Displacement operator for quantum systems with position-dependent mass

TL;DR

This paper introduces a new translation operator for quantum particles with position-dependent mass, leading to a generalized momentum operator and Schrödinger-like equations, with analytical solutions for various potential scenarios.

Contribution

It develops a novel formalism for quantum systems with position-dependent mass, including a unique commutation relation and analytical solutions for specific potentials.

Findings

Derived a generalized momentum operator for position-dependent mass

Established a unique commutation relation for position and momentum

Provided analytical solutions for particles in various potential scenarios

Abstract

A translation operator is introduced to describe the quantum dynamics of a position-dependent mass particle in a null or constant potential. From this operator, we obtain a generalized form of the momentum operator as well as a unique commutation relation for $\hat x$ and $\hat p_\gamma$. Such a formalism naturally leads to a Schr\"odinger-like equation that is reminiscent of wave equations typically used to model electrons with position-dependent (effective) masses propagating through abrupt interfaces in semiconductor heterostructures. The distinctive features of our approach is demonstrated through analytical solutions calculated for particles under null and constant potentials like infinite wells in one and two dimensions and potential barriers.