Boris RUBINSTEIN
Department of Mathematics, University of California,
Davis, CA 95616, USA
E-mail: rubibor@yahoo.com, boris@math.ucdavis.edu

Dynamical symmetries and well-localized hydrogenic wave packets

Abstract:
In recent years, new experimental techniques have made it possible to create and study the high energy (Rydberg) states in atoms. This states can be 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 semiclassical approximations, which is naturally provided by a coherent states (CS) formalism. The hydrogenic CS wave functions have a complicated form, so it is expediently to use in practice simplified asymptotic expressions. Starting from the O(4,2) dynamical group approach [2] and using three schemes of reduction to a subgroup [3]: O(4,2) ÉO(4)~O(3) ÄO(3), O(4,2) ÉO(2,2)~O(2,1) ÄO(2,1), O(4,2) ÉO(3) ÄO(2,1), we construct different CS in physical and auxiliary ("tilted") representations [4]. Using the saddle-point method, we develop general procedures for derivation of a wide class of well-localized (Gaussian) hydrogenic wave packets for circular and elliptic orbits. In addition, we investigate the semiclassical properties of Perelomov SO(3) and SO(2,1) CS, Barut-Girardello SO(2,1) CS, generalized hypergeometric CS [5], Brif SO(3) and SO(2,1) algebra eigenstates [6]. This analysis is directly related to the problem of construction of the CS path integrals.
 

References:

1.  Bellomo P., Stroud C.R., Farrelly D., User T. Phys. Rev. 1998. V.A58. P.3896.
2.  Barut A.O., Raczka R. Theory of Group Representations and Applications, World Scientific, Singapore, 1986.
3.  Zverev V.V., Rubinstein B.Ya. Soviet Physics - Lebedev Institute Reports. 1982. N.11. P.4.
4.  Bechler A. Ann. Phys. 1977. V.108. P.49.
5.  Zverev V.V., Abstract book of EASTMAG-2001, Institute of Metal Physics, Ekaterinburg, 2001, P.345.
6.  Brif C. Int. J. Theor. Phys. 1997. V.36. P.1651.