Classification of perturbations of the hydrogen atom by small static electric and magnetic fields

Abstract

We consider perturbations of the hydrogen atom by sufficiently small homogeneous static electric and magnetic fields of all possible mutual orientations. Normalizing with regard to the Keplerian symmetry, we uncover resonances and conjecture that the parameter space of this family of dynamical systems is stratified into zones centred on the resonances. The 1 : 1 resonance corresponds to the orthogonal field limit, studied earlier by Cushman & Sadovskií (Cushman & Sadovskií 2000 Physica 142 , 166–196). We describe the structure of the 1 : 1 zone, where the system may have monodromy of different kinds, and consider briefly the 1 : 2 zone.

Dynamical monodromy

“…The global perspective was brought into AMO physics especially by Sadovskii, Zhilinskii, and their colleagues. They have used the new methods to show the presence of monodromy and related phenomena in the hydrogen atom in perpendicular fields [37][38][39], the CO 2 molecule [40][41][42], HCN [43], LiCN [44], systems with coupled angular momenta [45,46], and a number of model oscillator systems [47][48][49]. Quasilinear molecules are discussed at length in [50][51][52], diatomic molecules in fields in [53], and the hydrogen atom in tilted fields in [54,55].…”

Most Typical1∶2Resonant Perturbation of the Hydrogen Atom by Weak Electric and Magnetic Fields

“…Results.-Compared to the 1:1 zone, the 1:2 zone corresponds to the next most important resonance, and it is expected and has been found to contain more dynamically complex systems [7] due to the higher order of the resonance.…”

“…[6] and independently in Ref. [7]. In the latter work the concept of resonance zones was introduced to describe systems near an exact resonance.…”

“…Presently, systems which are well understood dynamically (at the level of their integrable approximations) include the two single field limits (Stark and Zeeman effect) and practically all (near-)orthogonal configurations [7][8][9]. The latter constitute the 1:1 resonance zone.…”

“…For a given value n ! 0 of the Keplerian integral N, we introduce the n-scaled field amplitudes g ¼ Gn 2 , and f ¼ ðf e ; f b Þ ¼ 3ðF e ; F b Þn 3 and related field configuration parameters [7,8,11],…”