Higher-order multipole expansion in the Dirac equation.

Abstract

We investigate the Dirac Hamiltonian for an atomic electron coupled to a classical radiation field. Our starting point is the exact retardation expansion of the electromagnetic potentials in the Poincar\'e gauge. Specific prescriptions, which are the same for the Dirac and Pauli equations, are stated for obtaining, to any order of retardation, the genuine electric, magnetic, and displacement-current multipole contributions to the Hamiltonian. Accordingly, one has to carry out an appropriate phase-factor transformation of the wave function for every retardation correction that is linear in the field except for the first-order one. Along with the electric multipole interactions, the corresponding phase factors have the same functional dependence on coordinates and time in relativistic as well as in nonrelativistic quantum mechanics. We have derived and written explicitly the three different kinds of multipole terms in both the Dirac and Pauli Hamiltonians, within a fourth-order retardation approach. In the nonrelativistic limit, the Dirac magnetic and displacement-current multipole corrections split termwise into linear orbital contributions and their spin counterparts. Moreover, they combine to yield the quadratic orbital terms of the Pauli Hamiltonian. \textcopyright{} 1996 The American Physical Society.