Canonical transformations in quantum mechanics

TL;DR

This paper develops a comprehensive theory of canonical transformations in quantum mechanics within phase space formalism, showing how observables and states transform similarly to classical mechanics, and illustrates the theory with specific examples.

Contribution

It introduces a unified formalism for quantum canonical transformations in phase space and standard quantum mechanics, expanding the understanding of coordinate changes in quantum systems.

Findings

Transformations of star-products correspond to coordinate changes in phase space.

Quantum observables and states transform via composition with coordinate transformations.

The formalism is demonstrated through specific examples of quantum canonical transformations.

Abstract

This paper presents the general theory of canonical transformations of coordinates in quantum mechanics. First, the theory is developed in the formalism of phase space quantum mechanics. It is shown that by transforming a star-product, when passing to a new coordinate system, observables and states transform as in classical mechanics, i.e., by composing them with a transformation of coordinates. Then the developed formalism of coordinate transformations is transferred to a standard formulation of quantum mechanics. In addition, the developed theory is illustrated on examples of particular classes of quantum canonical transformations.