For a system consisting of a classical electromagnetic field and a charged quantum-mechanical particle, Poynting's theorem holds for the energy density of the electromagnetic field. For local energy conservation to hold, the power lost by the field is the power gained by the quantum-mechanical particle. The power per unit volume transferred from the field to the particle is $\overrightarrow{\mathrm{E}}·\overrightarrow{\mathrm{J}}$, which is shown to be the density of the quantum-mechanical power operator. For the particle an energy theorem, similar to Poynting's theorem for the field, is satisfied in such a way that the energy of the total isolated system is conserved. The energy density of the particle is the density of the energy operator for the particle, which in this time-dependent problem is not the same as the Hamiltonian. The energy operator is gauge invariant, while the Hamiltonian is not. When the combined conditions of gauge invariance and local energy conservation are used, the appropriate basis functions to use are the eigenstates of the energy operator. Transitions between these eigenstates are determined by matrix elements of the power operator.

• Received 14 January 1981

DOI:https://doi.org/10.1103/PhysRevD.26.1927