An effective Hamiltonian approach is developed for the perturbative calculation of bound state energies in quantum electrodynamics. In this approach a recoil correction of order ${\alpha }^2{E}_F$ to the hyperfine structure of hydrogenic systems with the arbitrary mass ratio is obtained. In the case of the positronium it amounts to $\Delta E=$$\frac{1}{3}$$m{\alpha }^6\left[1.130\right(5)\mathrm{}-$$\frac{1}{2}$$\ln \alpha ]$.

• Received 17 March 1997

DOI:https://doi.org/10.1103/PhysRevA.56.297