Quantum chaos is a branch of
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
which studies how
chaotic
Chaotic was originally a Danish trading card game. It expanded to an online game in America which then became a television program based on the game. The program was able to be seen on 4Kids TV (Fox affiliates, nationwide), Jetix, The CW4Kid ...
classical
dynamical systems
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a p ...
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
Chaos theory is an interdisciplinary area of scientific study and branch of mathematics focused on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions, and were once thought to have co ...
?" The
correspondence principle
In physics, the correspondence principle states that the behavior of systems described by the theory of quantum mechanics (or by the old quantum theory) reproduces classical physics in the limit of large quantum numbers. In other words, it says t ...
states that classical mechanics is the
classical limit
The classical limit or correspondence limit is the ability of a physical theory to approximate or "recover" classical mechanics when considered over special values of its parameters. The classical limit is used with physical theories that predict n ...
of quantum mechanics, specifically in the limit as the ratio of
Planck's constant to the
action
Action may refer to:
* Action (narrative), a literary mode
* Action fiction, a type of genre fiction
* Action game, a genre of video game
Film
* Action film, a genre of film
* ''Action'' (1921 film), a film by John Ford
* ''Action'' (1980 fil ...
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?
[''Quantum Signatures of Chaos'', Fritz Haake, Edition: 2, Springer, 2001, , .][ Michael Berry, "Quantum Chaology", pp 104-5 of ''Quantum: a guide for the perplexed'' by ]Jim Al-Khalili
Jameel Sadik "Jim" Al-Khalili ( ar, جميل صادق الخليلي; born 20 September 1962) is an Iraqi-British theoretical physicist, author and broadcaster. He is professor of theoretical physics and chair in the public engagement in scien ...
(Weidenfeld and Nicolson
Weidenfeld & Nicolson Ltd (established 1949), often shortened to W&N or Weidenfeld, is a British publisher of fiction and reference books. It has been a division of the French-owned Orion Publishing Group since 1991.
History
George Weidenfeld a ...
2003), http://www.physics.bristol.ac.uk/people/berry_mv/the_papers/Berry358.pdf .
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
In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle ...
and where quantum numbers are large.
# Correlating statistical descriptions of eigenvalues (energy levels) with the classical behavior of the same
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
(system).
# Study of probability distribution of individual eigenstates (see
scar
A scar (or scar tissue) is an area of fibrous tissue that replaces normal skin after an injury. Scars result from the biological process of wound repair in the skin, as well as in other organs, and tissues of the body. Thus, scarring is a na ...
s and
Quantum ergodicity
In quantum chaos, a branch of mathematical physics, quantum ergodicity is a property of the Quantization (physics), quantization of classical mechanics, classical mechanical systems that are chaos theory, chaotic in the sense of exponential sensit ...
).
#
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
![3dbifmap](https://upload.wikimedia.org/wikipedia/commons/7/79/3dbifmap.jpg)
During the first half of the twentieth century, chaotic behavior in mechanics was recognized (as in the
three-body problem
In physics and classical mechanics, the three-body problem is the problem of taking the initial positions and velocities (or momenta) of three point masses and solving for their subsequent motion according to Newton's laws of motion and Newton's ...
in
celestial mechanics
Celestial mechanics is the branch of astronomy that deals with the motions of objects in outer space. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical objects, such as stars and planets, to ...
), 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
![Eps3kekq](https://upload.wikimedia.org/wikipedia/commons/0/01/Eps3kekq.jpg)
Questions related to the correspondence principle arise in many different branches of physics, ranging from
nuclear
Nuclear may refer to:
Physics
Relating to the nucleus of the atom:
* Nuclear engineering
*Nuclear physics
*Nuclear power
*Nuclear reactor
*Nuclear weapon
*Nuclear medicine
*Radiation therapy
*Nuclear warfare
Mathematics
*Nuclear space
*Nuclear ...
to
atomic,
molecular
A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and bioche ...
and
solid-state physics
Solid-state physics is the study of rigid matter, or solids, through methods such as quantum mechanics, crystallography, electromagnetism, and metallurgy. It is the largest branch of condensed matter physics. Solid-state physics studies how the l ...
