Ural State Technical University,

Ekaterinburg, 620002, RUSSIA

E-mail: zverev@dpt.ustu.ru

*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.