Generalized space and linear momentum operators in quantum mechanics

TL;DR

This paper introduces a deformed quantum framework with generalized operators, leading to new insights into position-dependent mass systems, phase space dynamics, and fundamental principles like uncertainty in a modified quantum setting.

Contribution

It develops a Hermitian deformed linear momentum operator and a corresponding position operator using a $q$-exponential, extending quantum mechanics with a generalized phase space.

Findings

Deformed operators satisfy Hermitian properties.

Classical phase space dynamics expressed via $q$-derivatives.

Position-dependent mass in a potential well analyzed.

Abstract

We propose a modification of a recently introduced generalized translation operator, by including a $q$-exponential factor, which implies in the definition of a Hermitian deformed linear momentum operator $\hat{p}_q$, and its canonically conjugate deformed position operator $\hat{x}_q$. A canonical transformation leads the Hamiltonian of a position-dependent mass particle to another Hamiltonian of a particle with constant mass in a conservative force field of a deformed phase space. The equation of motion for the classical phase space may be expressed in terms of the generalized dual $q$-derivative. A position-dependent mass confined in an infinite square potential well is shown as an instance. Uncertainty and correspondence principles are analyzed.