The Modified Quantum Mechanics from the Covariant Generalized Uncertainty Principle

TL;DR

This paper explores a covariant generalized uncertainty principle that introduces minimal length and time scales, modifies dispersion relations, and revisits quantum systems like the particle in a box and the Hydrogen atom.

Contribution

It develops a covariant GUP framework incorporating minimal time and length, leading to modified dispersion relations and quantum system analyses.

Findings

Modified energy-momentum dispersion relation derived.

Corrections to wave functions for quantum systems obtained.

Revised analysis of particle in a box and Hydrogen atom.

Abstract

One can use the generalized uncertainty principle (GUP) to incorporate the minimum measurable length in quantum gravity. It may be interesting to have a minimal time interval as well as the minimal length in the relativistic version of quantum mechanics in the presence of the gravitational effects. In this paper, we consider a covariant version of the generalized uncertainty principle to investigate the effects of both the minimal time and the minimal length. Using the covariant GUP, the energy-momentum dispersion relation is modified. Starting with the modified dispersion relation, the corrections to the wave function are obtained and the problems of the particle in a box and the Hydrogen atom are revisited.