Conclusion

This study is intended to make a contribution to the still developing theory of quantum chaos, by investigating an important model system that in classical mechanics is characterized by weak chaos: the kicked harmonic oscillator.

Well-known classical properties of this model system that also manifest themselves in quantum mechanics -- above all the stochastic webs generated by this system and the diffusive dynamics within the channels of these webs -- make the investigation of its quantum dynamics a nontrivial task.

For the numerical iteration of the quantum map I have chosen to use the matrix representation of the FLOQUET operator with respect to the eigenbasis of the unkicked, i.e. free, harmonic oscillator. While from a numerical point of view it is difficult to evaluate the expressions for the matrix elements of the FLOQUET operator for the kick potential used here, once this has been accomplished one has a very powerful and numerically stable method for iterating the quantum map even for very long times. In practically all cases considered, the numerical error could safely be attributed to the insufficient size of the basis used; other sources of numerical error -- which cannot be avoided for finite differences methods, for example -- did not play a major role.

In quantum phase space, the eigenstates
of the harmonic
oscillator are nicely localized around the corresponding classical
trajectories of the harmonic oscillator.
This also means that the whole basis
is localized in a circular region centered
about the origin of phase space. Therefore, by systematically increasing
, quantum states in an increasingly large region of
phase space can be completely described using this basis.
Furthermore, although the
are
tailor-made
for
the *free* harmonic oscillator,
they also turn out to be well suited to
describe the quantum states that are subject to the nontrivial dynamics of
the *kicked* harmonic oscillator.

The robust
numerical method,
using this basis with values of
up to 6000,
was fundamental to
exploring the quantum dynamics of the model
system both for the *resonance* and *nonresonance* cases.
It allowed to establish the result that, in the first case,
the quantum map generates quantum stochastic webs which are very
similar to their classical counterparts in several ways. Both types of
web are characterized by the same topology and symmetry, and in the
channels of both types of web the dynamics is unbounded. While in the
classical case the dynamics in the channels is diffusive, the
quantum dynamics
there
is either diffusive or even ballistic.
This result is especially interesting as it is in disagreement with the
general hypothesis on quantum chaos
that classically chaotic systems with diffusive dynamics are characterized
by quantum suppression of diffusion --
at least for initial states
that are located in those
regions of phase space (of a weakly chaotic system) being
characterized by classically chaotic dynamics.

It has to be mentioned that for obtaining this result the numerical algorithm had to be pushed to its limit, with the consequence of declining accuracy. It would have been desirable to further increase the basis size in order to improve on the accuracy and to be able to follow the unbounded dynamics for longer times, but this would have required a considerable additional numerical effort. On the other hand, the result of quantum non-suppression of chaos in the resonance case is supported by a convincing analytical argument, such that its validity can hardly be questioned. However, clarification of this issue might be an appropriate object of future work on the kicked harmonic oscillator.

The numerical results in the case of *non*resonance are in
agreement with the conventional quantum suppression of diffusion
hypothesis, as this is exactly what is found for the model system
considered here.
This numerical result of
quantum localization of the nonresonant kicked harmonic oscillator
is given an analytical explanation in terms of the theory of ANDERSON localization.
While the general idea of proving localization by mapping the problem
onto the well-understood ANDERSON model is
motivated by
the
theory of quantum localization in the kicked rotor,
it should be noted that establishing this mapping for the
kicked harmonic oscillator
was much less straight-forward and required
a number of
additional considerations.
For example,
the kicked harmonic oscillator's hopping matrix elements for interaction
between the sites of an ANDERSON-like lattice are much more intricate
than those of the kicked rotor.
Nevertheless, localization could be explained analytically in this way.

Resonances are of particular importance throughout this study. Classically, resonances with respect to are important because they give rise to classical stochastic webs. The fact that quantum theory predicts the existence of quantum stochastic webs, too, and that these quantum webs are obtained in exactly the same cases of resonance with respect to , is a nice example of quantum-classical correspondence, which has been confirmed numerically in this study. Finally, two theories have been discussed in this study that are entirely independent of each other: the theoretical foundation of quantum stochastic webs and the theory of ANDERSON localization. From a -resonance point of view it is a nice and reassuring feature of the latter that it is consistent with the former in that it predicts localization for all values of with the exception of the resonant that give rise to stochastic webs. So both theories combined give the complete quantum picture in a complementary way, and there is no contradiction between the two.

By correspondence, even in the quantum case the resonances with respect
to
might
be understood to be an essentially classical phenomenon,
because they are closely associated with the existence of
classical
stochastic webs.
However, these classical resonances
turn out
to be true *quantum resonances* as well,
since they are obtained through even two independent
truly quantum mechanical theories,
without explicitly assuming resonance in the classical case.
-- In addition to the resonances with respect to there exists another
class of resonances, with respect to , which is of importance
in the analytical discussion of quantum web formation.
These -resonances have no obvious classical counterpart, such that
they manifest another class of true quantum resonances in the
quantum kicked harmonic oscillator.

Finally, I want to point to the fact that the numerical work needed to come to the above conclusions was indeed extensive. First, for investigating the quantum webs above all computers with large memory were needed. Second, in order to study the localization phenomena veritable long-time simulations were performed for which high speed of computation was the prime requirement. Therefore, only the availability of a large number of fast workstations made it possible to study the model system for many different parameter values and initial conditions.

The results obtained here are but a small contribution to the theory of quantum chaos, concentrating on a particular -- though important -- model system. In order to get a better understanding of quantum chaos, more classically chaotic model systems need to be studied with respect to their quantum behaviour. It might be of particular interest to consider other web-generating systems, some of which have been discussed in chapter 1.

For now, a *general*
and *comprehensive*
theory of quantum chaos, applicable to all
systems, regardless of the degree of their classical chaoticity,
and describing all their essential features,
is still lacking.

Wie's dich auch aufzuhorchen treibt, |

Das Dunkel, das Rätsel, die Frage bleibt. |

Die Frage bleibt |
---|

THEODOR FONTANE |