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= Quantum chaos =
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Introduction
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Quantum chaos is a branch of physics which studies how chaotic
classical dynamical systems can be described in terms of quantum
theory. The primary question that quantum chaos seeks to answer is:
"What is the relationship between quantum mechanics and classical
chaos?" The correspondence principle states that classical mechanics
is the classical limit of quantum mechanics, specifically in the limit
as the ratio of Planck's constant to the action of the system tends to
zero. If this is true, then there must be quantum mechanisms
underlying classical chaos (although this may not be a fruitful way of
examining classical chaos). If quantum mechanics does not demonstrate
an exponential sensitivity to initial conditions, how can exponential
sensitivity to initial conditions arise in classical chaos, which must
be the correspondence principle limit of quantum mechanics?
In seeking to address the basic question of quantum chaos, several
approaches have been employed:
# Development of methods for solving quantum problems where the
perturbation cannot be considered small in perturbation theory and
where quantum numbers are large.
# Correlating statistical descriptions of eigenvalues (energy levels)
with the classical behavior of the same Hamiltonian (system).
# Semiclassical methods such as periodic-orbit theory connecting the
classical trajectories of the dynamical system with quantum features.
# Direct application of the correspondence principle.
History
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During the first half of the twentieth century, chaotic behavior in
mechanics was recognized (as in the three-body problem in celestial
mechanics), but not well understood. The foundations of modern quantum
mechanics were laid in that period, essentially leaving aside the
issue of the quantum-classical correspondence in systems whose
classical limit exhibit chaos.
Approaches
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Questions related to the correspondence principle arise in many
different branches of physics, ranging from nuclear to atomic,
molecular and solid-state physics, and even to acoustics, microwaves
and optics. Important observations often associated with classically
chaotic quantum systems are spectral level repulsion, dynamical
localization in time evolution (e.g. ionization rates of atoms), and
enhanced stationary wave intensities in regions of space where
classical dynamics exhibits only unstable trajectories (as in
scattering).
In the semiclassical approach of quantum chaos, phenomena are
identified in spectroscopy by analyzing the statistical distribution
of spectral lines and by connecting spectral periodicities with
classical orbits. Other phenomena show up in the time evolution of a
quantum system, or in its response to various types of external
forces. In some contexts, such as acoustics or microwaves, wave
patterns are directly observable and exhibit irregular amplitude
distributions.
Quantum chaos typically deals with systems whose properties need to be
calculated using either numerical techniques or approximation schemes
(see e.g. Dyson series). Simple and exact solutions are precluded by
the fact that the system's constituents either influence each other in
a complex way, or depend on temporally varying external forces.
Quantum mechanics in non-perturbative regimes
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For conservative systems, the goal of quantum mechanics in
non-perturbative regimes is to find
the eigenvalues and eigenvectors of a Hamiltonian of the form
: H = H_s + \varepsilon H_{ns}, \,
where H_s is separable in some coordinate system, H_{ns} is
non-separable in the coordinate system in which H_{s} is separated,
and \epsilon is a parameter which cannot be considered small.
Physicists have historically approached problems of this nature by
trying to find the coordinate system in which the non-separable
Hamiltonian is smallest and then treating the non-separable
Hamiltonian as a perturbation.
Finding constants of motion so that this separation can be performed
can be a difficult (sometimes impossible) analytical task. Solving
the classical problem can give valuable insight into solving the
quantum problem. If there are regular classical solutions of
the same Hamiltonian, then there are (at least) approximate constants
of motion, and by solving the classical problem, we gain clues how to
find them.
Other approaches have been developed in recent years. One is to
express the Hamiltonian in
different coordinate systems in different regions of space, minimizing
the non-separable part of the Hamiltonian in each region.
Wavefunctions are obtained in these regions, and eigenvalues are
obtained by matching boundary conditions.
Another approach is numerical matrix diagonalization. If the
Hamiltonian matrix is computed in any complete basis, eigenvalues and
eigenvectors are obtained by diagonalizing
the matrix. However, all complete basis sets are infinite, and we
need to truncate the basis and still obtain accurate results. These
techniques boil down to choosing a truncated basis from which accurate
wavefunctions can be constructed. The computational time required to
diagonalize a matrix scales as N^3, where N is the dimension of the
matrix, so it is important to choose the smallest basis possible from
which the relevant wavefunctions can be constructed. It is also
convenient to choose a basis in which the matrix
is sparse and/or the matrix elements are given by simple algebraic
expressions because computing matrix elements can also be a
computational burden.
