Muonium and positronium potentials with radiative corrections are derived from the scattering operator, and general results for the spectra of these systems to order ${\ensuremath{\alpha}}^{3}{R}_{\ensuremath{\infty}}$ are obtained. We also compare our results with those obtained earlier with the Bethe-Salpeter approach, and find a discrepancy in the case of the $1S$ state. Our theoretical results are in good agreement with the available experimental data.