# The fine structure of Weber's hydrogen atom -- Bohr-Sommerfeld approach

## TL;DR

This paper calculates the energy levels of Weber's hydrogen atom using a second-order Bohr-Sommerfeld approach, matching results from Schrödinger equation methods, thus providing a semi-classical perspective on its fine structure.

## Contribution

It introduces a second-order Bohr-Sommerfeld quantization method for Weber's Hamiltonian, aligning with Schrödinger-based results, offering a semi-classical analysis of the atom's fine structure.

## Key findings

- Energy levels computed match Schrödinger equation results
- Second-order corrections in the fine structure constant are derived
- Provides a semi-classical perspective on Weber's hydrogen atom

## Abstract

In this paper we determine in second order in the fine structure constant the energy levels of Weber's Hamiltonian admitting a quantized torus. Our formula coincides with the formula obtained by Wesley using the Schr\"odinger equation for Weber's Hamiltonian.