A given Hamiltonian shares the same constants of motion for both
classical and quantum
dynamics. Quantum systems can also have additional quantum numbers
corresponding to discrete symmetries (such as parity conservation from
reflection symmetry). However, if we merely find quantum solutions of
a Hamiltonian which is not approachable by perturbation theory, we may
learn a great deal about quantum solutions, but we have learned little
about quantum chaos. Nevertheless, learning how to solve such quantum
problems is an important part of answering the question of quantum
chaos.
Correlating statistical descriptions of quantum mechanics with classical behavio
r
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Statistical measures of quantum chaos were born out of a desire to
quantify spectral features of complex systems. Random matrix theory
was developed in an attempt to characterize spectra of complex nuclei.
The remarkable result is that the statistical properties of many
systems with unknown Hamiltonians can be predicted using random
matrices of the proper
symmetry class. Furthermore, random matrix theory also correctly
predicts statistical properties
of the eigenvalues of many chaotic systems with known Hamiltonians.
This makes it useful as a tool for characterizing spectra which
require large numerical efforts to compute.
A number of statistical measures are available for quantifying
spectral features in a simple way. It is of great interest whether or
not there are universal statistical behaviors of classically chaotic
systems. The statistical tests mentioned here are universal, at least
to systems with few degrees of freedom (Berry and Tabor have put
forward strong arguments for a Poisson distribution in the case of
regular motion and Heusler et al. present a semiclassical explanation
of the so-called Bohigas-Giannoni-Schmit conjecture which asserts
universality of spectral fluctuations in chaotic dynamics). The
nearest-neighbor distribution (NND) of energy levels is relatively
simple to interpret and it has been widely used to describe quantum
chaos.
Qualitative observations of level repulsions can be quantified and
related to the classical dynamics
using the NND, which is believed to be an important signature of
classical dynamics in quantum systems. It is thought that regular
classical dynamics is manifested by a Poisson distribution of energy
levels:
:P(s) = e^{-s}.\
In addition, systems which display chaotic classical motion are
expected to be characterized by the statistics of random matrix
eigenvalue ensembles. For systems invariant under time reversal, the
energy-level statistics of a number of chaotic systems have been shown
to be in good agreement with the predictions of the Gaussian
orthogonal ensemble (GOE) of random matrices, and it has been
suggested that this phenomenon is generic for all chaotic systems with
this symmetry. If the normalized spacing between two energy levels is
s, the normalized distribution of spacings is well approximated by
:P(s) = \frac{\pi}{2}se^{-\pi s^2/4}.
Many Hamiltonian systems which are classically integrable
(non-chaotic) have been found to have quantum solutions that yield
nearest neighbor distributions which follow the Poisson distributions.
Similarly, many systems which exhibit classical chaos have been found
with quantum solutions yielding a Wigner-Dyson distribution, thus
supporting the ideas above. One notable exception is diamagnetic
lithium which, though exhibiting classical chaos, demonstrates Wigner
(chaotic) statistics for the even-parity energy levels and nearly
Poisson (regular) statistics for the odd-parity energy level
distribution.
Periodic orbit theory
=======================
Periodic-orbit theory gives a recipe for computing spectra from the
periodic orbits of a system. In contrast to the
Einstein-Brillouin-Keller method of action quantization, which applies
only to integrable or near-integrable systems and computes individual
eigenvalues from each trajectory, periodic-orbit theory is applicable
to both integrable and non-integrable systems and asserts that each
periodic orbit produces a sinusoidal fluctuation in the density of
states.
The principal result of this development is an expression for the
density of states which is the trace of the semiclassical Green's
function and is given by the Gutzwiller trace formula:
: g_c(E) = \sum_k T_k \sum_{n=1}^\infty
\frac{1}{2\sinh{(\chi_{nk}/2)}}\,
e^{i(nS_k - \alpha_{nk} \pi/2)}.
Recently there was a generalization of this formula for arbitrary
matrix Hamiltonians that involves a Berry phase-like term stemming
from spin or other internal degrees of freedom. The index k
distinguishes the primitive periodic orbits: the shortest period
orbits of a given set of initial conditions. T_k is the period of the
primitive periodic orbit and S_k is its classical action. Each
primitive orbit retraces itself, leading to a new orbit with action
nS_k and a period which is an integral multiple n of the primitive
period. Hence, every repetition of a periodic orbit is another
periodic orbit. These repetitions are separately classified by the
intermediate sum over the indices n. \alpha_{nk} is the orbit's Maslov
index.
