Dynamical Symmetries and Well-Localized Hydrogenic Wave Packets

In recent years, new experimental techniques opened way to creation and study of high energy (Rydberg) states in atoms. These states are described by approximate hydrogenic wave functions with very large principal quantum numbers. Some new effects, as the dynamical localization and the dynamical chaos, have attracted considerable interest. Explanation of these phenomena uses classical equations of motion [1]. It is reasonable to look for an alternative quantum description on the basis of semi-classical approximations, which is naturally provided by a coherent states (CS) formalism. In Section 2, starting from the O(4, 2) dynamical group approach [2] and using three schemes of reduction to subgroups [3]: O(4, 2) ⊃ O(4) ∼ O(3) ⊗ O(3), O(4, 2) ⊃ O(2, 2) ∼ O(2, 1) ⊗ O(2, 1), O(4, 2) ⊃ O(3)⊗O(2, 1), we construct composite CS in physical and auxiliary (“tilted”) representations [4]. We use two types of generating operators of CS with different procedures of transition to a classical limit. In particular, the generating operators for Perelomov SO(3) and SO(2, 1) CS [5], Barut–Girardello SO(2, 1) CS [2,6], generalized hypergeometric CS [7], Brif SO(3) and SO(2, 1) algebra eigenstates [8], may be used for this purpose. The CS are separated into two classes with different semi-classical behavior. The hydrogenic CS wave functions have a complicated form, so it is reasonable to use simplified asymptotic expressions. In Section 3 we describe a method for asymptotic estimate and obtain well-localized hydrogenic wave packets for circular and elliptic orbits. A similar asymptotic estimate method is used in the theory of CS path integrals. We believe that the approach discussed in this paper can be applied to computation of the CS path integrals for the hydrogen atom and other systems with known dynamical symmetry.