Quantum-classical correspondence in the hydrogen atom in weak external fields
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.