7.1 Introduction
197
Two-dimensional quantum billiards have been very important for studying the
manifestations of chaos in quantum systems and are the subject of Sect. 7.5. The first
direct evidence that underlying classical chaos might affect the spectral properties
of systems was obtained by McDonald and Kaufman (1979), who found that an
integrable circle billiard had a high probability of small spacings, while the nonintegrable stadium billiard showed definite evidence of level repulsion. These results
were further confirmed for the Sinai billiard, which was found to have a 3 -
statistic that was very close to GOE (Bohigas et al. 1984), and for an integrable
billiard, which was found to have a spectral spacing distribution that satisfied a
Poisson distribution (Casati et al. 1985; Cheng and Lebowitz 1991). All three of
these billiards have now been realized in laboratory experiments using microwave
cavities (Stöckmann and Stein 1990; Graf et al. 1992). The phenomenon of scarring
on energy eigenstates was first observed by McDonald (1983), but was studied
systematically by Heller (1984) for the energy eigenstates of the stadium billiard.
Scars have also been observed in microwave experiments.
Another billiard that has been the focus of study recently is the ripple billiard
(Luna-Acosta et al. 1996; Akguc and Reichl 2000; Li et al. 2002). The ripple billiard
is special because it is possible to construct a Hamiltonian matrix that governs
the quantum dynamics. The dynamics of the ripple billiard can be adjusted to be
integrable, mixed, or fully chaotic as we show in Sect. 7.5.
In classical mechanics, if we are given a dynamical system with two degrees of
freedom, we can often determine if it is integrable or not by numerically computing
the Poincaré surface of section. An alternative method to test for integrability,
that appears to work both for classical and quantum systems was found by Peres
(1984), and we discuss it in Sect. 7.6. For a system with two degrees of freedom
whose energy is conserved, Peres constructs a (hypothetical) second invariant in
the following way. He first finds an operator that does not commute with the
Hamiltonian and is independent of it. He then obtains the long-time average of this
operator. The long-time average is the second invariant. It is independent of the
Hamiltonian and commutes with it. One can then obtain simultaneous eigenvalues
of the Hamiltonian and of this second invariant and form a quantum web. Studies
that have been done to date indicate that for a system that is classically integrable,
this web will be regular, while for systems that are classically nonintegrable, the
web can become irregular. In Sect. 7.6 we show how this method has been used to
study the nature of the quantum dynamics of the D-billiard (Ree and Reichl 1999).
In Sect. 7.7, we show how this method has been used study the behavior of quantum
XY spin models (Srivastava and Muller 1990; Srivastava et al. 1990; Robb and
Reichl 1998).
Anharmonic oscillators provide an alternative type of system in which to study
the transition to chaos. In Sect. 7.8, we consier the dynamics one-dimensional triatomic molecule with interatomic coupling given by the Morse potential (Terasaka
and Matsushita 1985). We show that, as the classical system undergoes a transition
from regular to chaotic behavior, the spectral statistics of the quantum system
undergoes a parallel transition from Poisson-like to Wigner-like behavior.
197
Two-dimensional quantum billiards have been very important for studying the
manifestations of chaos in quantum systems and are the subject of Sect. 7.5. The first
direct evidence that underlying classical chaos might affect the spectral properties
of systems was obtained by McDonald and Kaufman (1979), who found that an
integrable circle billiard had a high probability of small spacings, while the nonintegrable stadium billiard showed definite evidence of level repulsion. These results
were further confirmed for the Sinai billiard, which was found to have a 3 -
statistic that was very close to GOE (Bohigas et al. 1984), and for an integrable
billiard, which was found to have a spectral spacing distribution that satisfied a
Poisson distribution (Casati et al. 1985; Cheng and Lebowitz 1991). All three of
these billiards have now been realized in laboratory experiments using microwave
cavities (Stöckmann and Stein 1990; Graf et al. 1992). The phenomenon of scarring
on energy eigenstates was first observed by McDonald (1983), but was studied
systematically by Heller (1984) for the energy eigenstates of the stadium billiard.
Scars have also been observed in microwave experiments.
Another billiard that has been the focus of study recently is the ripple billiard
(Luna-Acosta et al. 1996; Akguc and Reichl 2000; Li et al. 2002). The ripple billiard
is special because it is possible to construct a Hamiltonian matrix that governs
the quantum dynamics. The dynamics of the ripple billiard can be adjusted to be
integrable, mixed, or fully chaotic as we show in Sect. 7.5.
In classical mechanics, if we are given a dynamical system with two degrees of
freedom, we can often determine if it is integrable or not by numerically computing
the Poincaré surface of section. An alternative method to test for integrability,
that appears to work both for classical and quantum systems was found by Peres
(1984), and we discuss it in Sect. 7.6. For a system with two degrees of freedom
whose energy is conserved, Peres constructs a (hypothetical) second invariant in
the following way. He first finds an operator that does not commute with the
Hamiltonian and is independent of it. He then obtains the long-time average of this
operator. The long-time average is the second invariant. It is independent of the
Hamiltonian and commutes with it. One can then obtain simultaneous eigenvalues
of the Hamiltonian and of this second invariant and form a quantum web. Studies
that have been done to date indicate that for a system that is classically integrable,
this web will be regular, while for systems that are classically nonintegrable, the
web can become irregular. In Sect. 7.6 we show how this method has been used to
study the nature of the quantum dynamics of the D-billiard (Ree and Reichl 1999).
In Sect. 7.7, we show how this method has been used study the behavior of quantum
XY spin models (Srivastava and Muller 1990; Srivastava et al. 1990; Robb and
Reichl 1998).
Anharmonic oscillators provide an alternative type of system in which to study
the transition to chaos. In Sect. 7.8, we consier the dynamics one-dimensional triatomic molecule with interatomic coupling given by the Morse potential (Terasaka
and Matsushita 1985). We show that, as the classical system undergoes a transition
from regular to chaotic behavior, the spectral statistics of the quantum system
undergoes a parallel transition from Poisson-like to Wigner-like behavior.