Comment on `Exact solution of the position-dependent effective mass and angular frequency Schr\"odinger equation: harmonic oscillator model with quantized confinement parameter'

TL;DR

This paper critiques a recent study on a position-dependent mass harmonic oscillator, demonstrating alternative derivations that avoid quantizing the confinement parameter and extending the method to a shifted oscillator.

Contribution

It provides a simpler derivation of the confined harmonic oscillator model using point canonical transformation, avoiding quantization of the confinement parameter, and extends the approach to a shifted oscillator.

Findings

Alternative derivation without quantizing the confinement parameter

Extension to a confined shifted harmonic oscillator

Simplification of the original model's methodology

Abstract

In a recent paper by Jafarov, Nagiyev, Oste and Van der Jeugt (2020 {\sl J.\ Phys.\ A} {\bf 53} 485301), a confined model of the non-relativistic quantum harmonic oscillator, where the effective mass and the angular frequency are dependent on the position, was constructed and it was shown that the confinement parameter gets quantized. By using a point canonical transformation starting from the constant-mass Schr\"odinger equation for the Rosen-Morse II potential, it is shown here that similar results can be easily obtained without quantizing the confinement parameter. In addition, an extension to a confined shifted harmonic oscillator directly follows from the same point canonical transformation.