The relativistic energy spectrum of hydrogen

Abstract

AbstractThe 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.