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##

,

Figures
C.9-C.17,
4.3
and
4.7.

In the first three figures
no web-like structure can be seen developing, probably due to the large
value of , which makes the phase space structures quite
coarse-grained. This, together with the fact that the initial position
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 ,
takes on
rather small values for larger or , because
in the channels
the
phase space density
rapidly gets transported away to even more distant parts of the web.

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Martin Engel 2004-01-01