Rydberg states of the hydrogen atom in the instantaneous van der Waals potential: quantum mechanical, classical and semiclassical treatment

We present a quantum mechanical, classical and semiclassical study of the energy spectrum of a Rydberg hydrogen atom in the instantaneous van der Waals potential for the L z = 0 case. The semiclassical results are in good agreement with the results of the quantum mechanical calculations within the first-order perturbation theory as well as with the quantum mechanical calculations of other authors. The classical analysis shows that the phase space of the system is separated into the regions of vibrational and rotational motions, which are connected, respectively, with the lower-lying energy levels of doublet symmetry, and with the non-degenerate higher-energy levels. Finally, we compare the classical (eigen)trajectories with the corresponding eigenstates and find that both of them show the same symmetry patterns.