The amplitude factor, 1/\sinh{(\chi_{nk}/2)}, represents the square
root of the density of neighboring orbits. Neighboring trajectories
of an unstable periodic orbit diverge exponentially in time from the
periodic orbit. The quantity \chi_{nk} characterizes the instability
of the orbit. A stable orbit moves on a torus in phase space, and
neighboring trajectories wind around it. For stable orbits,
\sinh{(\chi_{nk}/2)} becomes \sin{(\chi_{nk}/2)}, where \chi_{nk} is
the winding
number of the periodic orbit. \chi_{nk} = 2\pi m, where m is the
number of times that neighboring orbits intersect the periodic orbit
in one period. This presents a difficulty because \sin{(\chi_{nk}/2)}
= 0 at a classical bifurcation. This causes that orbit's contribution
to the energy density to diverge. This also occurs in the context of
photo-absorption spectrum.
Using the trace formula to compute a spectrum requires summing over
all of the periodic orbits of a system. This presents several
difficulties for chaotic systems: 1) The number of periodic orbits
proliferates exponentially as a function of action. 2) There are an
infinite number of periodic orbits, and the convergence properties of
periodic-orbit theory are unknown. This difficulty is also present
when applying periodic-orbit theory to regular systems. 3) Long-period
orbits are difficult to compute because most trajectories are unstable
and sensitive to roundoff errors and details of the numerical
integration.
Gutzwiller applied the trace formula to approach the anisotropic
Kepler problem (a single particle in a 1/r potential with an
anisotropic mass tensor)
semiclassically. He found agreement with quantum computations for low
lying (up to n = 6) states for small anisotropies by using only a
small set of easily computed periodic orbits, but the agreement was
poor for large anisotropies.
The figures above use an inverted approach to testing periodic-orbit
theory. The trace formula asserts that each periodic orbit
contributes a sinusoidal term to the spectrum. Rather than dealing
with the computational difficulties surrounding long-period orbits to
try to find the density of states (energy levels), one can use
standard quantum mechanical perturbation theory to compute eigenvalues
(energy levels) and use the Fourier transform to look for the periodic
modulations of the spectrum which are the signature of periodic
orbits. Interpreting the spectrum then amounts to finding the orbits
which correspond to peaks in the Fourier transform.
Rough sketch on how to arrive at the Gutzwiller trace formula
===============================================================
# Start with the semiclassical approximation of the time-dependent
Green's function (the Van Vleck propagator).
# Realize that for caustics the description diverges and use the
insight by Maslov (approximately Fourier transforming to momentum
space (stationary phase approximation with h a small parameter) to
avoid such points and afterwards transforming back to position space
can cure such a divergence, however gives a phase factor).
# Transform the Greens function to energy space to get the energy
dependent Greens function ( again approximate Fourier transform using
the stationary phase approximation). New divergences might pop up that
need to be cured using the same method as step 3
# Use d(E)=-\frac{1}{\pi}\Im(\operatorname{Tr}(G(x,x^\prime,E))
(tracing over positions) and calculate it again in stationary phase
approximation to get an approximation for the density of states d(E).
Note: Taking the trace tells you that only closed orbits contribute,
the stationary phase approximation gives you restrictive conditions
each time you make it. In step 4 it restricts you to orbits where
initial and final momentum are the same i.e. periodic orbits. Often it
is nice to choose a coordinate system parallel to the direction of
movement, as it is done in many books.
Closed orbit theory
=====================
Closed-orbit theory was developed by J.B. Delos, M.L. Du, J. Gao, and
J. Shaw. It is similar to
periodic-orbit theory, except that closed-orbit theory is applicable
only to atomic and molecular spectra and yields the oscillator
strength density (observable photo-absorption spectrum) from a
specified initial state whereas periodic-orbit theory yields the
density of states.
Only orbits that begin and end at the nucleus are important in
closed-orbit theory. Physically, these are associated with the
outgoing waves that are generated when a tightly bound electron is
excited to a high-lying state. For Rydberg atoms and molecules, every
orbit which is closed at the nucleus is also a periodic orbit whose
period is equal to either the closure time or twice the closure time.
According to closed-orbit theory, the average oscillator strength
density at constant \epsilon is given by a smooth background plus an
oscillatory sum of the form
:
f(w) = \sum_k \sum_{n=1}^{\infty} D^{i}_{\it nk}
\sin(2\pi nw\tilde{S_k} - \phi_{\it nk}).
