PDM creation and annihilation operators of the harmonic oscillators and the emergence of an alternative PDM-Hamiltonian

TL;DR

This paper constructs and compares two approaches to define creation and annihilation operators for the position-dependent mass harmonic oscillator, revealing a new, simpler PDM-Hamiltonian with potential pedagogical and theoretical benefits.

Contribution

It introduces a novel, simplified position dependent mass Hamiltonian and develops two distinct methods for defining PDM creation and annihilation operators, expanding the theoretical framework.

Findings

New position dependent mass Hamiltonian identified

Two approaches yield consistent PDM creation and annihilation operators

Proposed Hamiltonian is simpler and more user-friendly

Abstract

The exact solvability and impressive pedagogical implementation of the harmonic oscillator creation and annihilation operators make it a problem of great physical relevance and the most fundamental one in quantum mechanics. So would be the position dependent mass oscillator for the position dependent mass quantum mechanics. We, hereby, construct the position dependent mass creation and annihilation operators for the position dependent mass oscillator via two different approaches. First, via von Roos position dependent mass Hamiltonian and show that the commutation relation between the position dependent mass creation and annihilation operators not only offers a unique position dependent mass Hamiltonian but also suggests a position dependent mass deformation in the coordinate system. Next, we use a position dependent mass point canonical transformation of the textbook constant mass harmonic oscillator analog and obtain yet another set of position dependent mass creation and annihilation operators, hence an apparently new position dependent mass Hamiltonian is obtained. The new position dependent mass Hamiltonian turned out to be not only correlated but also represents an alternative and most simplistic user-friendly position dependent mass Hamiltonian that has never been reported before.