Eigenvalues of the Breit Equation

Eigenvalues of the Breit Equation {eqnarray*} [(\vecα_{1} \vec{p} + β_{1}m)_{αα^{\prime}} δ_{ββ^{\prime}} + δ_{αα^{\prime}} (-\vecα_{2} \vec{p} + β_{2}M)_{ββ^{\prime}} - \frac{e^{2}}{r} δ_{αα^{\prime}} δ_{ββ^{\prime}}] Ψ_{α^{\prime} β^{\prime}} = E Ψ_{αβ}, {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 $α^2$ 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.

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