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Точно розв'язувана модель квантової частинки з сингулярною і періодично залежною від координат масою

V. K. Kasyanenko, V. M. Burdeynyy, В. М. Бурдейний, В. Х. Касияненко

Published Jan 1, 2021• 0 citations

Abstract

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.

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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.

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(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

V_m

which produced from the dependence of mass on the coordinate, compared with that 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. 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.

6 p.) -РОЗКЛАД ДЛЯ РIВНЯННЯ ШРЕДИНҐЕРА З МАСОЮ, ЗАЛЕЖНОЮ ВIД КООРДИНАТ Д. О. Кулiков 1 , В. М. Шаповал 2 1 Днiпропетровський нацiональний унiверситет iм.Олеся Гончара, кафедра теоретичної фiзики, пр.Гагарiна, 72, Днiпропетровськ, 49010, Україна, 2 Iнститут теоретичної фiзики iм.М.М.Боголюбова НАНУ, вул.Метрологiчна, 14-б, Київ, 03680, Україна (Отримано 18 жовтня 2010 р.; в остаточному виглядi 14 листопада 2011 р.) Представлено рекурентну технiку отримування асимптотичних розкладiв для власних значень енерґiї зв'язаних станiв радiального рiвняння Шрединґера з масою, залежною вiд координат.Як приклад її застосування розраховано власнi значення енерґiї для кулонiвського потенцiалу за наявностi залежної вiд координат маси та виведено нерiвностi, що реґулюють змiщення енерґетичних рiвнiв вiдносно їх положення за сталої маси.

An approximate method is proposed to solve position dependent mass Schrodinger equation. The procedure suggested here leads to the solution of the PDM Schrodinger equation without trans- forming the potential function to the mass space or vice verse. The method based on asymptotic Taylor expansion of the function, produces an approximate analytical expression for eigenfunction and numerical results for eigenvalues of the PDM Schrodinger equation. The results show that PDM and constant mass Schrodinger equations are not isospectral. The calculations are carried out with the aid of a computer system of symbolic or numerical calculation by constructing a simple algorithm.

We show that the trigonometric Pöschl-Teller potential well problem of the non-relativistic quantum mechanics is equivalent to a certain model of a linear singular oscillator with the position-dependent mass of the form 𝑀 𝑥 𝑎 𝑚 𝑎 𝑥 ⁄ , 0 𝑥 𝑎.We found an explicit form of the functions and energy of the wave functions and discrete energy spectrum for this model.Wave functions are expressed through the Jacobi polynomials.At the limit when 𝑎 → ∞ equation of the motion, wave functions and energy spectrum of the model correctly reduce to corresponding results of the usual non-relativistic linear singular oscillator with a constant mass 𝑚 .