Stark Energy Shifts due to Quantum Gravity in RGUP Algebra

TL;DR

This paper explores how quantum gravitational effects, modeled via RGUP, modify the Stark effect in hydrogen atoms, deriving energy shifts that incorporate minimal length scale corrections in a relativistic context.

Contribution

It introduces a relativistic GUP framework to analyze the Stark effect, deriving energy spectrum modifications and setting bounds on the RGUP parameter.

Findings

Quantum gravitational corrections to Stark energy spectrum are derived.

Energy shifts depend on the RGUP parameter $eta$.

Results recover standard Stark effect and non-relativistic GUP in appropriate limits.

Abstract

In this paper, we investigate the Stark effect in the hydrogen atom under an external electric field, incorporating relativistic generalized uncertainty principle (RGUP) corrections within Minkowskian spacetime and calculate the upper bound on $\beta$ the RGUP parameter. Employing RGUP algebra and the Stetsko-Tkachuk approximation, we derive modifications to the energy spectrum for degenerate and non-degenerate states. The perturbed Hamiltonian, modified by RGUP, enfold quantum gravitational effects. Our results reveal quantum gravitational corrections to the Stark energy spectrum in the relativistic regime, with energy shifts for non-degenerate ($n=1$) and degenerate ($n \neq 1$) cases showing additional terms proportional to $\beta$. These findings reduce to standard Stark effect results and non-relativistic GUP frameworks in the limits $\beta\rightarrow 0$ and $c \rightarrow \infty $, establishing our model as a generalized framework for analyzing minimal length effects in relativistic quantum systems.