,

In the first three figures no web-like structure can be seen developing, probably due to the large value of $\hbar =1.0$ , which makes the phase space structures quite coarse-grained. This, together with the fact that the initial position $(0,\pi)^t$ lies in the heart of the classically stochastic region, may account for the lack of structure in these pictures.

The rest of the figures in this subsection and figure 4.3 convincingly show how the central portions of the stochastic web get filled by the phase space density evolving with time. This process is accelerated for larger values of

, as can be seen in figure C.16, for example. Here, this process takes place very fast, such that after only approximately 500 kicks the algorithm needs to be stopped, due to declining accuracy. The last figure is also interesting in that it quite exactly reproduces the form and orientation of the meshes of the classical stochastic webs as displayed in figure 1.8. In particular, the waviness of the classical webs, i.e. the sinusoidal deviations of the classical skeleton from the rectilinear grid lines (1.45) -- see figure 1.12 for an example -- are clearly visible quantum mechanically. The comparison of the classical figure 1.8 and the figures of the present section also shows that apparently the web-like structures in phase space survive for larger values of

in quantum mechanics.

Note that not only the central four meshes of the stochastic webs are outlined by the evolving quantum states, although some of the figures seem to give this impression. Rather, the stochastic channels further away are explored, as well: see figures 4.7, C.13 (for

) and C.14 (

), for example. But for finite

, $F^{\rm H}(x,p,nT-0;1)$ takes on rather small values for larger $\vert x\vert$ or $\vert p\vert$ , because in the channels the phase space density rapidly gets transported away to even more distant parts of the web.

$\begin{figure} % latex2html id marker 31982 \par \vspace*{-0.0cm} \par \hspace*{... ...Initial state: $\left\vert \psi_0 \right>=\left\vert 0 \right>$. } \end{figure}$

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