HIGHER-ORDER MULTIPOLE EXPANSION IN THE DIRAC-EQUATION

We investigate the Dirac Hamiltonian for an atomic electron coupled to a classical radiation field. Our starting point is the exact retardat ion expansion of the electromagnetic potentials in the Poincare gauge. Specific prescriptions, which are the same for the Dirac and Pauli eq uations, are stated for obtaining, to any order of retardation, the ge nuine electric, magnetic, and displacement-current multipole contribut ions to the Hamiltonian. Accordingly, one has to carry out an appropri ate phase-factor transformation of the wave function for every retarda tion correction that is linear in the field except for the first-order one. Along with the electric multipole interactions, the correspondin g 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 m ultipole terms in both the Dirac and Pauli Hamiltonians, within a four th-order retardation approach. In the nonrelativistic limit, the Dirac magnetic and displacement-current multipole corrections split termwis e into linear orbital contributions and their spin counterparts. Moreo ver, they combine to yield the quadratic orbital terms of the Pauli Ha miltonian.