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 nonrelativistic 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 wavefunctions of the stationary states is expressed by means of generalized Laguerre polynomials, too. Unlike the case without any external field, when the energy spectrum completely overlaps with the energy spectrum of the standard nonrelativistic 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, wavefunctions and probabilities of transitions, when the semiconfinement parameter a goes to infinity.

The Husimi function of a semiconfined harmonic oscillator model with a position-dependent effective mass

“…We have to remind one of the main properties of the semiconfined quantum harmonic oscillator model [1,2] that is exhibited in the x-configuration space: it was observed from the distribution of the probability densities computed from the wavefunctions (3.1) and (3.3) that when the semiconfinement parameter a is close to zero, then the quantum system under study stays close to the infinitely high wall and as the value of the parameter a increased, the effect of the semiconfinement gradually disappeared and the behavior became harmonic oscillatorlike. One observes from fig.…”

“…Later, this model also was generalized to the case of the applied external homogeneous field [2] and the following analytical expression of the wavefunctions of the stationary states in terms of the generalized Laguerre polynomials have been obtained:…”

“…Its wavefunctions of the stationary states are expressed via the generalized Laguerre polynomials, but, surprisingly, its energy spectrum completely overlaps with the energy spectrum of the standard non-relativistic canonical quantum harmonic oscillator described above. Later, the same quantum oscillator model was generalized to the case of the applied external homogeneous field [2]. We observed that the wavefunctions of the oscillator model generalized via the applied external homogeneous field are still expressed through the generalized Laguerre polynomials, its energy spectrum is still equidistant but depends on the semiconfinement parameter a.…”

The Husimi function of a semiconfined harmonic oscillator model with a position-dependent effective mass

“…Next, this model also was generalized to the case of the applied external homogeneous field [19] and the following analytical expression of the wavefunctions of the stationary states in terms of the generalized Laguerre polynomials have been obtained:…”

“…Few papers discussing phase-space behavior of the oscillator-like quantum systems with the position-dependent mass also exist [14][15][16]. In [17], we computed the simplest Gaussian smoothed Wigner function for the oscillator model with a position-dependent effective mass exhibiting semiconfinement effect [18,19]. That simplest definition of the computed Gaussian smoothed Wigner function of the joint quasiprobability of momentum and position also called as Husimi function is well known, too [20].…”