Properties of Phase Space Wavefunctions and Eigenvalue Equation of Momentum Dispersion Operator

TL;DR

This paper explores the properties of phase space wavefunctions and derives the eigenvalue equation for the momentum dispersion operator within quantum mechanics, extending previous theoretical work.

Contribution

It introduces the eigenvalue equation for the momentum dispersion operator in phase space and analyzes properties of phase space wavefunctions, building on prior research.

Findings

Most phase space wavefunctions satisfy the eigenvalue equation.

Properties of wavefunctions in phase space are systematically characterized.

Examples illustrate the theoretical results.

Abstract

This paper is a continuation of our previous works about coordinate, momentum, dispersion operators and phase space representation of quantum mechanics. It concerns a study on the properties of wavefunctions in the phase space representation and the momentum dispersion operator, its representations and eigenvalue equation. After the recall of some results from our previous papers, we give most of the main properties of the phase space wavefunctions and consider some examples of them. Then we establish the eigenvalue equation for the differential operator corresponding to the momentum dispersion operator in the phase space representation. It is shown in particular that any phase space wavefunction is solution of this equation.