Diffusing Wave Packets

In subsection 1.2.4 I have discussed the diffusive
energy growth in the channels of *classical* stochastic webs,
i.e. in the cases of resonance given by equation
(1.33).
In the present subsection I study the *quantum* energy growth in
these cases leading to the emergence of quantum stochastic webs, as
shown in subsection 4.1.1 and
in sections C.1 and
C.2 of the appendix.

as a function of discretized time for rectangular and hexagonal stochastic webs with and . The results of numerical simulations for several values of are shown, where the initial states are located at the unstable periodic points and of intersections of separatrices, as seems suitable for studying unbounded dynamics. The figures are for -- other values of lead to similar results; but note that very large values of can lead to a very fast spreading of the states and thus may necessitate to stop the algorithm after only a short period of time. For comparison, results for , i.e. for the ensemble averaged classical energies (1.74), of classical computations with corresponding parameter values are also shown.

Within the accuracy of the computation, the figures indicate that
for *generic* values of
-- here: for ; nongenericity of in the present
context is defined in equations (4.7) below --
the quantum energy grows in the same way as the classical energy average
does: in the doubly logarithmic plots, the slopes of the respective
graphs indicate an asymptotically *linear* dependence on time,

Many more results of this kind, for other values of generic and
, have been obtained numerically, but are not shown here.
Similarly, numerical results with respect to the third
nontrivial
type of resonance, given by , are not shown here either, but
have been obtained in large numbers; they lead to
similar observations as described above.
Summarizing I have the result that, generically, in quantum mechanics
*diffusive energy growth* within stochastic webs is obtained, just as
in the classical case.

The figures also show results for some *nongeneric* values of
. In the present context, is called nongeneric if there is
an integer such that

This definition of nongenericity of is based on equations
(4.44) and (4.49) below, which are obtained
in a natural way
in subsection 4.2.2
when analytically discussing
dynamical consequences of the symmetries of quantum stochastic webs.
The cases of nongeneric as given by equations
(4.7) are examples of *quantum resonances*;
this is explained in subsection 4.2.2 as well.

With more than convincing numerical accuracy, the figures
-- with and
in figure
4.8, and
and
in figure
4.9 --
indicate that
these
nongeneric
values of asymptotically lead to faster, namely quadratic,
energy growth,

Both energy growth rate results, diffusive (4.6) and ballistic (4.8), are given a theoretical explanation in section 4.2. Note that, irrespective of the (non-) genericity of the value of , for a given resonant always the same symmetric phase space patterns are obtained. In other words, depending on the value of , the same quantum stochastic web may be subject to diffusive or ballistic energy growth of the quantum state moving within the web.

Considering the diffusive energy growth of the quantum states
is one way of discussing the spreading of the states
in the phase plane with time.
Another way is to discuss their VON NEUMANN entropy