Relativistic Ritz approach to hydrogen-like atoms I: theoretical considerations

TL;DR

This paper introduces a relativistic effective theory for hydrogen-like atoms that extends the Ritz approach, providing improved accuracy over nonrelativistic methods and enabling new applications in spectroscopic data analysis.

Contribution

It develops a relativistic extension of the Ritz approach for hydrogen-like systems, linking bound-state QED predictions with a new effective theory.

Findings

Relativistic theory fits hydrogen energy levels better than nonrelativistic models.

Analytic relations connect higher-order QED corrections to lower-order terms.

Application to spectral data enables determination of the fine-structure constant.

Abstract

The Rydberg formula along with the Ritz quantum defect ansatz has been a standard theoretical tool used in atomic physics since before the advent of quantum mechanics, yet this approach has remained limited by its non-relativistic foundation. Here I present a long-distance relativistic effective theory describing hydrogen-like systems with arbitrary mass ratios, thereby extending the canonical Ritz-like approach. Fitting the relativistic theory to the hydrogen energy levels predicted by bound-state QED indicates that it is superior to the canonical, nonrelativistic approach. An analytic analysis reveals nonlinear consistency relations within the bound-state QED level predictions that relate higher-order corrections to those at lower order, providing guideposts for future perturbative calculations as well as insights into the asymptotic behavior of Bethe logarithms. Applications of the approach include fitting to atomic spectroscopic data, allowing for the determination the fine-structure constant from large spectral data sets and also to check for internal consistency of the data independently from bound-state QED.