Equivalent multipole operators for degenerate Rydberg states

Abstract

quantum number n. We seek to develop similar rules for higher multipole operators by expressing equivalent operators in terms only of the two vector constants of motion—the orbital angular momentum L and the Runge-Lenz vector A—appropriate to the degenerate hydrogenic shell. Equivalence of two operators means here that they yield identical matrix elements within a subspace of Hilbert space that corresponds to fixed n. Such equivalent-operator techniques permit direct algebraic calculation of perturbations of Rydberg atoms by external fields and often exact analytical results for transition probabilities. Explicit expressions for equivalent quadrupole and octupole operators are derived, examples are provided, and general aspects of the problem are discussed. Highly excited Rydberg states with the same principal quantum number n have small deviations from pure hydro- genic behavior. The degenerate shell of these states forms the basis of a representation of the O4 symmetry group 1 associated with the 1 /r Coulomb potential governing the dy- namics of the Rydberg electron. Many structural properties of the Rydberg atom can then be calculated by using alge- braic rules and group representation techniques. These fea- tures combine mathematical beauty with pragmatic useful- ness. Moreover, such algebraic techniques facilitate direct quantal and classical solution of Rydberg atoms in static ex- ternal electric and magnetic fields 2, slow collisions with Rydberg atoms 3-5, and intrashell dynamics of a Rydberg atom in time-dependent electric and magnetic fields 6. For example, analytical probabilities have been derived 3-5, without the need for any perturbative and numerical analysis, for the full array of l →l transitions in atomic hydrogen Hnl induced by a time-varying weak electric field gener- ated by adiabatic collision with slow ions. The dimension of the degenerate subspace grows as n 2 without electron spin and traditional close-coupling R-matrix calculations using spatial wave functions become prohibitively difficult and ultimately impractical, either be- cause of the sheer dimension of the space or because of the large number of oscillations. Rydberg states with n as large as several hundred are now accessible to observations and experiments. The group representation technique may there- fore offer the only practical and effective way of solving problems involving such Rydberg states. In so doing, some essential underlying physics can be exposed, as an additional asset.