Dmitrii A. Sadovskii and Boris I. Zhilinskii
Tuning the hydrogen atom in crossed fields between the Zeeman and Stark limits
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We consider the hydrogen atom in the orthogonal electric and magnetic
fields whose strength is assumed to be small enough for the Coulomb
$n$-shell perturbation theory to apply. Appropriate scaling of the two
fields leads to a uniform parameterization of the problem by $S$, the
combined strength of the two fields, and $\alpha$, the ratio of the two
field strengths. The initial six dimensional phase space $R_6$ is
lifted to the standard Kustaanheimo-Stiefel 8-space and then reduced
explicitly to the $S2xS2$ reduced space of the $n$-shell using the Lie
transformation to the 3-rd order in $S$. At fixed $S$ the system is
uniformly tuned between the Zeeman and the Stark limits using the
analytic formulae of the perturbation theory. The approach requires
application of the invariant theory, group theory and topology to the
analysis of the dynamics on the reduced space $S2xS2$, and subsequent
explicit transition to the original $R_6$. In particular we follow
the evolution of the four principal periodic orbits (nonlinear normal
modes) and corresponding four relative equilibria on $S2xS2$.
PACS 32.60.+i, 03.20.+i, 03.65.Sq, 46.10.+z
Phys. Rev. A 57(4), 2867--84 (1998)