Exact solution of position dependent mass Schrödinger equation by supersymmetric quantum mechanics

Abstract

A supersymmetric technique for the solution of the effective mass Schrödinger equation is proposed. Exact solutions of the Schrödinger equation corresponding to a number of potentials are obtained. The potentials are fully isospectral with the original potentials. The conditions for the shape invariance of the potentials are discussed.

-fold supersymmetry in quantum systems with position-dependent mass

“…For references, see e.g., Refs. [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26] and those cited therein. Due to the fact that the position-dependent mass m(q) does not commute with the momentum operator p = −id/dq, ambiguity arises in defining a quantum kinetic operator which is formally Hermitian and reduces to the classical kinetic term T = p 2 /2m(q).…”

“…A different choice of the parameters results in a different correction to the original potential profile V (q), and the above Hamiltonian always has the following form: Hence, the typical investigations into (quasi-)exact solvability of PDM quantum systems consist in finding simultaneously a pair of an effective potential U(q) and a mass function m(q) for which the PDM Hamiltonian (1.2) admits (a number of) exact eigenfunctions in closed form. Up to now, two different methods have been frequently employed, namely, coordinate transformations including point canonical transformations [1,2,5,6,7,8,9,12,16,18,20,25], and supersymmetric methods [3,4,7,8,11,13,14,19,20,21,22,24,25]. The latter approaches were also applied to many-body PDM quantum systems [26].…”

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.…”

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.…”