Analytical solution to position dependent mass Schrödinger equation

Abstract

Using a recently developed technique to solve Schrödinger equation for constant mass, we studied the regime in which mass varies with position i.e position dependent mass Schrödinger equation(PDMSE). We obtained an analytical solution for the PDMSE and applied our approach to study a position dependent mass m(x) particle scattered by a potential V(x). We also studied the structural analogy between PDMSE and two-level atomic system interacting with a classical field. PACS numbers: 03.65.Ge; 03.65.Fd; 03.65.-w Schrödinger equation with position dependent mass is one of the areas of research which has gained great attention in the past decades. Position-dependent mass Schrödinger equation(PDMSE) has been applied to several physical systems. For example PDMSE is applied in electronic properties of semiconductors [1], quantum dots and quantum wells [2, 3], semiconductors heterostructures [4], supper-lattice band structures [5], He-Clusters [6] quantum liquids [7], the dependence of energy gap on magnetic field in semiconductor nano-scale rings [8], the solid state problem with Dirac equation [9]etc. One of the motivation behind investigating these system with position dependent mass is, how do these mass variation effects the dynamics of the quantum-mechanical system. In such systems the energy is described by a Hamiltonian which contains the kinetic energy and the potential energy opeators H = T + V , where special care is taken for the kinetic term.O. von Roos [10] was the first to suggest the following generalized form of the kinetic energy operator for position-dependent mass modelwhere m = m(r) is the position-dependent mass. The constants η, and ρ, which are also knows as the von Roos ambiguity parameters can be assumed to be arbitrary but they obey the constraint equation η + + ρ = −1. Before von Roos several forms of operator T has been used to solve this problem [11][12][13][14]. Different approaches have been used to find analytical solution to PDMSE like point-canonical transformation [15], Green's function [16], Heun equation [17, 18], Group-Theoretical method [19], potential algebra [20], Lie-algebraic [21] and supersymmetric [22] approach etc. In this Brief Report, we obtained an analytical solution for the position dependent mass Schrödinger equation with general mass variation m(x) by transforming the PDMSE to Riccati equation. Analytical solution for Schrödinger equation with constant mass, beyond adiabatic approximation, has been recently investigated extensively [23] where we applied our approach to 1-D [23] and 3-D [24] scattering problem and showed that our method gives better accuracy than the well-known JWKB method. Here in this article we have extended that approach to variable mass regime and obtained very accurate results for PDMSE. The main result of this paper is the analytical solution given by Eq. (19). To illustrate how well our method works we considered a position dependent mass m(x) particle scattered by potential V (x) and obtained the wave function for 1D case (which could easily be...

Effect of Intense Laser Field in Gaussian Quantum Well With Position‐Dependent Effective Mass

“…In order to determine the basic physical properties of these systems, the Schrödinger equation together with PDM equation must be solved. In this context, to solve this problem analytically, different approaches have been employed . Given a PDM distribution, Alhajdari has obtained potential functions for exactly solvable nonrelativistic problems .…”

“…It is emphasized that the developed approach can be applied to examine the various effects in nano‐systems. Also, Jha et al have studied the dynamics of the PDM Schrödinger equation to find an analytical solution . They have also investigated the structural analogy between the PDM Schrödinger equation and the two‐level atomic system driven by a resonant classical field that establishes a connection between quantum optics and condensed matter physics.…”

Точно Решаемые Потенциалы И Решения Для Связанных Состояний Уравнения Шредингера В $D$-Мерном Пространстве С Зависящей От Координат Массой

“…Для решения уравнения Шредин-гера с массой, зависящей от координат, применяются различные методы, в ли-тературе можно найти каноническое точечное преобразование [9], [18]- [22], метод Никифорова-Уварова [23]- [25], суперсимметричный подход [26], [27], метод квадра-тичной алгебры [28], аналитический метод [29], поиск решений в виде ряда [30], ме-тоды преобразований Дарбу [31], [32], сплетающих операторов [33], восстановления волновых пакетов [34], -разложение [35], метод расширенного преобразования [12] и т. д. Во всех этих случаях волновые функции выражаются через классические ортогональные полиномы (КОП). Фактически оказывается, что КОП играют важ-ную роль с самого зарождения квантовой механики, поскольку через эти полиномы выражаются собственные функции связанных состояний.…”

Exact solutions of the position-dependent-effective mass Schrödinger equation

“…[42][43][44] We can cite the point-canonical transformation, 24,25,45,46 Nikiforov-Uvarov (NU) method [45][46][47][48][49] Green's function, 50 the Heun equation, 51 the potential algebra 52 and the supersymmetric approach 53,54 as analytical methods to generate solutions for the PDEM Schrödinger equation. However, the exact solutions are limited to a small set of systems.…”

“…The concept of a position-dependent effective mass (PDEM) in the Schrödinger equation has gained much interest [1][2][3][4][5][6][7] in the last two decades, because of its applications in several fields of physics [8][9][10][11][12][13][14] from semiconductors 15,16 to quantum fluids. 17,18 For example, the transport proprieties in semiconductors, 19 the effective interaction in nuclear physics, 20 and the dynamical properties of a neutron superfluid in a neutron star 21 are described by the PDEM Schrödinger equation.…”