Effective mass of the discrete values as a hidden feature of the one-dimensional harmonic oscillator model: Exact solution of the Schrödinger equation with a mass varying by position

Abstract

In this paper, exactly solvable model of the quantum harmonic oscillator is proposed. Wave functions of the stationary states and energy spectrum of the model are obtained through the solution of the corresponding Schrödinger equation with the assumption that the mass of the quantum oscillator system varies with position. We have shown that the solution of the Schrödinger equation in terms of the wave functions of the stationary states is expressed by the pseudo Jacobi polynomials and the mass varying with position depends from the positive integer [Formula: see text]. As a consequence of the positive integer [Formula: see text], energy spectrum is not only non-equidistant, but also there are only a finite number of energy levels. Under the limit, when [Formula: see text], the dependence of effective mass from the position disappears and the system recovers known non-relativistic quantum harmonic oscillator in the canonical approach where wave functions are expressed by the Hermite polynomials.