General solution of the coulomb–dirac problem: calculation of a divergence‐free lamb shift

Paul Dirac's time‐dependent equation is inferred from the scalar product of an electron's four‐momentum and an electromagnetic four‐potential on identifying an electromagnetic carrier‐wave energy with the electron's rest‐mass energy (Maxwell–Dirac equivalency). Dirac's Schroedinger‐like temporally harmonic solution is not the general solution to his time‐dependent equation, because it constrains all four components of his vector wave function to oscillate in time at a single frequency. In fact, it is equivalent to an approximation method known as adiabatic elimination, which is widely used in the optical‐physics literature to solve temporally coupled equations. The general time‐dependent solution for the Coulomb problem includes coupled positive‐ and negative‐energy states whose wave function is a mixture of bound and unbound components. The Maxwell–Dirac equivalency permits the unbound component, which is known as Zitterbewegung in the free‐elecron problem, to be interpreted as a photonic component, which conserves energy for a ground state in which the electron simultaneously occupies two states whose separation is of order 2 mc 2 . Equations of motion for a photon, which are formed from the scalar product of a photon's four‐momentum and an electromagnetic four‐potential, are also presented and used to calculate a divergence‐free Lamb shift. Finally, Dirac's general time‐dependent solution is shown to contain subatomic bound states for the Coulomb problem, even though such states are forbidden in Dirac's standard solution. These states exist for an electron, whose positive‐ (negative‐) energy motion is attractive (repulsive), or for a positron, whose positive‐ (negative‐) energy motion is repulsive (attractive), respectively. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2012