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

TL;DR

This paper develops a method to analytically propagate nondispersive Gaussian wave packets in the hydrogen atom using a fictitious time variable, enabling complete basis expansion and periodic behavior analysis.

Contribution

It introduces restricted Gaussian wave packets consistent with the Coulomb problem's constraints, providing a new analytical approach for wave packet dynamics in hydrogen.

Findings

Wave packets exhibit nondispersive periodic behavior in fictitious time.

Restricted Gaussian wave packets form a complete basis for hydrogen atom states.

Method enables analytical propagation of localized wave packets.

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 n umbers 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\v{c}i\v{c} et al., submitted].