Generalized Harmonic Oscillator and the Schr\"{o}dinger Equation with Position-Dependent Mass

TL;DR

This paper explores a generalized harmonic oscillator with position-dependent mass, deriving eigenvalues, eigenfunctions, and coherent states, revealing systems that are isospectral to the standard harmonic oscillator.

Contribution

It provides explicit solutions and analyzes properties of a harmonic oscillator with position-dependent mass, extending the traditional model to more general mass functions.

Findings

Eigenvalues and eigenfunctions similar to constant mass oscillator

Coherent states analyzed for PDM systems

Certain mass functions lead to isospectral potentials

Abstract

We study the generalized harmonic oscillator which has both the position-dependent mass and the potential depending on the form of mass function in a more general framework. The explicit expressions of the eigenvalue and eigenfunction for such system are given, they have the same forms as those for the usual harmonic oscillator with constant mass. The coherent state and the its properties for the system with PDM are also discussed. We give the corresponding effective potentials for several mass functions, the systems with such potentials are isospectral to the usual harmonic oscillator.