Explicit solution of the position-dependent mass Schrödinger equation with Gora-Williams kinetic energy operator: Confined harmonic oscillator model

Exactly-solvable confined model of the non-relativistic quantum harmonic oscillator is proposed. Free Hamiltonian of the system under study has a form of the Gora-Williams kinetic energy operator. Explicit solution of this confined harmonic oscillator Schrödinger equation in the canonical approach has achieved thanks to effective mass changing with position. Confinement effect also appears as a result of certain behaviour of the position-dependent effective mass depending from confinement parameter a. It is shown that the discrete energy spectrum of the confined harmonic oscillator with position-dependent mass also depends from confinement parameter and has a non-equidistant form. Wavefunctions of the stationary states of the confined oscillator with position-dependent mass are expressed in terms of the Gegenbauer polynomials. At limit a → ∞, both energy spectrum and wavefunctions recover well-known equidistant energy spectrum and wavefunctions of the stationary non-relativistic harmonic oscillator expressed by Hermite polynomials. Position-dependent effective mass also becomes homogeneous under this limit. © 2020, Politechnica University of Bucharest. All rights reserved.