Position-dependent mass momentum operator and minimal coupling: point canonical transformation and isospectrality

TL;DR

This paper explores the position-dependent mass quantum Hamiltonians, resolving ordering ambiguities, and investigates how minimal coupling with electromagnetic fields differs in classical and quantum contexts using point canonical transformations.

Contribution

It introduces a method to determine the ordering ambiguity parameters of the von Roos Hamiltonian and identifies the form of minimal coupling in quantum PDM systems.

Findings

The von Roos ambiguity parameters are strictly determined.

The form of minimal coupling differs between classical and quantum PDM Hamiltonians.

Only one of the two common vector potentials is suitable under the transformation.

Abstract

The classical and quantum mechanical correspondence for constant mass settings is used, along with some point canonical transformation, to find the position-dependent mass (PDM) classical and quantum Hamiltonians. The comparison between the resulting quantum PDM-Hamiltonian and the von Roos PDM-Hamiltonian implied that the ordering ambiguity parameters of von Roos are strictly determined. Eliminating, in effect, the ordering ambiguity associated with the von Roos PDM-Hamiltonian. This, consequently, played a vital role in the construction and identification of the PDM-momentum operator. The same recipe is followed to identify the form of the minimal coupling of electromagnetic interactions for the classical and quantum PDM-Hamiltonians. It turned out that whilst the minimal coupling may very well inherit the usual form in classical mechanics, it admits a necessarily different and vital form in quantum mechanics. Under our point transformation settings, only one of the two commonly used vector potentialsis found eligible and is considered for our Illustrative examples.