On the energy spectrum of the hydrogen atom in a photon field. I

Abstract

In 1930 Weisskopf and Wigner gave an account, based on the Maxwell–Schrödinger equations, of the natural spectral line broadening of radiation emitted by a hydrogen atom. Their calculations were based on an approximation involving certain single-photon transitions in the perturbation series for the solutions of these equations. In Part I of this series of papers the exact expressions for both the line shift and the line broadening are obtained from the Maxwell–Dirac equations in such a way that the Weisskopf–Wigner results appear as a second order approximation. The Maxwell–Dirac Hamiltonian for the coupled fields is first shown to admit a complex analytic dilation in the energy variables. The Fredholm–Born series for the resolvent is shown to converge uniformly when certain high-energy cutoff factors are included in the interaction and the photons are given a small mass. The series is then rearranged to show that the spectrum of the modified dilated Hamiltonian, which consists of a complete set of complex eigenvalues, thresholds, and branch cuts, is only a slight perturbation of the known spectrum of the dilated Hamiltonian for the uncoupled fields. The real part of the shift of each complex eigenvalue then accounts for the spectral line shift, and the complex part accounts for the associated line broadening. Finally, the implications for the scattering matrix and the various phenomena of resonance scattering are discussed. In Part II of this series these results are shown to remain valid when the cutoff factors and the photon mass are removed.