Hydrogen atom in a strong magnetic field: Semiclassical quantization using classical adiabatic invariance

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Abstract
The method of adiabatic switching (AS) is applied to the problem of a hydrogen atom in a strong magnetic field, i.e., the quadratic Zeeman effect (QZE). The QZE is one of the simplest realistic physical problems exhibiting classical chaos which presents conceptual and computational obstacles to the implementation of many semiclassical quantization methods, while the highly nonseparable nature of the problem makes exact quantum treatments problematic. The AS method is straightforward, mainly involving integration of Hamilton’s equations of motion, and in addition, unlike most other trajectory-based semiclassical methods, works even in mildly chaotic volumes of classical phase space. In AS, a zeroth-order classical torus is quantized and then the perturbation is switched on adiabatically using a time-dependent or time-independent switching function which is incorporated into the Hamiltonian. A central problem in AS is the choice of the most appropriate zeroth-order tori. Based on a comprehensive study of the classical dynamics of the QZE it is shown that the best zeroth-order tori for AS are those obtained by quantizing the zeroth-order Hamiltonian (i.e., the hydrogen atom) and simultaneously an adiabatic invariant found by Solov’ev. AS is performed for a wide variety of magnetic fields and energies including states lying in mildly chaotic regions of phase space where Solov’ev’s invariant is no longer conserved. Results are generally in excellent agreement with exact quantum results, and additionally the method is self-diagnostic, yielding large standard deviations in energies should it begin to fail. However, AS is seen to break down in the strongly chaotic regions of phase space where the quantum levels display multiple avoided crossings (strongly n-mixing regime). Application of AS to the QZE leads to a number of new developments in the theory of AS, including AS in extended phase space where the Hamiltonian is explicitly independent of time.

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