Analytical solutions of the one-dimensional Schr\"{o}dinger equation with position-dependent mass

TL;DR

This paper derives exact analytical solutions for the one-dimensional Schrödinger equation with position-dependent mass, transforming it into a Riccati equation and exploring integrability cases to find seven solution classes.

Contribution

It introduces a method to solve the Schrödinger equation with variable mass by converting it into a Riccati equation and identifying seven classes of solutions.

Findings

Seven classes of exact solutions obtained

Solutions relate mass functions and potentials under consistency conditions

Method applicable to various physical systems with position-dependent mass

Abstract

The study of the Schr\"{o}dinger equation with the position-dependent effective mass has attracted a lot of attention, due to its applications in many fields of physics, including the properties of the semiconductors, semiconductor heterostructures, graded alloys, quantum liquids, Helium-3 clusters, quantum wells, wires and dots etc. In the present work we obtain several classes of solutions of the one-dimensional Schr\"{o}dinger equation with position-dependent particle mass. As a first step the single particle Schr\"{o}dinger equation with position-dependent mass is transformed into an equivalent Riccati type equation. By considering some integrability cases of the Riccati equation, seven classes of exact analytical solutions of the Schr\"{o}dinger equation are obtained, with the particle mass function and the external potential satisfying some consistency conditions.