\phi_{\it nk} is a phase that depends on the Maslov index and other
details of the orbits. D^i_{\it nk} is the recurrence amplitude of a
closed orbit for a given initial state (labeled i). It contains
information about the stability of the orbit, its initial and final
directions, and the matrix element of the dipole operator between the
initial state and a zero-energy Coulomb wave. For scaling systems such
as Rydberg atoms in strong fields, the Fourier transform of an
oscillator strength spectrum computed at fixed \epsilon as a function
of w is called a recurrence spectrum, because it gives peaks which
correspond to the scaled action of closed orbits and whose heights
correspond to D^i_{\it nk}.
Closed-orbit theory has found broad agreement with a number of chaotic
systems, including diamagnetic hydrogen, hydrogen in parallel electric
and magnetic fields, diamagnetic lithium, lithium in an electric
field, the H^{-} ion in crossed and parallel electric and magnetic
fields, barium in an electric field, and helium in an electric field.
One-dimensional systems and potential
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For the case of one-dimensional system with the boundary condition
y(0)=0 the density of states obtained from the Gutzwiller formula is
related to the inverse of the potential of the classical system by
\frac{d^{1/2}}{dx^{1/2}} V^{-1}(x)=2 \sqrt \pi \frac{dN(x)}{dx} here
\frac{dN(x)}{dx} is the density of states and V(x) is the classical
potential of the particle, the half derivative of the inverse of the
potential is related to the density of states as in the Wu-Sprung
potential
Recent directions
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The traditional topics in quantum chaos concerns spectral statistics
(universal and non-universal features), and the study of
eigenfunctions (Quantum ergodicity, scars) of various chaotic
Hamiltonian H(x,p;R).
Further studies concern the parametric (R) dependence of the
Hamiltonian, as reflected in e.g. the statistics of avoided crossings,
and the associated mixing as reflected in the (parametric) local
density of states (LDOS). There is vast literature on wavepacket
dynamics, including the study of fluctuations, recurrences, quantum
irreversibility issues etc. Special place is reserved to the study of
the dynamics of quantized maps: the standard map and the kicked
rotator are considered to be prototype problems.
Works are also focused in the study of driven chaotic systems, where
the Hamiltonian H(x,p;R(t)) is time dependent, in particular in the
adiabatic and in the linear response regimes. There is also
significant effort focused on formulating ideas of quantum chaos for
strongly-interacting 'many-body' quantum systems far from
semiclassical regimes.
Berry�Tabor conjecture
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In 1977, Berry and Tabor made a still open "generic" mathematical
conjecture which, stated roughly, is: In the "generic" case for the
quantum dynamics of a geodesic flow on a compact Riemann surface, the
quantum energy eigenvalues behave like a sequence of independent
random variables provided that the underlying classical dynamics is
completely integrable.
See also
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* Scar (physics)
Further resources
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*
* Martin C. Gutzwiller, 'Chaos in Classical and Quantum Mechanics',
(1990) Springer-Verlag, New York .
* Hans-Jürgen Stöckmann, 'Quantum Chaos: An Introduction', (1999)
Cambridge University Press .
*
*Fritz Haake, 'Quantum Signatures of Chaos' 2nd ed., (2001)
Springer-Verlag, New York .
*Karl-Fredrik Berggren and Sven Aberg, "Quantum Chaos Y2K Proceedings
of Nobel Symposium 116" (2001)
*L. E. Reichl, "The Transition to Chaos: In Conservative Classical
Systems : Quantum Manifestations", Springer (2004),
External links
======================================================================
* [
http://www.sciam.com/article.cfm?id=quantum-chaos-subatomic-worlds
Quantum Chaos] by Martin Gutzwiller (1992 and 2008, 'Scientific
American')
* [
http://www.scholarpedia.org/article/Quantum_chaos Quantum Chaos]
Martin Gutzwiller Scholarpedia 2(12):3146.
doi:10.4249/scholarpedia.3146
*[
http://www.scholarpedia.org/article/Category:Quantum_Chaos
Category:Quantum Chaos Scholarpedia]
* [
http://www.ams.org/notices/200801/tx080100032p.pdf What is...
Quantum Chaos] by Ze'ev Rudnick (January 2008, 'Notices of the
American Mathematical Society')
*
[
http://web.williams.edu/go/math/sjmiller/public_html/RH/Hayes_spectrum_riemanni
um.pdf
Brian Hayes, "The Spectrum of Riemannium"; 'American Scientist' Volume
91, Number 4, July-August, 2003 pp. 296-300]. Discusses relation to
the Riemann zeta function.
* [
http://nbn-resolving.de/urn:nbn:de:bsz:14-ds-1213275874643-50420
Eigenfunctions in chaotic quantum systems] by Arnd Bäcker.
*
[
https://web.archive.org/web/20110725154552/http://www.chaosbook.org/
ChaosBook.org]
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