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Results in Physics

A novel approach for constructing kinetic energy operators with position dependent mass

YASSINE CHARGUI, A. Dhahbi, A. Trabelsi

Published May 8, 2019• 7 citations
Publisher

Abstract

The interest in physical systems with position-dependent mass (PDM) is always topical. In this work we present a new way to construct kinetic energy operators (KEO’s) for particles endowed with a position varying mass within the Schrödinger equation. Taking recent developments of non-Hermitian quantum mechanics into account, we start with a non-Hermitian Hamiltonian and we recall that it is quasi-Hermitian and can be always associated to a Hermitian partner. The latter is then used along with the formalism of supersymmetric quantum mechanics (SUSY QM) to generate exactly solvable models on the basis of shape invariance condition. Some known and new concrete examples are considered as illustrations of the proposed scheme.

DOI
10.1016/j.rinp.2019.102329

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The interest in physical systems with position-dependent mass (PDM) is always topical. In this work we present a new way to construct kinetic energy operators (KEO’s) for particles endowed with a position varying mass within the Schrödinger equation. Taking recent developments of non-Hermitian quantum mechanics into account, we start with a non-Hermitian Hamiltonian and we recall that it is quasi-Hermitian and can be always associated to a Hermitian partner. The latter is then used along with the formalism of supersymmetric quantum mechanics (SUSY QM) to generate exactly solvable models on the basis of shape invariance condition. Some known and new concrete examples are considered as illustrations of the proposed scheme.
The study of physical systems endowed with a position-dependent mass (PDM) remains a fundamental issue of quantum mechanics. In this paper we use a new approach, recently developed by us for building the quantum kinetic energy operator (KEO) within the Schrodinger equation, in order to construct a new class of exactly solvable models with a position varying mass, presenting a harmonic-oscillator-like spectrum. To do so we utilize the formalism of supersymmetric quantum mechanics (SUSY QM) along with the shape invariance condition. Recent outcomes of non-Hermitian quantum mechanics are also taken into account.
In this paper we present a complete classification scheme for kinetic energy operators (KEO's) describing a particle endowed with position-dependent mass (PDM). We first present a generalized formulation of KEO's with PDM and show that it is equivalent to three-parameter linear formulation. This reduces further to two independent parameters under Hermiticity condition. Based on this linear formulation we prove that, contrary to what was widely believed, von Roos family is not the most general ordering. We found an entire new family of Hermitian KEO's that does not fit into von Roos ordering. We were able to construct all the Hermitian KEO's and classify them into two-parameter classes. As one application, we solve the puzzling case of Yang and Yee KEO. We also find, under certain conditions, some kind of duality between von Roos ordering and one of the new classes.
A new method for generating exactly solvable Schrödinger equations with a position-dependent mass is proposed. It is based on a relation with some deformed Schrödinger equations, which can be dealt with by using a supersymmetric quantum mechanical approach combined with a deformed shape-invariance condition. The solvability of the latter is shown to impose the form of both the deformed superpotential and the position-dependent mass. The conditions for the existence of bound states are determined. A lot of examples are provided and the corresponding bound-state spectrum and wavefunctions are reviewed.
Quantum dots, liquid crystals, compositionally graded crystals, and other condensed matter systems all use position dependent effective mass (PDEM) Hamiltonians to describe the dynamics of electrons. Because the PDEM quantum Hamiltonians are not Hermitian, we employ the effective mass kinetic energy operator in Von Ross?s two-parameter form, which is Hermitian by default and contains additional reasonable forms as special instances. It is shown that Hamiltonians of the form H(s)=(1-s)H_-+sH_+,0?s?1 where H_? are supersymmetric partner Hamiltonians corresponding to position dependent mass Schr?dinger equations are exactly solvable for a number of deformed shape invariant 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.
We consider the application of the supersymmetric quantum-mechanical formalism to the Schr\"odinger equation describing a particle characterized by a position-dependent effective mass m(x).m(x).m(x). We show that any one-dimensional quantum system with effective mass has a supersymmetric partner system characterized by the same position dependence of the mass, but with a new potential function. The form of this supersymmetric partner potential V2(x){V}_{2}(x)V2​(x) depends on both the form of the original potential V1(x){V}_{1}(x)V1​(x) and the form of the mass x dependence. We also generalize the concept of shape invariance to the nonconstant mass scenario. As illustrative examples we provide, for a given form m(x)m(x)m(x) of the effective mass, shape-invariant potentials exhibiting (a) harmonic-oscillator-like spectra and (b) Morse-like spectra. In both cases the energy eigenvalues and eigenfunctions can be obtained in algebraic fashion.