6
1 Overview
When we study quantum systems, we have no phase space in which to describe
the evolution of individual orbits because of the Heisenberg uncertainty relations.
A single quantum state occupies volume of order ¯
h d in the classical phase space,
where ¯
h is Planck’s constant and d is the number of degrees of freedom. We are
forced from the outset to study quantum systems at the level of a linear probability
(probability amplitude to be more precise) equation, namely the Schrödinger
equation.
Most of the mechanisms at work in nonlinear classical systems are also at
work in their quantum counterparts. For example, nonlinear resonances exist in
quantum systems and can destroy constants of the motion (good quantum numbers)
in local regions of the Hilbert space. They form self-similar structures, but only
down to scales of order ¯
h d and not to infinitely small scales as they do in classical
systems. However, because the Schrödinger equation is an equation for probability
amplitudes rather than probabilities, we will find some new phenomena that can
occur in quantum systems but not in classical systems.
One of the most important discoveries of quantum chaos theory is that the
statistical properties of energy spectra and scattering delay times indicate that the
information content of a quantum system is extremized (minimized) as its classical
counterpart undergoes a transition to chaos. The idea of studying the spectral
statistics of quantum systems is largely due to Wigner (1951, 1957), who in the
1950s analyzed the statistical properties of nuclear scattering resonances. It was
found that the nearest neighbor spacing of scattering resonances, for some nuclear
scattering processes, has a distribution that agrees with the distribution of spacings
of eigenvalues of ensembles of random Hermitian matrices (the Gaussian ensemble)
whose matrix elements extremize information. The work of Wigner led Dyson
(1962) to study the statistical properties of ensembles of random unitary matrices
(the circular ensembles) that extremize information.
The connection between chaos theory and random matrix theory was made in
1979 by McDonald and Kaufman (1979), who found that classically chaotic quantum billiards have spectral spacing distributions given by the Gaussian ensembles.
Comparison between statistical properties of deterministic quantum systems with
underlying classical chaos and predictions of random matrix theories that extremize
information is now a standard tool of quantum mechanics.
In the early days of quantum mechanics, before the work of Heisenberg and
Schrödinger, the quantum version of a classical system was obtained by quantizing
the action variables. This is straightforward if the classical system is integrable and
one can find the action variables. However, Einstein, who knew of the work of
Poincaré, as early as 1917 (Einstein 1917) pointed out that there may be difficulties
with this method of quantization if invariant tori do not exist in the classical phase
space, as is the case with chaotic systems.
Indeed, until the work of Gutzwiller in the early 1980s (Gutzwiller 1982), there
was no way to link classically chaotic systems to their quantum counterparts.
However, Gutzwiller showed that Feynman path integrals, in the semiclassical limit,
provide such a link, and the spectral properties of a quantum system, whose classical
counterpart is chaotic are determined largely in terms of an infinite sum over the
unstable periodic orbits of the classical system.
1 Overview
When we study quantum systems, we have no phase space in which to describe
the evolution of individual orbits because of the Heisenberg uncertainty relations.
A single quantum state occupies volume of order ¯
h d in the classical phase space,
where ¯
h is Planck’s constant and d is the number of degrees of freedom. We are
forced from the outset to study quantum systems at the level of a linear probability
(probability amplitude to be more precise) equation, namely the Schrödinger
equation.
Most of the mechanisms at work in nonlinear classical systems are also at
work in their quantum counterparts. For example, nonlinear resonances exist in
quantum systems and can destroy constants of the motion (good quantum numbers)
in local regions of the Hilbert space. They form self-similar structures, but only
down to scales of order ¯
h d and not to infinitely small scales as they do in classical
systems. However, because the Schrödinger equation is an equation for probability
amplitudes rather than probabilities, we will find some new phenomena that can
occur in quantum systems but not in classical systems.
One of the most important discoveries of quantum chaos theory is that the
statistical properties of energy spectra and scattering delay times indicate that the
information content of a quantum system is extremized (minimized) as its classical
counterpart undergoes a transition to chaos. The idea of studying the spectral
statistics of quantum systems is largely due to Wigner (1951, 1957), who in the
1950s analyzed the statistical properties of nuclear scattering resonances. It was
found that the nearest neighbor spacing of scattering resonances, for some nuclear
scattering processes, has a distribution that agrees with the distribution of spacings
of eigenvalues of ensembles of random Hermitian matrices (the Gaussian ensemble)
whose matrix elements extremize information. The work of Wigner led Dyson
(1962) to study the statistical properties of ensembles of random unitary matrices
(the circular ensembles) that extremize information.
The connection between chaos theory and random matrix theory was made in
1979 by McDonald and Kaufman (1979), who found that classically chaotic quantum billiards have spectral spacing distributions given by the Gaussian ensembles.
Comparison between statistical properties of deterministic quantum systems with
underlying classical chaos and predictions of random matrix theories that extremize
information is now a standard tool of quantum mechanics.
In the early days of quantum mechanics, before the work of Heisenberg and
Schrödinger, the quantum version of a classical system was obtained by quantizing
the action variables. This is straightforward if the classical system is integrable and
one can find the action variables. However, Einstein, who knew of the work of
Poincaré, as early as 1917 (Einstein 1917) pointed out that there may be difficulties
with this method of quantization if invariant tori do not exist in the classical phase
space, as is the case with chaotic systems.
Indeed, until the work of Gutzwiller in the early 1980s (Gutzwiller 1982), there
was no way to link classically chaotic systems to their quantum counterparts.
However, Gutzwiller showed that Feynman path integrals, in the semiclassical limit,
provide such a link, and the spectral properties of a quantum system, whose classical
counterpart is chaotic are determined largely in terms of an infinite sum over the
unstable periodic orbits of the classical system.