Quantum-classical correspondence in the hydrogen atom in weak external fields
Paolo Bellomo, C. R. Stroud, Jr., David Farrelly, and T. Uzer
Phys. Rev. A 58, 3896 (1998).
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The complex processes leading to the collisional population of
ultra-long-lived Rydberg states with very high angular momentum can be
explained surprisingly well using classical mechanics. In this paper, we
explain the reason behind this striking agreement between classical theory
and experiment by showing that the classical and quantum dynamics of
Rydberg electrons in weak, slowly varying external fields agree beyond the
mandates of Ehrenfest's theorem. In particular, we show that the
expectation values of angular momentum and Runge-Lenz vectors in hydrogenic
eigenstates obey exactly the same perturbative equations of motion as the
time averages of the corresponding classical variables. By time averaging
the quantum dynamics over a Kepler period, we extend this special
quantum-classical equivalence to Rydberg wave packets relatively well
localized in energy. Finally, the perturbative equations hold well also
for external fields beyond the Inglis-Teller limit, and in the case of
elliptic states, which yield the appropriate quasiclassical initial
conditions, the matching with classical mechanics is complete.
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