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

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. Fabčič, J. Main, and G. Wunner, Phys. Rev. A 79, 043417 (2009)].

Figure 1

(Color online) Fictitious time propagation of the initially Gaussian wave packet (41) located at ${\mathbf{x}}_0=\left(8,0,0\right)$ with the momentum ${\mathbf{p}}_0=\left(1,2,0\right)$ plotted in the plane $z=0$. The classical Kepler ellipse with the same initial conditions is indicated by the dotted curve on the bottom of each panel. For time resolved comparison, that part of the ellipse that has been traversed by the particle so far in each plot is shown by a solid black line. Although the Gaussian wave packet does not stay Gaussian during the period it follows in general the classical path and is recovered after one period $\tau =\pi$, indicating the periodicity of the wave packet. Lengths are given in scaled atomic unit $n{a}_0$ with $a_0$ the Bohr radius [see Eq. (10)].

Figure 2

(Color online) Fictitious-time propagation of GWP (57) with magnetic quantum number $m=0$. The part on the negative $\rho$ axis is obtained by reflecting the positive part at the $z$ axis. The wave packet runs along the classical Kepler ellipse with the corresponding initial values. For details see text.

Figure 3

(Color online) Fictitious-time propagation of the wave function (69) with $\psi \left(r\right)$ given by the Gaussian (68) with $r_0=10.0$, $p_{r_0}=-0.5$, and $\sigma =3.0$ for angular momentum quantum numbers (a) $l=0$, $m=0$ and (b) $l=5$, $m=0$. An interference pattern is observed at the turning points at approximately $\tau =2\pi /5$ in both panels. The initial wave function is expanded in $N=10000$ basis states [Eq. (62)] with $ϵ=0.2$.