Exact solution of the position-dependent effective mass and angular frequency Schrödinger equation: harmonic oscillator model with quantized confinement parameter

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 ∞, 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.

Comment on `Exact solution of the position-dependent effective mass and angular frequency Schrödinger equation: harmonic oscillator model with quantized confinement parameter'

“…We conclude that the results of Ref. [1] can be derived by a PCT applied to the constantmass Schrödinger equation for the Rosen-Morse II potential with B = 0 and any A = l = 2, 3, . .…”

“…In conclusion, we have shown that the confined harmonic oscillator with positiondependent mass and angular frequency of Ref. [1] can be extended to the case where the confinement parameter is not quantized, its spectrum remaining finite, non-equidistant, and dependent on the confinement parameter. A further extension to a confined shifted harmonic oscillator with similar characteristics has also been constructed.…”

On the Direct Limit from Pseudo Jacobi Polynomials to Hermite Polynomials

“…Before presenting the main theorem, we have a lemma with two transformation formulas (which appeared already in [11], but were not derived there).…”

Generalized semiconfined harmonic oscillator model with a position-dependent effective mas

“…One of the most powerful techniques for such a purpose consists in applying a point canonical transformation (PCT) to an exactly solvable constant-mass Schrödinger equation [21,22]. Recently, such an approach has proved its efficiency again by providing a straightforward generalization [23] of a harmonic oscillator model wherein both the mass and the angular frequency are dependent on the position [24].…”