Relativistic precession and spin dynamics of an elliptic Rydberg wave packet

Time evolution of wave packets built from the eigenstates of the Dirac equation for a hydrogenic system is considered. We investigate the space and spin motion of wave packets which, in the non-relativistic limit, are stationary states with a probability density distributed uniformly along the classical, elliptical orbit (elliptic WP). We show that the precession of such a WP, due to relativistic corrections to the energy eigenvalues, is strongly correlated with the spin motion. We show also that the motion is universal for all hydrogenic systems with an arbitrary value of the atomic number Z. The detailed study of the time evolution of quantum wave packets (WPs) in simple atomic or molecular systems has been the object of growing attention for more than ten years [1]. Most of the previous theoretical studies were done in non-relativistic framework. In the field of relativistic quantum mechanics most of the efforts have been focused on the problem of the interaction between the atoms and a mixture of static fields with, most of the time, intense laser fields [2–11]. Under these conditions the use of a relativistic theory is fully justified since the external field is then able to bring considerable energy to the WP. For isolated atoms, however, the use of relativistic dynamics is more questionable, if the WP is followed or observed only during a short period of time. In ref. [12] relativistic wave packets, corresponding to circular orbits, have been constructed for hydrogenic atoms with a large Z, and propagated over time according to the Dirac equation. Particular attention was paid to the spin collapse event, i.e. to the maximum entanglement, in the course of time, of the spin degree of freedom with the spatial ones. This phenomenon was first shown to take place for a WP in a harmonic oscillator with a spin-orbit force [13], where it is periodic. For this reason it has been called the spin-orbit pendulum. In the Dirac equation with a Coulomb Relativistic precession 2 potential, it is produced by the built-in spin-orbit force and it is not periodic. The time scale where this effect can manifest itself was discussed in ref. [12], as a function of the charge Z of the atom and the average principal quantum number N of the WP. This phenomenon is expected to take place on a longer time scale like the other time dependent quantum effects of spreading and of …