Hydrogen atom in momentum space with a minimal length

TL;DR

This paper analyzes the hydrogen atom in momentum space considering a minimal length from a generalized uncertainty principle, deriving analytical solutions and bounds on the minimal length.

Contribution

It introduces a momentum space approach to the hydrogen atom with a minimal length, providing analytical solutions and bounds on the minimal length scale.

Findings

Analytical solution for s-wave bound states with minimal length

Leading energy correction depends on the square root of the minimal length parameter

Upper bound on minimal length estimated at about 10^{-9} fm

Abstract

A momentum representation treatment of the hydrogen atom problem with a generalized uncertainty relation,which leads to a minimal length ({\Delta}X_{i})_{min}= \hbar \sqrt(3{\beta}+{\beta}'), is presented. We show that the distance squared operator can be factorized in the case {\beta}'=2{\beta}. We analytically solve the s-wave bound-state equation. The leading correction to the energy spectrum caused by the minimal length depends on \sqrt{\beta}. An upper bound for the minimal length is found to be about 10^{-9} fm.