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

Abstract In this study, we present expressions for the full effective potential corresponding to the one-photon exchange interaction between two fermions within the framework of the effective Schrödinger-like equation, derived exactly from the Bethe-Salpeter equation in quantum electrodynamics. The final effective potential is expressed in terms of eight scalar functions. When these scalar functions are expanded order by order in terms of velocities, we systematically recover the non-relativistic effective potential organized in terms of 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. A comparison is made between our numerical results and those obtained using conventional the bound-state perturbative theory. Our calculations suggest that this method can accurately account for all relativistic contributions. It would be interesting to extend these calculations to positronium, muonic hydrogen, and scenarios involving nuclear structure and radiative corrections.