Exact solution of the position-dependent mass Schrödinger equation with the completely positive oscillator-shaped quantum well potential

Abstract

Two exactly-solvable confined models of the completely positive oscillator-shaped quantum well are proposed. Exact solutions of the position-dependent mass Schrödinger 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 non-equidistant 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.

11 pages, 4 figures

References in corpus (7)

• Inverted Oscillator
• Generalized nonlinear oscillators with quasi-harmonic behaviour: classical solutions
• Quantum solvability of a general ordered position dependent mass system: Mathews-Lakshmanan oscillator
• A finite oscillator model related to sl(2|1)
• Discrete series representations for sl(2|1), Meixner polynomials and oscillator models
• On the exactly-solvable semi-infinite quantum well of the non-rectangular step-harmonic profile
• The Husimi function of a semiconfined harmonic oscillator model with a position-dependent effective mass

Cited by in corpus (2)

• Deformation of the Heisenberg-Weyl algebra and the Lie superalgebra $\mathfrak{osp}\left(1|2\right)$: exact solution for the quantum harmonic oscillator with a position-dependent mass
• Spectral and thermodynamic properties of supersymmetric quantum systems with self-adjoint deformed momentum