Revisiting the displacement operator for quantum systems with position-dependent mass

TL;DR

This paper revisits the formalism of position-dependent mass quantum systems, critiques previous approaches, and proposes an alternative method, demonstrating its application through a particle in a square well potential.

Contribution

It offers a critical analysis of existing formalisms for position-dependent mass quantum mechanics and introduces an alternative approach with comparative application results.

Findings

The new formalism provides consistent results for a particle in a square well.

Comparison shows improvements over previous methods.

The approach clarifies aspects of the position-dependent translation operator.

Abstract

Recently R. N. Costa Filho et al. (PRA 84, 050102(R) (2011)) have introduced a position dependent infinitesimal translation operator which corresponds to a position dependent linear momentum and consequently to a position dependent effective mass quantum particle. Although there is no doubt in novelty of the idea and the formalism, we believe that some aspects of the quantum mechanics could be complemented in their original work. Here in this letter first we address those points and then an alternative will be introduced. Finally we apply the formalism for a quantum particle under a null potential confined in a square well and the results will be compared with those in the paper mentioned above.