Eigenvalues of the Breit Equation

TL;DR

This paper derives exact energy eigenvalues for the Breit Equation with static Coulomb potential, analyzing specific cases and using an alpha-squared expansion to find radial wave functions, providing a foundation for further applications.

Contribution

It presents a detailed derivation of eigenvalues for the Breit Equation considering static Coulomb potential, including simple cases and wave function analysis, which was not previously available.

Findings

Exact eigenvalues for specific cases of the Breit Equation derived.

Radial wave functions obtained using alpha-squared expansion.

Classical Coulomb wave function appears as leading term.

Abstract

Eigenvalues of the Breit Equation {eqnarray*} [(\vec{\alpha}_{1} \vec{p} + \beta_{1}m)_{\alpha \alpha^{\prime}} \delta_{\beta \beta^{\prime}} + \delta_{\alpha \alpha^{\prime}} (-\vec{\alpha}_{2} \vec{p} + \beta_{2}M)_{\beta \beta^{\prime}} - \frac{e^{2}}{r} \delta_{\alpha \alpha^{\prime}} \delta_{\beta \beta^{\prime}}] \Psi_{\alpha^{\prime} \beta^{\prime}} = E \Psi_{\alpha \beta}, {eqnarray*} in which only the static Coulomb potential is considered, have been found. Here the detailed discussion on the simple caces, $^{1}S_{0},\ m=M$ and $m \neq M$ is given deriving the exact energy eigenvalues. The $\alpha^2$ expansion is used to find radial wave functions. The leading term is given by classical Coulomb wave function. The technique used here can be applied to other cases.