  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 article, 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 quasi-classical initial conditions, the matching with classical
mechanics is complete.
