Bethe--Salpeter equation: numerical experience with a hydrogenlike atom

As a test of the accuracy of numerical techniques used in solving the Bethe-Salpeter equation, we compute the binding energy of a hydrogenlike atom including one-photon exchange only. A 5% deviation from the result predicted from the Dirac equation with a Coulomb potential is in agreement with a result which we derive analytically, $E_b=\left(\frac{1}{4}\right)m{\alpha }^2[1-(\frac{4\alpha }{\pi }\left)\ln \right(\frac{2}{\alpha })+\cdots ]$. $E_b$ cannot be expanded in a power series in α² similar to the series expansion of the Sommerfeld expression for $E_b$ resulting from the Dirac theory. This apparently startling result is not due to a failure of quantum electrodynamics, of course, but to the fact that we have used the relativistic interaction ${\gamma }_{\mu }^{\left(1\right)}\left(\frac{1}{k^2}\right){\gamma }_{\mu }^{\left(2\right)}$ rather than working in the Coulomb gauge as is usually done. Salpeter remarked in 1952 that doing the calculation as we have done it would result in a slowly convergent series for $E_b$; nevertheless the precise magnitudes involved are surprising.