Non-hermitian techniques of canonical transformations in quantum mechanics

TL;DR

This paper explores non-Hermitian operator methods to implement classical canonical transformations in quantum mechanics, demonstrating their utility through harmonic oscillator eigenvalue problems and quantum Hamilton-Jacobi theory.

Contribution

It introduces a non-Hermitian operator approach to quantum canonical transformations and applies classical Hamilton-Jacobi theory to quantum evolution problems.

Findings

Eigenvalue problem of harmonic oscillator solved using non-Hermitian transformations

Quantum propagator derived via classical action as generating function

Method simplifies solving time-dependent Schrödinger equations

Abstract

The quantum mechanical version of the four kinds of classical canonical transformations is investigated by using non-hermitian operator techniques. To help understand the usefulness of this appoach the eigenvalue problem of a harmonic oscillator is solved in two different types of canonical transformations. The quantum form of the classical Hamiton-Jacobi theory is also employed to solve time dependent Schr\"odinger wave equations, showing that when one uses the classical action as a generating function of the quantum canonical transformation of time evolutions of state vectors, the corresponding propagator can easily be obtained.