Collapse and revival in the dynamics of a spin with the spin-orbit potential

We have calculated the time evolution of the density matrix reduced to the spin degree of freedom for a particle in a spherical harmonic oscillator with a constant spin-orbit interaction. We have studied two classes of spatial wave functions in the initial state: pure orbital states |lm right-angle and coherent states. When the initial spin is a pure state, its evolution in time produces a mixed state during a time called the collapse time. This behavior of the spin is similar to that observed in the Jaynes-Cummings model. A similar dynamics is produced for a broad range of geometrical initial conditions for the spin as well as for the wave packet. The system is exactly recurrent but we have also found the existence of approximate recurrences corresponding to a reversal of the spin direction. We have tentatively applied our result to mimic the evolution of the spin in the H atom. In this case only a few spin-orbit partners play a role and, after a long time, produce a state where the admixture is not complete. The average quantum behavior is always very different from that of a classical particle having a spin as a vector of constant length. The increase of the number 2S+1 of spin-orbit partners changes the time scale of the collapse by the factor √S , and a strong mixing is always observed