Relativistic corrections to energy spectrum of hydrogen due to the full one-photon-exchange interaction

Abstract: In this work, we present expressions for the full effective potential corresponding to the one-photon exchange interaction within the framework of an effective Schr\"{o}dinger-like equation, which is derived exactly from the Bethe-Salpeter equation in quantum electrodynamics. The final effective potential is expressed in terms of eight scalar functions. When these eight scalar functions are expanded in terms of velocities order by order, we can return to the non-relativistic effective potential systematically organized by velocities. By retaining the exact momentum dependence in the effective potential, we estimate its corrections to the energy spectrum of hydrogen using a highly precise numerical method. The comparison of the numerical results with those obtained using conventional bound-state perturbative theory is discussed. Our calculations suggest that it is possible to accurately account for all relativistic contributions using this method. It would be interesting to extend these calculations to positronium, muonic hydrogen, and scenarios involving nuclear structure and radiative corrections.