Motion of a Charged Particle in a Uniform Magnetic Field

Abstract

The quantum-mechanical study of the motion of a charged spin-\textonehalf{} particle in a uniform magnetic field has a number of features which are of interest to theorists. Not only can the Dirac equations be solved completely and analytical solutions be obtained, but unlike the Coulomb problem, the expressions for absorption and emission of radiation can be evaluated exactly with these exact Dirac wave functions without resorting to a multipole expansion of the usual sort that is made in the theory of the hydrogen atom. In the ultrarelativistic limit, reasonably simple, closed expressions can be obtained for synchrotron radiation, or photoproduction of electron-positron pairs in a magnetic field. The latter expression allows one to associate an absorption coefficient with the magnetic field in vacuo, and through the use of dispersion theory, to obtain the real (dispersive) part of the index of refraction. The use of exact wave functions, rather than plane waves, to describe the electron may simplify the anomalous-moment calculations and shed some light on the computation of the complete power series in $\ensuremath{\alpha}$.