8
1 Overview
analysis of nuclear scattering processes in terms of random matrix theory. We then
use these tools to define scattering phenomena such as resonance, quasibound states,
and delay times for scattering processes. Finally, in Chap. 8, we show a variety of
experimental and numerical data on nuclear and molecular energy-Ievel sequences
and show that these systems are exhibiting the manifestations of chaos.
Another connection between classically chaotic systems and their quantum
counterpart involves the use of semiclassical path integrals. In Chap. 9, we use the
semiclassical limit of Feynman path integrals to derive the Gutzwiller trace formula,
which expresses the trace of the Green’s function of a quantum system in terms of
periodic orbits of the classical system. We show that the trace formula gives very
good results for the energy levels of the anisotropic Kepler system, a classically
chaotic system. Finally, we conclude Chap. 9 with a numerical and experimental
study of the influence of periodic orbits on the absorption spectrum of diamagnetic
hydrogen.
Chapter 10 is devoted to periodically driven quantum systems, which can be
described using Floquet theory. We show that nonlinear resonances exist in the
Hilbert space of quantum systems, and we use Floquet theory to interpret the
results of dynamic tunneling experiments using cold atoms confined to optical
lattices. We decribe the behavior of the quantum delta-kicked rotor, which was
the first system in which dynamic Anderson localization was observed numericaly.
We also describe extensive experiments on microwave-driven hydrogen that give
experimental confirmation of the existence of higher-order nonlinear resonances
in quantum systems. Finally, we show that the Arnol’d web exists in quantum
systems and plays an important role in destablizing their dynamics, and we show
the influence of chaos on quantum control.
This book contains several appendices that give background on subjects of
importance to this book. For example, there is a review of the effect of symmetries
on the structure of Hamiltonian matrices. There is a derivation of the measures for
Hermitian and unitary matrices used in random matrix theory. There is a derivation
of the normalization constants and expressions for probability distributions of the
Gaussian and circular ensembles in terms of quaternion matrices. There are other
appendices as weIl that will aid the reader with some of the theory concepts in this
book.
We do not have room in this book to discuss in detail all of the interesting
applications of classical and quantum chaos theory, so in the concluding section
of each chapter we have given references to additional topics of interest.
References
Arnol’d VI (1963) Russ Math Surv 189:1885
Driebe DJ (1999) Fully chaotic maps and broken time symmetry. Kluwer Academic Publishers,
Dordrecht
Dyson FJ (1962) J Math Phys 3:140
1 Overview
analysis of nuclear scattering processes in terms of random matrix theory. We then
use these tools to define scattering phenomena such as resonance, quasibound states,
and delay times for scattering processes. Finally, in Chap. 8, we show a variety of
experimental and numerical data on nuclear and molecular energy-Ievel sequences
and show that these systems are exhibiting the manifestations of chaos.
Another connection between classically chaotic systems and their quantum
counterpart involves the use of semiclassical path integrals. In Chap. 9, we use the
semiclassical limit of Feynman path integrals to derive the Gutzwiller trace formula,
which expresses the trace of the Green’s function of a quantum system in terms of
periodic orbits of the classical system. We show that the trace formula gives very
good results for the energy levels of the anisotropic Kepler system, a classically
chaotic system. Finally, we conclude Chap. 9 with a numerical and experimental
study of the influence of periodic orbits on the absorption spectrum of diamagnetic
hydrogen.
Chapter 10 is devoted to periodically driven quantum systems, which can be
described using Floquet theory. We show that nonlinear resonances exist in the
Hilbert space of quantum systems, and we use Floquet theory to interpret the
results of dynamic tunneling experiments using cold atoms confined to optical
lattices. We decribe the behavior of the quantum delta-kicked rotor, which was
the first system in which dynamic Anderson localization was observed numericaly.
We also describe extensive experiments on microwave-driven hydrogen that give
experimental confirmation of the existence of higher-order nonlinear resonances
in quantum systems. Finally, we show that the Arnol’d web exists in quantum
systems and plays an important role in destablizing their dynamics, and we show
the influence of chaos on quantum control.
This book contains several appendices that give background on subjects of
importance to this book. For example, there is a review of the effect of symmetries
on the structure of Hamiltonian matrices. There is a derivation of the measures for
Hermitian and unitary matrices used in random matrix theory. There is a derivation
of the normalization constants and expressions for probability distributions of the
Gaussian and circular ensembles in terms of quaternion matrices. There are other
appendices as weIl that will aid the reader with some of the theory concepts in this
book.
We do not have room in this book to discuss in detail all of the interesting
applications of classical and quantum chaos theory, so in the concluding section
of each chapter we have given references to additional topics of interest.
References
Arnol’d VI (1963) Russ Math Surv 189:1885
Driebe DJ (1999) Fully chaotic maps and broken time symmetry. Kluwer Academic Publishers,
Dordrecht
Dyson FJ (1962) J Math Phys 3:140