Exact solution of the semiconfined harmonic oscillator model with a position-dependent effective mass in an external homogeneous field

Abstract

We extend exactly solvable model of a one-dimensional non-relativistic canonical semiconfined quantum harmonic oscillator with a mass that varies with position to the case where an external homogeneous field is applied. The problem is still exactly solvable and the analytic expression of the wave functions of the stationary states is expressed by means of generalised Laguerre polynomials, too. Unlike the case without any external field, when the energy spectrum completely overlaps with the energy spectrum of the standard non-relativistic canonical quantum harmonic oscillator, the energy spectrum is now still equidistant but depends on the semiconfinement parameter a. We also compute probabilities of the transitions for the model under the external field and discuss limit cases for the energy spectrum, wave functions and probabilities of transitions, when the semiconfinement parameter a goes to infinity.