Trajectories of an atomic electron in a magnetic field

Classical trajectories of an atomic electron in a magnetic field are calculated for various values of the field strength $B$. Qualitative properties of these trajectories are examined. With use of a scaling law, it is shown that the equations of motion can be written in a form such that they depend upon only one parameter, which may be regarded as a reduced angular momentum (proportional to $L_z{B}^{\frac{1}{3}}$). For small values of this parameter there is an "elliptical regime" in which the trajectory may be regarded as a Kepler ellipse with orbital parameters that evolve slowly in time. For large values of the parameter there is a "helical regime" in which the electron circles rapidly around a magnetic field line and bounces slowly back and forth along the field. Between these two regimes there is an irregular regime, with "chaotic" orbits and a "transition regime" in which the trajectories can be described in oblate spheroidal coordinates. Bound states persist even at energies above the escape energy, provided that the angular momentum (or field strength) is sufficiently large. With use of action-variable quantization, some formulas for semiclassical energy eigenvalues are given for regimes where the trajectories are regular.