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

摘要

Nondispersive wave packets in a fictitious time variable are calculated analytically for the field-free hydro-gen atom. As is well known by means of the Kustaanheimo-Stiefel transformation the Coulomb problem canbe converted into that of a four-dimensional harmonic oscillator, subject to a constraint. This regularizationmakes use of a fictitious time variable, but arbitrary Gaussian wave packets in that time variable in generalviolate that constraint. The set of "restricted Gaussian wave packets" consistent with the constraint is con-structed and shown to provide a complete basis for the expansion of states in the original three-dimensionalcoordinate space. Using that expansion arbitrary localized Gaussian wave packets of the hydrogen atom can bepropagated 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 staticexternal fields in the subsequent paper [T. Fabcic, J. Main, and G. Wunner, Phys. Rev. A 79, 043417 (2009)].