The continuity equation and its applications in phase space quantum mechanics

TL;DR

This paper derives a phase space version of the continuity equation for quantum systems with internal degrees of freedom, introduces a phase space Dirac equation, and analyzes properties of Wigner functions in various scenarios.

Contribution

It presents a novel phase space formulation of the Dirac equation and continuity equation, expanding quantum mechanics tools for systems with internal degrees of freedom.

Findings

Derived phase space continuity equation for quantum systems.

Formulated phase space Dirac equation.

Analyzed Wigner functions of unbound states.

Abstract

A quantum phase space version of the continuity equation for systems with internal degrees of freedom is derived. The $1$ -- D Dirac equation is introduced and its phase space counterpart is found. The phase space representation of free motion and of scattering in a nonrelativistic and relativistic case for setups with internal degrees of freedom is discussed and illustrated. Properties of Wigner functions of unbound states are analysed.