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One example of exactly solvable quantum mechanics problem of position dependent mass

V. M. Burdeynyy, В. М. Бурдейний

Published Jan 1, 2017• 0 citations
OpenAlex

Abstract

This communication has been dedicated to quantum dynamics of particle whose mass depends on coordinate. We considered one dimensional model which admits to obtain the exact solution of wave equation. The position dependent mass was represented as the periodic function. Inside of period the mass varies accordingly to the inverse proportionality. We have found wave functions in their explicit form as well as energy eiguen values. It has been shown that the energy spectrum manifests properties typical for periodic nanostructures.

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This communication has been dedicated to quantum dynamics of particle whose mass depends on coordinate. We considered one dimensional model which admits to obtain the exact solution of wave equation. The position dependent mass was represented as the periodic function. Inside of period the mass varies accordingly to the inverse proportionality. We have found wave functions in their explicit form as well as energy eiguen values. It has been shown that the energy spectrum manifests properties typical for periodic nanostructures.
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.
Using the coordinate transformation method, we solve the one-dimensional Schrödinger equation with position-dependent mass(PDM). 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, Hemite and generalized Laguerre polynomials. All potentials for these solvable systems have an extra term VmV_mVm​ which produced from the dependence of mass on the coordinate, compared with that for the systems of constant mass. The properties of VmV_mVm​ for several mass functions are discussed.
One special case of the quantum particle with the effective mass which depends on the coordinate has been considered. The coordinate dependence of the particle mass is expressed in terms of trigonometric functions, namely, inverse square of the cosine. The model we have chosen has the following main features: (1) position dependent mass is periodic function of the coordinate; (2) the coordinate dependence has singular points of inverse squares type which can be classified as the centres of falling; (3) the eigenfunctions in their explicit exact form can be established. Having determined the eigenfunctions by applying procedure of regulazation proposed in the given article and Flock`s theorem the equation for eigenvalues has been found. Its approximate solution allows to find and analyse the law of dispersion that, as it turned out, is continuous in function of wave number and forms a continuous spectrum.
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