Canonical Transformations in Quantum Mechanics

TL;DR

This paper extends quantum canonical transformations algebraically beyond Hilbert spaces, including non-unitary transformations, and demonstrates their role in solving Schrödinger equations through elementary transformations and examples.

Contribution

It introduces a generalized algebraic framework for quantum canonical transformations, encompassing non-unitary cases, and links them to solving differential equations in quantum mechanics.

Findings

Elementary transformations have quantum implementations as finite transformations.

General transformations can be constructed as products of elementary ones.

The method simplifies solving Schrödinger equations using canonical transformations.

Abstract

Quantum canonical transformations are defined algebraically outside of a Hilbert space context. This generalizes the quantum canonical transformations of Weyl and Dirac to include non-unitary transformations. The importance of non-unitary transformations for constructing solutions of the Schr\"odinger equation is discussed. Three elementary canonical transformations are shown both to have quantum implementations as finite transformations and to generate, classically and infinitesimally, the full canonical algebra. A general canonical transformation can be realized quantum mechanically as a product of these transformations. Each transformation corresponds to a familiar tool used in solving differential equations, and the procedure of solving a differential equation is systematized by the use of the canonical transformations. Several examples are done to illustrate the use of the canonical transformations. [This is an extensively revised version of hep-th-9205080: the first third of the paper is new material; the notation has been simplified, and further discussion has been added to the remainder.]