Time dependence of operators in minimal and multipolar nonrelativistic quantum electrodynamics. II. Analysis of the functional forms of operators in the two frameworks

The relationship between the multipolar and minimal Hamiltonians through a canonical transformation is used to analyze the time dependence of quantum operators in the two formalisms. It is shown that operators dependent on particle position, the vector potential, and their time derivatives have the same time dependence. However, operators such as the photon number and the atomic population number evolve differently in the two cases. Expressions correct to second order in the electric-dipole moments for these are given. The expectation values of the photon number operator are calculated in the two formalisms and are used to predict natural line shapes. These theoretical shapes differ. The observed shape will depend on the particular setup of the experiment involving the radiative decay of an excited atom. A comparison with the theoretical predictions will determine which of the two frameworks is most appropriate to describe the decay. Finally, the energy density of the electromagnetic field in the neighborhood of an atom is calculated within the two formalisms. [S1050-2947(99)04411-X].