Kicked hydrogen

The electron is restricted to move in the half-space x > 0. There is a totally reflecting wall at x = 0. Since the Hamiltonian (8.1.1) of the kicked hydrogen atom and the Hamiltonian of microwave-driven surface state electrons are so similar, we can use many of the results that were derived in Chapter 6. The most important result is the transformation to action and angle variables I and 6, respectively, defined in (6.1.18). The... 

For the kicked rotor, too, we were able to scale out one of the control parameters, leaving only the single control parameter K. A further analogy to the kicked rotor is the fact that in the case of the quantized one-dimensional kicked hydrogen atom (see, e.g., Bliimel and Smilansky (1984)) a reduction to one control parameter is not possible. 

Fig. 8.2. Three stages in the construction of the one-dimensional kicked hydrogen fractal after a) N = 1, h) N = 2 and c) N = S kicks.

Fig. 8.3. Powerlaw decay of the phase-space probability for the kicked hydrogen atom.

At this point our restriction to the positively kicked hydrogen atom enters. If we confine our attention to > 0 only, we can easily obtain a... 

This estimate is independent of a particular location in phase space. This implies that all points of phase space, in particular the period-1 points, are linearly unstable. Thus we have proved that the positively kicked hydrogen atom does indeed not possess any first order eUiptic islands in phase space. It is possible to extend this proof to period-iV points and to show that all period-N points, N integer, are Unearly unstable (Bliimel (1993c)). This implies that the positively kicked hydrogen atom is completely chaotic. We emphasize that, as far as we know, the kicked hydrogen atom is the only model for a physically realizable system where a numerically motivated chaos conjecture was followed up by an analytical proof. In this sense the kicked hydrogen atom is a most remarkable system. 

Prom the physical point of view the absence of stable islands means that all the phase-space probability eventually ionizes. Since all the atomic physics systems investigated to date possess a mixed phase space that shows regular islands embedded in a chaotic sea, the absence of stable islands in the kicked hydrogen atom is a very unique property. 

A more detailed investigation has been carried out [44,45] for the periodically kicked hydrogen atom, in which the sinusoidal dependence of the external field in Eq. (10) is replaced by a periodic sequence of instantaneous kicks. 

Fig. 7. Doubly logarithmic plot of the survival probabilities N(n) in kicked hydrogen (Eq. (11)) as functions of the number n of cks for scaled field strength s = 1. The various curves are obtained with initial conditions on different cuts through phase space corresponding to fixed values of the scaled action f = no and uniformly distributed angles. (From [45])...

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