Exact Solution of the Klein-Gordon Equation for the Hydrogen Atom Including Electron Spin

TL;DR

This paper derives an exact, spin-corrected Klein-Gordon equation for the hydrogen atom that reproduces Dirac energy levels without using matrix formalism.

Contribution

It introduces a spin-inclusive Klein-Gordon equation that matches Dirac energy eigenvalues, simplifying the mathematical framework.

Findings

Reproduces Dirac energy levels for hydrogen

Provides a matrix-free second order differential equation

Offers a new approach to include spin in Klein-Gordon framework

Abstract

The term describing the coupling between total angular momentum and energy-momentum in the hydrogen atom is isolated from the radial Dirac equation and used to replace the corresponding orbital angular momentum coupling term in the radial K-G equation. The resulting spin-corrected K-G equation is a second order differential equation that contains no matrices. It is solved here to generate the same energy eigenvalues for the hydrogen atom as the Dirac equation.