234
7 Bounded Quantum Systems
E = 100, are fairly flat and there are many avoided crossings. Since the slope of
the E-versus-k curve is related to the velocity of the particles, in the lower energy
states where the lattice is mostly chaotic, we find that particles are moving slowly
and there are numerous avoided crossings. For energies E > 100, the slopes of the
bands become greater, and the number of avoided crossings starts to decrease with
increasing energy. Thus, the band structure also contains signatures of chaos in the
unit cells of the lattice.
7.10 Conclusions
In this chapter, we showed that the spectrum of a quantum system can change its
character in energy regions where it is classically chaotic. Indeed, one of the main
tools now used to distinguish integrable from nonintegrable quantum systems is to
study the statistical properties of the energy spectrum of the quantum system. It has
been found that the spectral properties of quantum systems whose classical limit is
completely chaotic are very similar to those obtained from random matrix theory,
indicating that the information content is extremized.
In previous sections, we have found that some dynamical systems have an
eigenvalue spacing distribution that is well-described by the Gaussian orthogonal
ensemble. It is of interest to note that the Riemann zeta function, ζ(
1
2 − iE), which
can be written in the form
ζ(z) =
1
1 − 2 1−z
∞
n=1
(−1)
n+1 n
−z for Re < 0,
(7.58)
has an infinite number of zeros that are complex numbers whose imaginary parts,
E, form a spectrum whose nearest neighbor spectral spacing distribution appears to
correspond to the Gaussian unitary ensemble (Berry 1986).
The effect of symmetry on level spacing statistics has been studied in some detail
by Haake, Kus, and Scharf for the case of kicked tops (Haake et al. 1987a,b; Kus
et al. 1987) (see also Scharf et al. 1988 and Haake 2001). They found a dramatic
change in the level spacing statistics when the symmetry is changed, in agreement
with random matrix predictions.
Nonintegrable quantum systems do not become chaotic (even though their
classical counterpart may be chaotic) in the sense that their wave functions do
not show sensitive dependence on initial conditions. However, Feingold and Peres
(1986) have shown that quantum perturbation expansions are divergent when the
quantum system is classically chaotic. Furthermore, Peres (1991) has found that a
slight change in the Hamiltonian of a quantum system generates only slight changes
in the final state (starting from the same initial state) if the system is classically
regular, but it changes the final state completely if the system is classically chaotic.
7 Bounded Quantum Systems
E = 100, are fairly flat and there are many avoided crossings. Since the slope of
the E-versus-k curve is related to the velocity of the particles, in the lower energy
states where the lattice is mostly chaotic, we find that particles are moving slowly
and there are numerous avoided crossings. For energies E > 100, the slopes of the
bands become greater, and the number of avoided crossings starts to decrease with
increasing energy. Thus, the band structure also contains signatures of chaos in the
unit cells of the lattice.
7.10 Conclusions
In this chapter, we showed that the spectrum of a quantum system can change its
character in energy regions where it is classically chaotic. Indeed, one of the main
tools now used to distinguish integrable from nonintegrable quantum systems is to
study the statistical properties of the energy spectrum of the quantum system. It has
been found that the spectral properties of quantum systems whose classical limit is
completely chaotic are very similar to those obtained from random matrix theory,
indicating that the information content is extremized.
In previous sections, we have found that some dynamical systems have an
eigenvalue spacing distribution that is well-described by the Gaussian orthogonal
ensemble. It is of interest to note that the Riemann zeta function, ζ(
1
2 − iE), which
can be written in the form
ζ(z) =
1
1 − 2 1−z
∞
n=1
(−1)
n+1 n
−z for Re < 0,
(7.58)
has an infinite number of zeros that are complex numbers whose imaginary parts,
E, form a spectrum whose nearest neighbor spectral spacing distribution appears to
correspond to the Gaussian unitary ensemble (Berry 1986).
The effect of symmetry on level spacing statistics has been studied in some detail
by Haake, Kus, and Scharf for the case of kicked tops (Haake et al. 1987a,b; Kus
et al. 1987) (see also Scharf et al. 1988 and Haake 2001). They found a dramatic
change in the level spacing statistics when the symmetry is changed, in agreement
with random matrix predictions.
Nonintegrable quantum systems do not become chaotic (even though their
classical counterpart may be chaotic) in the sense that their wave functions do
not show sensitive dependence on initial conditions. However, Feingold and Peres
(1986) have shown that quantum perturbation expansions are divergent when the
quantum system is classically chaotic. Furthermore, Peres (1991) has found that a
slight change in the Hamiltonian of a quantum system generates only slight changes
in the final state (starting from the same initial state) if the system is classically
regular, but it changes the final state completely if the system is classically chaotic.