The relativistic energy spectrum of hydrogen

The method originally used by Dirac to derive his equation for a single electron is here applied to the two-particle system electron plus proton, considered as an elementary fermion, to obtain a relativistic two-particle Dirac-Breit-type Hamiltonian for the hydrogen atom. This problem can be solved exactly. Thus the relativistic energy levels of the hydrogen atom as a system bound by the static Coulomb force are obtained. The radial part of the resulting Hamiltonian operates in a four-dimensional space describing particles, respectively antiparticles, i. e. electron and positron as well as proton and antiproton. The spin of a particle is described by the normal two-component Pauli spinor, and therefore standard theoretical tools for dealing with angular-momentum coupling can be exploited. The classical energy states of the hydrogen atom are retained in the appropriate non-relativistic limit, in particular the energy levels resulting from Schrödinger's equation. The exact energy spectrum shows the expected dependence on the reduced mass of the two-particle system, and thus describes the recoil of the core properly. The fine structure of the hydrogen spectrum arises from a dependence of the energy levels upon the quantum number of the total angular momentum.