Generalized Uncertainty Principle, Modified Dispersion Relation and Barrier penetration by a Dirac particle

TL;DR

This paper investigates how a generalized uncertainty principle modifies the energy band structure of a Dirac particle, revealing new forbidden bands and affecting barrier penetration and Klein's paradox.

Contribution

It introduces a modified dispersion relation and band structure for Dirac particles under GUP, showing new forbidden bands and altered barrier reflection properties.

Findings

Existence of additional forbidden energy bands at spectrum edges

Dirac particles cannot penetrate high potential barriers due to GUP effects

Massless particles can reflect from sufficiently high barriers under GUP

Abstract

We have studied the energy band structure of a Dirac particle in presence of a generalised uncertainty principle (GUP). We start from defining a modified momentum operator and derive corresponding modified dispersion relation (MDR) and GUP. Apart from the forbidden band within the range $\pm m$, $m$ being the mass of the particle, we find the existence of additional forbidden bands at the both ends of the spectrum. Such band structure forbids a Dirac particle to penetrate a potential step of sufficient height ($\sim E_P$, $E_P$ being Planck energy). This is also true for massless particle. Unlike the relativistic case, a massless particle also can reflect from a barrier of sufficient height. Finally we discuss about the Klein's paradox in presence of the GUP.