Quadratic Zeeman Spectra for the Hydrogen Atom by Means of Semiclassical Quantization

The elliptic cylindrical coordinates of type I adapted to the Fock hypersphere in momentum space of the Kepler motion and their canonical momenta are used to construct an analytic form of the classical action integrals which yield an adequate parametrization of the KAM (Kolmogorov-Arnold-Moser) tori of the Kepler trajectories weakly perturbed by a uniform magnetic field. The semiclassical quantization formula so provided presents a prototype of the exact EBK (Einstein-Brillouin-Kepler) quantization scheme, and the resulting quantized energies vs the magnetic field strength correspond to the quadratic Zeeman spectra of each Rydberg multiplet liftedby the perturbation.