Hydrogen-atom spectrum under a minimal-length hypothesis

TL;DR

This paper investigates how a minimal-length hypothesis affects the hydrogen atom's energy spectrum, using numerical and perturbative methods, and finds constraints on the minimal length scale from spectroscopy data that are less restrictive than earlier estimates.

Contribution

It provides a detailed analysis of the hydrogen spectrum under minimal-length commutation relations, including numerical and perturbative results for arbitrary parameters.

Findings

Minimal length scale constrained to a few GeV^{-1} from spectroscopy data

Numerical and perturbative methods applied to spectrum calculations

Weaker constraints than previously claimed on minimal length

Abstract

The energy spectrum of the Coulomb potential with minimal length commutation relations $[X_i, P_j] = i\hbar\{\delta_{ij}(1+\beta P^2) + \beta'P_iP_j\}$ is determined both numerically and perturbatively for arbitrary values of $\beta'/\beta$ and angular momenta $\ell$. The constraint on the minimal length scale from precision hydrogen spectroscopy data is of order of a few GeV$\null^{-1}$, weaker than previously claimed.