THE MULTIPOLE EXPANSION IN QUANTUM THEORY

The Hamiltonian of a system of charged particles interacting with the electromagnetic field is investigated. For an arbitrary system the multipole expansion of the interaction between the system and the field is derived by means of a suitable canonical transformation. The transformed Hamiltonian is obtained from the Hamiltonian of the system by replacing the momenta by the transformed kinetic momenta and by adding to the Hamiltonian a term representing the interaction of the system with the electric component of the field. By expanding this interaction term, as well as the transformed momenta, in powers of the dimension of the system over the wavelength, the multipole expansion of the Hamiltonian is obtained. For a system interacting with a classical field the multipole form of the Hamiltonian is exactly equivalent to the original Hamiltonian. For a quantized field this is not true, and the multipole form of the transformed Hamiltonian is shown to be equivalent to the original Hamiltonian only for first-order radiation processes.