Non-Hermitian quantum canonical variables and the generalized ladder operators

TL;DR

This paper explores non-Hermitian quantum canonical variables and generalized ladder operators, showing how they resolve operator ordering issues and facilitate solving eigenvalue problems in quantum mechanics.

Contribution

It introduces a framework for non-Hermitian realizations of canonical variables that generalize ladder operators and address operator ordering ambiguities.

Findings

Operator ordering problem is resolved with non-Hermitian realizations.

Generalized ladder operators enable neat solutions to eigenvalue problems.

Non-Hermitian representations allow canonical quantizations of non-measurable variables.

Abstract

Quantum canonical transformations of the second kind and the non-Hermitian realizations of the basic canonical commutation relations are investigated with a special interest in the generalization of the conventional ladder operators. The operator ordering problem is shown to be resolved when the non-Hermitian realizations for the canonical variables which can not be measured simultaneously with the energy are chosen for the canonical quantizations. Another merit of the non-Hermitian representations is that it naturally allows us to introduce the generalized ladder operators with which one can solve eigenvalue problems quite neatly.