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arXiv:2010.04477 · Journal of Physics A Mathematical and Theoretical

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

E I Jafarov, Sh. M. Nagiyev, R Oste, J Van der Jeugt

Published Sep 30, 2020• 30 citations
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Abstract

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.

DOI
10.1088/1751-8121/abbd1a

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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.
Abstract In a recent paper by Jafarov et al (2020 J. Phys. A 53 485301), a confined model of the non-relativistic quantum harmonic oscillator, where the effective mass and the angular frequency are dependent on the position, was constructed and it was shown that the confinement parameter gets quantized. By using a point canonical transformation starting from the constant-mass Schrödinger equation for the Rosen–Morse II potential, it is shown here that similar results can be easily obtained without quantizing the confinement parameter. In addition, an extension to a confined shifted harmonic oscillator directly follows from the same point canonical transformation.
In a recent paper by Jafarov, Nagiyev, Oste and Van der Jeugt (2020 {\sl J.\ Phys.\ A} {\bf 53} 485301), a confined model of the non-relativistic quantum harmonic oscillator, where the effective mass and the angular frequency are dependent on the position, was constructed and it was shown that the confinement parameter gets quantized. By using a point canonical transformation starting from the constant-mass Schrödinger equation for the Rosen-Morse II potential, it is shown here that similar results can be easily obtained without quantizing the confinement parameter. In addition, an extension to a confined shifted harmonic oscillator directly follows from the same point canonical transformation.
An external electric field is provided to the familiar one-dimensional quantum harmonic oscillator model with a various position mass m(x)=(a m_0)/(a+x). The Nikiforov-Uvarov approach is involved to investigate a particular solution to the exact Schrödinger equation. The exactly solvable confined model of the quantum harmonic oscillator in an external electric field was proposed. Starting with the BenDanial-Duke kinetic energy operator approximations, the construction of the position-dependent mass Schrödinger equation is studied. The analytic representation of the wave functions of the stationary states is expressed analytically and graphically using modified Laguerre polynomials as well as the energy spectrum. In contrast with the absence of an external electric field, when the energy spectrum totally overlaps with that of the harmonic oscillator potential in an external electric field: the energy spectrum becomes non-equidistant and varies depending on some factors. The Nikiforov-Uvarov approach succeeds broadly in demonstrating the wave function and energy spectrum and showing good sense.
We constructed a new exactly solvable model of the non-relativistic quantum linear harmonic oscillator with the position-dependent mass Mx=2m0/eαx+1. We solved the Schrödinger equation for this model in the absence and presence of an external homogeneous field. We showed that in both cases the wave functions of the bound states are expressed through Jacobi polynomials, and the corresponding energy spectra are non-equidistant. We also showed that in the limit α→0 the wave functions and energy spectra coincide with the corresponding expressions for a linear harmonic oscillator with the constant mass m0 in the absence and presence of an external homogeneous field. We present also two new limit relations that reduce the Jacobi polynomials directly to the Hermite polynomials with shifted and non-shifted arguments. The proofs of these limit relations are based on the method of mathematical induction.
"Two exactly-solvable confined models of the completely positive oscillator-shaped quantum well are proposed. Exact solutions of the position-dependent mass Schrodinger equation corresponding to the proposed quantum well potentials are ¨ presented. It is shown that the discrete energy spectrum expressions of both models depend on certain positive confinement parameters. The spectrum exhibits positive equidistant behavior for the model confined only with one infinitely high wall and nonequidistant behavior for the model confined with the infinitely high wall from both sides. Wavefunctions of the stationary states of the models under construction are expressed through the Laguerre and Jacobi polynomials. In general, the Jacobi polynomials appearing in wavefunctions depend on parameters a and b, but the Laguerre polynomials depend only on the parameter a. Some limits and special cases of the constructed models are discussed."