A contribution to the dynamical theory of radiating electrons

The dynamical problem of a harmonically bound nonrelativistic electron with standard dipole model coupling to the electromagnetic field in a finite region of three-dimensional space is solved exactly in a simple and (in the realm) novel manner. The emphasis is on a Lagrangian real space formulation. An effective electron radius is introduced mathematically rather than physically. The solution is obtained in terms of the natural modes of the coupled system. The phase shifts are easily found from the dynamical boundary condition (on the wave equation) at the electron. The electron's mass renormalization arises naturally. Several aspects of the exact eigensolutions are discussed, such as their nonorthogonality and the completeness relations. The classical initial value problem is solved explicitly. The quantum theory of the model is shown to involve a logarithmic infinity in the electron's momentum fluctuations. All formulae are exact for arbitrary finite spatial extension of the field (of spherical geometry). In the limit of infinite field extension the classical equation of motion for the electron is obtained. It contains more general damping terms than the standard phenomenological result. It allows for a simple discussion of the phenomenon of self-acceleration. Further, attention is given to such features as: the essential singularity of the eigenmatrix of the electromagnetic field, emission and scattering processes, and thermal radiation.