The Hydrogen Atom: Consideration of the Electron Self-Field

Abstract

We substantiate the need for a “nonperturbative” account of the self-interaction of the electron with its own electromagnetic field in the canonical hydrogen problem in relativistic quantum mechanics. Mathematically, the problem is reduced to determination of the spectrum of everywhere regular axially symmetric solutions to the self-consistent system of Dirac and Maxwell equations, the classical analog of operator equations of quantum electrodynamics, in the presence of an external Coulomb potential. We demonstrate that only particular classes of solutions, “nonlinear” analogs of s- and p-states, can be obtained via expansion of a solution in a series over the fine structure constant $\alpha$. In the zero approximation for $\alpha \to 0$, we have the reduction to the self-consistent nonrelativistic system of Schrödinger–Poisson equations. The solutions corresponding to the ground state and a large set of excited states are obtained for this system using both numerical and variational methods. The spectrum of binding energies with remarkable accuracy reproduces the “Bohrian” dependence ${{W}_{n}} = {W \mathord{\left/ {\vphantom {W {{{n}^{2}}}}} \right. \kern-0em} {{{n}^{2}}}}$. In this case, the ionization energy $W$ proves to be universal, yet about twice as small as its observed value. The problem of calculation of relativistic corrections to the binding energies and a relation between the model and the ideas and methods of quantum electrodynamics are discussed.