Approximate solution of the Dirac equation using the Foldy-Wouthuysen hamiltonian

Abstract

The Foldy-Wouthuysen hamiltonian is used to obtain the energies of various states of the hydrogen-like atom. For the 1s and 2s states to order mc ² α⁶ and for the 2p states to order mc ² α⁸ divergent contributions to the energy arise, but it is shown that provided these divergences are handled properly they cancel one another and the correct finite result is obtained. An attempt to carry the calculations to higher orders using delta function recipes fails, but it proves possible to evaluate the mc ² α⁸ energy contribution for the 1s state using a different method.