Klein–Gordon equation for atoms

Abstract

The relativistic energy–momentum relation for a free atom in flight is translated into a free Klein–Gordon equation, in which the atomic mass is replaced by a differential operator M for the total centre-of-mass energy levels E. As the Klein–Gordon operator contains M 2, it gives the squares of E. When all atomic constituents are treated relativistically, the squares appear automatically after elimination of the components of opposite ‘total chirality’ from the wave function, and after a scaling of variables. For hydrogenic atoms, the new equations are nearly identical with the single-electron equation including hyperfine interaction. The time dependence of the Klein–Gordon operator implies exact energy conservation in radiative decays. The history of these atomic equations is reviewed. For quarkonium, the origin of the large hyperfine splitting is discussed, and a speculation about bound quark masses is mentioned.