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International Journal of Science and Research (IJSR)

Supersymmetric Approach to Solve Interpolated Position - Dependent Mass Hamiltonians

Tapas Kumar Jana

Published Aug 5, 2022• 0 citations
Publisher

Abstract

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.

DOI
10.21275/sr22817170424

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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.
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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.
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.
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.
In this work, the quantum operator approach is applied to both, the position-dependent mass Schrödinger equation (PDMSE) and the Schrodinger equation with constant mass (CMSE). This fact enable us to find the factorization operators that relates both Hamiltonians by means of a kinetic energy operator that comes from the proposal of Morrow and Brownstein. With this approach is possible to find the exactly-solvable PDMSE, for any value of the parameters α and γ in the von Roos's Hamiltonian. For that, our proposal can be considered as a unified treatment of the PDMSE because it contains as particular cases, the kinetic energy operators of various authors such as BenDaniel-Duke, Gora-Williams, Zhu-Kroemer and Li-Kuhn among others. To show the usefulness of our result, we show the solvable PDMSE that comes from the harmonic oscillator potential model for the CMSE. The proposal is general and can easily be extended to other potential models and mass distributions which will be given in the extended paper.
The position dependent mass Schrödinger equation (PDMSE) has a wide range of quantum applications such as the study of semiconductors, quantum wells, quantum dots and impurities in crystals, among many others. On the other hand, the Morse potential is one of the most important potential models used to study the electronic properties of diatomic molecules. In this work, the solution of the effective mass one-dimensional Schrödinger equation for the Morse potential is presented. This is done by means of the canonical transformation method in algebraic form. The PDMSE is solved for any model of the proposed kinetic energy operators as for example the BenDaniel-Duke, Gora-Williams, Zhu-Kroemer or Li-Kuhn. Also, in order to solve the PDMSE with Morse potential, we consider a superpotential leading to a special form of the exactly solvable Schrödinger equation of constant mass for a class of multiparameter exponential-type potential along with a proper mass distribution. The proposed approach is general and can be applied in the search of new potentials suitable on science of materials by looking into the viable choices of the mass function.