Position-dependent mass harmonic oscillator: classical-quantum mechanical correspondence and ordering-ambiguity

TL;DR

This paper revisits the classical-quantum correspondence for position-dependent mass harmonic oscillators, clarifies the ordering ambiguity, and identifies a unique quantum Hamiltonian using point canonical transformation.

Contribution

It amends previous results on PDM oscillators, establishing a unique Hamiltonian and ordering parameters, and explores classical-quantum correspondence for PDM particles.

Findings

Unique quantum PDM oscillator Hamiltonian identified

Ordering ambiguity parameters determined as j=l=-1/4, k=-1/2

Classical-quantum correspondence established for PDM systems

Abstract

We recycle Cruz et al.'s (Phys. Lett. A 369 (2007) 400) work on the classical and quantum position-dependent mass (PDM) oscillators. To elaborate on the ordering ambiguity, we properly amend some of the results reported in their work and discuss the classical and quantum mechanical correspondence for the PDM harmonic oscillators. We use a point canonical transformation and show that one unique quantum PDM oscillator Hamiltonian (consequently, one unique ordering-ambiguity parametric set j=l=-1/4 and k=-1/2) is obtained. To show that such a parametric set is not just a manifestation of the quantum PDM oscillator Hamiltonian, we consider the classical and quantum mechanical correspondence for quasi-free PDM particles moving under the influence of their own PDM force fields.