Study on linear canonical transformation in a framework of a phase space representation of quantum mechanics

TL;DR

This paper explores linear canonical transformations within a phase space framework of quantum mechanics, detailing their effects on quantum states and introducing the concept of isodispersion transformations.

Contribution

It extends phase space quantum mechanics by defining transformation laws and introducing isodispersion linear canonical transformations, enriching the theoretical understanding.

Findings

Transformation laws for mean values and dispersions derived

Basis states and wave functions transformation laws established

Introduction of isodispersion linear canonical transformation

Abstract

We present a study on linear canonical transformation in the framework of a phase space representation of quantum mechanics that we have introduced in our previous work [1]. We begin with a brief recall about the so called phase space representation. We give the definition of linear canonical transformation with the transformation law of coordinate and momentum operators. We establish successively the transformation laws of mean values, dispersions, basis state and wave functions.Then we introduce the concept of isodispersion linear canonical transformation.