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arXiv:2208.07875

Nonrelativistic and Nonstationary Effective Mass Bound-Spectra Analysis of Squared Class Trigonometric Potentials Through the Point Canonical Formalism

Metin Aktaş

Published Aug 16, 2022• 0 citations
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Abstract

The present paper engages in a particular attempt to acquire exact analytical eigensolutions of the position-dependent effective mass (PDEM) Schrödinger equation for a variety of squared style trigonometric potentials. The algebraic process entitled as the point canonical transformation (PCT) approach is implemented in the course of study. Certain spatially varying effective mass configurations have been utilized in establishing of the target system. Then, performing the systematic computational procedures enables us to determine not only the possible explicit forms of both discrete energy spectra and their corresponding wavefunctions but also canonical counterparts of original potentials, involved in the framework of PDEM based quantum systems.

DOI
10.48550/arxiv.2208.07875

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The present paper engages in a particular attempt to acquire exact analytical eigensolutions of the position-dependent effective mass (PDEM) Schrödinger equation for a variety of squared style trigonometric potentials. The algebraic process entitled as the point canonical transformation (PCT) approach is implemented in the course of study. Certain spatially varying effective mass configurations have been utilized in establishing of the target system. Then, performing the systematic computational procedures enables us to determine not only the possible explicit forms of both discrete energy spectra and their corresponding wavefunctions but also canonical counterparts of original potentials, involved in the framework of PDEM based quantum systems.
The point canonical transformation (PCT) approach is used to solve the Schrödinger equation for an arbitrary dimension D with a power-law position-dependent effective mass (PDEM) distribution function for the pseudoharmonic and modified Kratzer (Mie-type) diatomic molecular potentials. In mapping the transformed exactly solvable D-dimensional (D ≥ 2) Schrödinger equation with constant mass into the effective mass equation by using a proper transformation, the exact bound state solutions including the energy eigenvalues and corresponding wave functions are derived. The well-known pseudoharmonic and modified Kratzer exact eigenstates of various dimensionality is manifested.
We study the one-dimensional non-Hermitian imaginary potential with a real energy spectrum in the framework of the position-dependent effective mass Dirac equation. The Dirac equation is mapped into the exactly solvable Schrödinger-like equation endowed with position-dependent effective mass that we present a new procedure to solve it. The point canonical transformation in non-relativistic quantum mechanics is applied as an algebraic method to obtain the mass function and then by using the obtained mass function, the imaginary potential can be obtained. The spinor wavefunctions for some of the obtained electrostatic potentials are given in terms of orthogonal polynomials. We also obtain the relativistic bound state spectrum for each case in terms of the bound state spectrum of the solvable potentials.
On using the known equivalence between the presence of a position-dependent mass (PDM) in the Schrödinger equation and a deformation of the canonical commutation relations, a method based on deformed shape invariance has recently been devised for generating pairs of potential and PDM for which the Schrödinger equation is exactly solvable.This approach has provided the bound-state energy spectrum, as well as the ground-state and the first few excited-state wavefunctions.The general wavefunctions have however remained unknown in explicit form because for their determination one would need the solutions of a rather tricky differential-difference equation.Here we show that solving this equation may be avoided by combining the deformed shape invariance technique with the point canonical transformation method in a novel way.It consists in employing our previous knowledge of the PDM problem energy spectrum to construct a constant-mass Schrödinger equation with similar characteristics and in deducing the PDM wavefunctions from the known constantmass ones.Finally, the equivalence of the wavefunctions coming from both approaches is checked.
A systematic procedure to study one-dimensional Schrodinger equation with a position-dependent effective mass (PDEM) in the kinetic energy operator is explored. The conventional free-particle problem reveals an interesting situation in that, in the presence of a mass background, formation of bound states is signalled. We also discuss coordinate-transformed, constant-mass Schrodinger equation, its matching with the PDEM form and the consequent decoupling of the ambiguity parameters. Many exact results are derived in this way.