Continuous quantum mechanics of single particles in closed and quasi-closed systems: Pt. III and IV

The rigorous solution to Schroedinger's nonrelativistic time-dependent equation of a single electron's spin-orbit or magnetic hyperfine interaction reveals the full dynamics of angular momentum coupling especially in the presence of an external, arbitrarily oriented magnetic field. Besides of making a description of major dynamical properties possible in simple classical terms, it also sheds new light on the seeming quantization of states: They turn out to be exceptional states of dynamical balance, which in the field-free case are only made possible by the respective pairs of Clebsch-Gordan coefficients. Moreover, the results not only show that integer and half-integer quantum numbers of the total angular momentum only apply to this special case but that in the general case they are also able to obtain the well-known splitting pattern of the stationary energy levels in the field as produced by separately identifiable, field-dependent contributions from the spin-orbit or the magnetic hyperfine interactions on the one hand, and the magnetic dipole interaction with external magnetic fields on the other in a way not accessible to the Breit-Rabi formula. The results further demonstrate that the total magnetic moment responds quite sensitively to even weak field strengths, i.e., with substantial changes although the linear Zeeman effect suggests their seeming constancy in low fields also for levels with vertical stroke MJ vertical stroke < l+1/2. Although Schroedinger's equation is equivalent to an energy representation this detailed description of the behavior of the total magnetic moment is made possible by the fortunate fact that the coupling magnetic moments enter the Hamiltonian linearly. This circumstance can be exploited to describe in full detail how the total magnetic moment behaves in an arbitrarily oriented magnetic field. It is found that its motion in this environment consists of a basic Larmor precession about the field direction superimposed on which is an oscillation of the coupling angle. The characteristic frequencies of the total magnetic moment's longitudinal and transverse motions relative to the field direction are in compliance with Bohr's frequency condition and the selection rules for magnetic dipole transitions in exactly the way their frequencies and polarizations are also observed experimentally. (orig.)