Hydrogen atom in a strong magnetic field: on the existence of the third integral of motion

Abstract

The problem of the existence of the third integral of motion for the classical Hamiltonian describing a hydrogen atom in a magnetic field is studied by numerical methods. It is found that the third integral is isolating for all initial conditions for which the energy is lower than a critical energy, beyond which the phase orbits are unstable and the Hamilton system can behave stochastically. This critical energy depends upon the strength of the magnetic field and the value of the z component of the angular momentum. The critical energy approaches the (classical) ionisation energy in the weak-field and strong-field limits, while it is lowest in the transition region. The consequences for the quantum mechanical energy spectrum of the hydrogen atom are discussed: the existence of this approximate dynamical symmetry would allow for close anti-crossings of levels, and might facilitate the analytic calculations of the energy levels below the critical energy. In discussion of the correspondence diagram a criticism of an earlier paper is given.