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Journal of Physics A Mathematical and General

Energy eigenvalues for the systems with position-dependent effective mass

Y C Ou, Zhuangqi Cao, Qishun Shen

Published Mar 23, 2004• 39 citations
Publisher

Abstract

Based on the analytical transfer matrix method (ATMM), the universal dispersion equation determining the energy spectrum for an arbitrary potential function experienced by a particle with arbitrary position-dependent effective mass distribution is obtained. As an illustrative example, for a given mass distribution m ( x ) and a potential function V ( x ), our predicted results are in complete agreement with the exact ones. The energy spectrum of a harmonic oscillator with nonconstant mass is obtained. Since the exact solutions to Schrödinger equation with nonconstant effective mass are a valuable means for checking and improving models and numerical methods in the study of electronic properties, the ATMM will find wide applications in material science and condensed matter.

DOI
10.1088/0305-4470/37/14/007

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Based on the analytical transfer matrix method (ATMM), the universal dispersion equation determining the energy spectrum for an arbitrary potential function experienced by a particle with arbitrary position-dependent effective mass distribution is obtained. As an illustrative example, for a given mass distribution m ( x ) and a potential function V ( x ), our predicted results are in complete agreement with the exact ones. The energy spectrum of a harmonic oscillator with nonconstant mass is obtained. Since the exact solutions to Schrödinger equation with nonconstant effective mass are a valuable means for checking and improving models and numerical methods in the study of electronic properties, the ATMM will find wide applications in material science and condensed matter.
The Schrodinger equation with position-dependent mass (PDM) becomes one of interesting subjects in the study of quantum systems, because of its wide applications in many physical problems. Meanwhile, harmonic oscillator becomes important model in most physical problems. In this paper, analytical solutions of the PDM Schrodinger equation, i.e. energy eigenvalues En and eigenfunctions , of a one dimensional harmonic oscillator system using transformation method has been studied. The results showed that there was no difference between the energy eigenvalues, En of the harmonic oscillator systems with and without PDM, whereas it is not for the eigenfunctions,  of these two harmonic oscillator systems. This difference was due to the presence of dimensionless mass function  and the changes of x becomes  variable for the eigenfunctions of the harmonic oscillator system with PDM.  Moreover, the differences in the eigenfunctions will affect the dynamical properties of the quantum systems. It can be concluded that the harmonic oscillator system with PDM is the generalization of the harmonic oscillator system with constant mass.
In this paper, two novel semiclassical methods including the standard and supersymmetric WKB quantization conditions are suggested to discuss the Schrödinger equation with position-dependent effective mass. From a proper coordinate transformation, the formalism of the Schrödinger equation with position-dependent effective mass is mapped into isospectral one with constant mass and therefore for a given mass distribution and physical potential function the bound state energy spectrum can be determined easily by above method associated with a simple integral formula. It is shown that our method can give the analytical results for some exactly-solvable quantum systems.
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.
Abstract The aim of the present study is to obtain exact solutions for the one‐dimensional position‐dependent effective mass Schrödinger equation. After the change of wave function on position‐dependent effective mass Schrödinger equation and comparison between the resulted equation with the PT‐symmetric invariant anharmonic potential, three distinct classes of exact solutions are obtained for the Schrödinger equation. As an illustrative example for a Gaussian form of m(x), energy spectrum, potential energy function and wave function are obtained explicitly. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004
Using the coordinate transformation method, we solve the one-dimensional Schrödinger equation with position-dependent mass. The explicit expressions for the potentials, energy eigenvalues, and eigenfunctions of the systems are given. The eigenfunctions can be expressed in terms of the Jacobi, Hermite, and generalized Laguerre polynomials. All potentials for these solvable systems have an extra term V m , which is produced from the dependence of mass on the position, compared with those for the systems of constant mass. The properties of V m for several mass functions are discussed.
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.