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7 Bounded Quantum Systems
systems driven by time-periodic forces in which energy is not conserved but Floquet
energy (quasienergy) is conserved.
In classical systems, constants of the motion constrain phase space trajectories
to surfaces of lower dimension (KAM tori). When these surfaces are destroyed by
resonances, trajectories are free to wander in a chaotic manner throughout regions
of higher dimension in the classical phase space. In quantum systems, we cannot
follow phase space trajectories because of the Heisenberg uncertainty principle. We
cannot simultaneously specify both the momentum and position of a particle with
arbitrarily high precision. However, the transition to chaos in a classical system
manifests itself in the quantum version of that system if the classical structure
occupies a volume of phase space larger than ¯
h d , where ¯
h is Planck’s constant and
d is the number of degrees of freedom.
One way to study the manifestation of chaos in a quantum system is to analyze
the statistical properties of the energy spectrum of the quantum system as its
classical counterpart undergoes a transition from regular (dominated by KAM
surfaces) to chaotic behavior. As discussed in Chap. 6, one manifestation of chaos
is a loss of information about the quantum system. When a constant of the motion
is destroyed, so is the quantum number associated with it (Percival 1973). In this
chapter, we will look at the statistical properties of spectra obtained from a variety
of sources, and we will find a direct connection between the statistical properties
of the energy eigenvalue spectrum of quantum systems and the dynamics of their
classical counterpart.
We begin in Sect. 7.2 by defining what we mean by integrability in quantum
systems. Stated briefly, a quantum system with N degrees of freedom is integrable
if it has N independent globally defined operators that commute with one another.
There is a whole class of quantum systems that have been shown to be integrable.
The Moyal bracket, which is derived in Appendix I, expresses the commutator of
two operators in terms of a sinusoidal function of differential operators acting on
scalar functions. Using this formulation of the commutation relations, Hietarinta
(1982, 1984) has developed a systematic method to search for additional constants
of the motion in a quantum system. Two examples of quantum integrable systems
with two degrees of freedom are given in Sect. 7.2.
In Sect. 7.3, we describe the effect of symmetries on the spectrum of Hamiltonian
matrices. Berry and Wilkenson (1984) have studied the spectrum of a triangular
billiard that can be made integrable or nonintegrable by varying the angles.
They focus not on spectral statistics but on degeneracies that might occur as an
angle is varied. They introduce an important criterion for distinguishing between
degeneracies due to symmetries and accidental degeneracies (diabolical points) that
can occur in the absence of symmetries.
Classical maps are important systems for studying the transition to chaos in
classical dynamics. The classical baker’s map is one of the simplest dynamical
systems that is also a K-flow. The baker’s map was quantized by Balazs and Voros
(1989) and has proven to be a useful system for studying the phenomenon of
scarring. The quantum baker’s map is discussed in Sect. 7.4.
7 Bounded Quantum Systems
systems driven by time-periodic forces in which energy is not conserved but Floquet
energy (quasienergy) is conserved.
In classical systems, constants of the motion constrain phase space trajectories
to surfaces of lower dimension (KAM tori). When these surfaces are destroyed by
resonances, trajectories are free to wander in a chaotic manner throughout regions
of higher dimension in the classical phase space. In quantum systems, we cannot
follow phase space trajectories because of the Heisenberg uncertainty principle. We
cannot simultaneously specify both the momentum and position of a particle with
arbitrarily high precision. However, the transition to chaos in a classical system
manifests itself in the quantum version of that system if the classical structure
occupies a volume of phase space larger than ¯
h d , where ¯
h is Planck’s constant and
d is the number of degrees of freedom.
One way to study the manifestation of chaos in a quantum system is to analyze
the statistical properties of the energy spectrum of the quantum system as its
classical counterpart undergoes a transition from regular (dominated by KAM
surfaces) to chaotic behavior. As discussed in Chap. 6, one manifestation of chaos
is a loss of information about the quantum system. When a constant of the motion
is destroyed, so is the quantum number associated with it (Percival 1973). In this
chapter, we will look at the statistical properties of spectra obtained from a variety
of sources, and we will find a direct connection between the statistical properties
of the energy eigenvalue spectrum of quantum systems and the dynamics of their
classical counterpart.
We begin in Sect. 7.2 by defining what we mean by integrability in quantum
systems. Stated briefly, a quantum system with N degrees of freedom is integrable
if it has N independent globally defined operators that commute with one another.
There is a whole class of quantum systems that have been shown to be integrable.
The Moyal bracket, which is derived in Appendix I, expresses the commutator of
two operators in terms of a sinusoidal function of differential operators acting on
scalar functions. Using this formulation of the commutation relations, Hietarinta
(1982, 1984) has developed a systematic method to search for additional constants
of the motion in a quantum system. Two examples of quantum integrable systems
with two degrees of freedom are given in Sect. 7.2.
In Sect. 7.3, we describe the effect of symmetries on the spectrum of Hamiltonian
matrices. Berry and Wilkenson (1984) have studied the spectrum of a triangular
billiard that can be made integrable or nonintegrable by varying the angles.
They focus not on spectral statistics but on degeneracies that might occur as an
angle is varied. They introduce an important criterion for distinguishing between
degeneracies due to symmetries and accidental degeneracies (diabolical points) that
can occur in the absence of symmetries.
Classical maps are important systems for studying the transition to chaos in
classical dynamics. The classical baker’s map is one of the simplest dynamical
systems that is also a K-flow. The baker’s map was quantized by Balazs and Voros
(1989) and has proven to be a useful system for studying the phenomenon of
scarring. The quantum baker’s map is discussed in Sect. 7.4.