κ-Deformed quantum and classical mechanics for a system with position-dependent effective mass

Bruno G Da Costa,Mariela Portesi,Ignacio S Gomez

doi:10.1063/5.0014553

Abstract

We present the quantum and classical mechanics formalisms for a particle with a position-dependent mass in the context of a deformed algebraic structure (named κ-algebra), motivated by the Kappa-statistics. From this structure, we obtain deformed versions of the position and momentum operators, which allow us to define a point canonical transformation that maps a particle with a constant mass in a deformed space into a particle with a position-dependent mass in the standard space. We illustrate the formalism with a particle confined in an infinite potential well and the Mathews–Lakshmanan oscillator, exhibiting uncertainty relations depending on the deformation.

Highlights

Minimum length scales are of crucial importance in several areas of physics, such as quantum gravity, string theory, and relativity, fundamentally due to the techniques developed for removing divergences in field theories maintaining the parameters lengths as universal constants of the theory in question
We present the quantum and classical mechanics formalisms for a particle with a position-dependent mass in the context of a deformed algebraic structure, motivated by the Kappa-statistics
The development of generalized translation operators motivated the introduction of a positiondependent linear momentum for characterizing a particle with a PDM7,48–56 that can be related to a generalized algebraic structure inherited from the mathematical background of nonextensive statistics.[58]

Summary

INTRODUCTION

Minimum length scales are of crucial importance in several areas of physics, such as quantum gravity, string theory, and relativity, fundamentally due to the techniques developed for removing divergences in field theories maintaining the parameters lengths as universal constants of the theory in question (for a review, see, for instance, Ref. 1) In this sense, the seek for these minimum lengths in quantum mechanics has been translated into generalizations of the standard commutation relationship between the position and the momentum.[2]. We employ the κ-algebra for generalizing classical and quantum mechanics with the aim of studying the properties of the resulting noncommuting space originated by the deformation.

REVIEW OF THE κ-ALGEBRA

Π2 2m0

PARTICLE IN AN INFINITE POTENTIAL WELL

Quantum case

CONCLUSIONS