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 provides an exact analytical solution for a quantum harmonic oscillator with position-dependent mass and frequency, revealing a quantized confinement parameter, a finite non-equidistant energy spectrum, and convergence to the standard oscillator in the limit.

Contribution

It introduces a novel exact solution for a confined harmonic oscillator with position-dependent parameters, preserving the force constant and revealing quantized confinement effects.

Findings

Finite, non-equidistant energy spectrum dependent on confinement parameter

Wave functions expressed in associated Legendre or Gegenbauer polynomials

Convergence to standard harmonic oscillator as confinement parameter approaches infinity

Abstract

We present an exact solution of a confined model of the non-relativistic quantum harmonic oscillator, where the effective mass and the angular frequency are dependent on the position. The free Hamiltonian of the proposed model has the form of the BenDaniel--Duke kinetic energy operator. The position-dependency of the mass and the angular frequency is such that the homogeneous nature of the harmonic oscillator force constant $k$ and hence the regular harmonic oscillator potential is preserved. As a consequence thereof, a quantization of the confinement parameter is observed. It is shown that the discrete energy spectrum of the confined harmonic oscillator with position-dependent mass and angular frequency is finite, has a non-equidistant form and depends on the confinement parameter. The wave functions of the stationary states of the confined oscillator with position-dependent mass and angular frequency are expressed in terms of the associated Legendre or Gegenbauer polynomials. In the limit where the confinement parameter tends to $\infty$, both the energy spectrum and the wave functions converge to the well-known equidistant energy spectrum and the wave functions of the stationary non-relativistic harmonic oscillator expressed in terms of Hermite polynomials. The position-dependent effective mass and angular frequency also become constant under this limit.