A classical treatment of the quadratic Zeeman effect in atomic hydrogen

Abstract

The classical Hamiltonian describing the hydrogen atom in the presence of a static magnetic field of arbitrary strength for arbitrary angular momentum is derived. For this Hamiltonian the transition from the regular to the chaotic motion is observed by means of the Poincare mappings. Two different classes of non-planar periodic orbits are traced in both regular and irregular regions. The bifurcations and variation of the periodic motion with the change of the total energy parameter throughout the regular regime and into the chaotic regime are given together with the relevant frequencies. For both classes the stability/instability of the periodic orbits is studied by calculating the linearization matrix in the neighbourhood of the corresponding fixed points of the Poincare mappings. In one class, the class of orbits that approach very close to the nucleus, we have surprisingly found that a set of periodic orbits bifurcate from the same periodic orbit along the field at various values of the energy. These values are determined numerically. A repeated pattern of stability and instability of these orbits exists over decreasing intervals of energy until the escape energy is approached. All these periodic orbits are unstable beyond the ionization limit. On the other hand we have found that the bifurcation of the second class of orbits is, generally, generic. Three sets of the energy separation lines due to three types of periodic motions are given when B = 60 kG with m - 0. Other sets of lines are given for B = 42 kG with m = 0, m = -1 and m = -2. Many of these lines coincide with the spectral lines obtained experimentally by A. Holle et al (1986).The energy spacing 0.64 near the ionisation limit, which has been found recently in the experiments of Holle et al (1986) is due to one of the non-planar orbits. Other new predicted spacings arising from other orbits have been seen in high resolution experiments on atoms in external fields (Main et al 1986).