Effects of an Electric Dipole Moment of the Electron on the Hydrogen Energy Levels

The bound states in a Coulomb field of a charged, spin ½, particle with an electric dipole moment are obtained. The nonrelativistic Schrödinger equation for such a particle is solved in closed form. The wave functions are generalizations of the Coulomb wave functions, involving Laguerre polynomials of nonintegral upper index. The accidental degeneracies of the Coulomb energy levels are removed by the dipole interaction. In particular, there will be an additional contribution to the splitting between the $2S_{\frac{1}{2}}$ and $2P_{\frac{1}{2}}$ states coming from the dipole moment. By requiring that this extra energy shift should not destroy the agreement between the theoretical and experimental values of the Lamb shift, it is found that the dipole moment of the electron must be less than ${10}^{-13\mathrm{}}$ cm times the electron charge. Other effects of the dipole moment on the hydrogen energy levels are discussed.