wigner function of interfering states are also investigated. Examples of this quasi - classical approximation in deformation quantization are analysed. A strict form of the Wigner function for states represented by tempered generalised functions has been derived. Wigner functions of unbound states in the Poeschl - Teller potential have been found.