Fictitious-time wave-packet dynamics. I. Nondispersive wave packets in the quantum Coulomb problem

Abstract

Nondispersive wave packets in a fictitious time variable are calculated analytically for the field-free hydrogen atom. As is well known by means of the Kustaanheimo-Stiefel transformation the Coulomb problem can be converted into that of a four-dimensional harmonic oscillator, subject to a constraint. This regularization makes use of a fictitious time variable, but arbitrary Gaussian wave packets in that time variable in general violate that constraint. The set of ``restricted Gaussian wave packets'' consistent with the constraint is constructed and shown to provide a complete basis for the expansion of states in the original three-dimensional coordinate space. Using that expansion arbitrary localized Gaussian wave packets of the hydrogen atom can be propagated analytically and exhibit a nondispersive periodic behavior as functions of the fictitious time. Restricted wave packets with and without well-defined angular momentum quantum numbers are constructed. They will be used as trial functions in time-dependent variational computations for the hydrogen atom in static external fields in the subsequent paper [T. Fab\ifmmode \check{c}\else \v{c}\fi{}i\ifmmode \check{c}\else \v{c}\fi{}, J. Main, and G. Wunner, Phys. Rev. A 79, 043417 (2009)].