Evaluation of the energy states of hydrogen atom using Schroedinger equation with a Coulomb potential modified by the interaction between the magnetic moments of the proton and electron

TL;DR

This paper calculates hydrogen atom energy levels by solving the Schrödinger equation with a modified Coulomb potential that includes magnetic moment interactions, successfully reproducing fine, hyperfine, and Lamb shift structures in agreement with experimental data.

Contribution

It introduces a simple formula derived from a direct Schrödinger equation solution that accounts for magnetic moment interactions to explain hydrogen energy structures.

Findings

Accurately predicts hyperfine splitting of 21 cm line

Reproduces Lamb shift between nP1/2 and nS1/2 states

Results align well with experimental measurements

Abstract

By using a Coulomb potential modified by the interaction between the magnetic moments of the electron and proton, we have calculated the energy levels of a hydrogen atom. We have obtained fine structure, hyperfine structure and the Lamb shift. All these are obtained from a simple formula which is a direct solution of the Schroedinger equation. The obtained results are in a good agreement with experimental data. For example, the hyperfine splitting between the energy levels of the states 1S1/2,1 and 1S1/2,0 is of the order of 5.6x10^(-6) eV, which is the source of the famous "21 cm line" which is strongly useful to radio astronomers for tracking hydrogen in the interstellar medium of galaxies. The energy of the states nP1/2 is lower than those of the states nS1/2 (Lamb shift), because in the first case the interaction between the magnetic moments of the proton and electron spins is diminished by the spin-orbit coupling.