, and even to
acoustics
Acoustics is a branch of physics that deals with the study of mechanical waves in gases, liquids, and solids including topics such as vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acoustician ...
,
microwave
Microwave is a form of electromagnetic radiation with wavelengths ranging from about one meter to one millimeter corresponding to frequencies between 300 MHz and 300 GHz respectively. Different sources define different frequency ran ...
s and
optics
Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultraviole ...
. However, classical-quantum correspondence in chaos theory is not always possible. Thus, some versions of the classical butterfly effect do not have counterparts in quantum mechanics.
Important observations often associated with classically chaotic quantum systems are spectral
level repulsion Level repulsion is the quantum mechanical equivalent to a repulsion effect in oscillators. A system of two coupled oscillators has two natural frequencies. As the coupling strength between the oscillators increases, the lower frequency decreases and ...
, 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
Scattering is a term used in physics to describe a wide range of physical processes where moving particles or radiation of some form, such as light or sound, are forced to deviate from a straight trajectory by localized non-uniformities (including ...
). In the semiclassical approach of quantum chaos, phenomena are identified in
spectroscopy
Spectroscopy is the field of study that measures and interprets the electromagnetic spectra that result from the interaction between electromagnetic radiation and matter as a function of the wavelength or frequency of the radiation. Matter wa ...
by analyzing the statistical distribution of spectral lines and by connecting spectral periodicities with classical orbits. Other phenomena show up in the
time evolution
Time evolution is the change of state brought about by the passage of time, applicable to systems with internal state (also called ''stateful systems''). In this formulation, ''time'' is not required to be a continuous parameter, but may be disc ...
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
The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of amplit ...
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
In scattering theory, a part of mathematical physics, the Dyson series, formulated by Freeman Dyson, is a perturbative expansion of the time evolution operator in the interaction picture. Each term can be represented by a sum of Feynman diagrams. ...
). 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
![hfspec1](https://upload.wikimedia.org/wikipedia/commons/f/fd/Hfspec1.jpg)
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
:
where
is separable in some coordinate system,
is non-separable in the coordinate system in which
is separated, and
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
, where
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 behavior
![Qdfels](https://upload.wikimedia.org/wikipedia/commons/5/53/Qdfels.jpg)
Statistical measures of quantum chaos were born out of a desire to quantify spectral features of complex systems.
Random matrix
In probability theory and mathematical physics, a random matrix is a matrix-valued random variable—that is, a matrix in which some or all elements are random variables. Many important properties of physical systems can be represented mathemat ...
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
A berry is a small, pulpy, and often edible fruit. Typically, berries are juicy, rounded, brightly colored, sweet, sour or tart, and do not have a stone or pit, although many pips or seeds may be present. Common examples are strawberries, raspb ...
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
In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known co ...
of energy levels:
:
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
, the normalized distribution of spacings is well approximated by
:
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.
Semiclassical methods
Periodic orbit theory
![Bhevendk1](https://upload.wikimedia.org/wikipedia/commons/1/1c/Bhevendk1.jpg)
Periodic-orbit theory gives a recipe for computing spectra from the periodic orbits of a system. In contrast to the
Einstein–Brillouin–Keller method
The Einstein–Brillouin–Keller method (EBK) is a semiclassical method (named after Albert Einstein, Léon Brillouin, and Joseph B. Keller) used to compute eigenvalues in quantum-mechanical systems. EBK quantization is an improvement from Bohr- ...
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:
:
Recently there was a generalization of this formula for arbitrary matrix Hamiltonians that involves a
Berry phase In classical and quantum mechanics, geometric phase is a phase difference acquired over the course of a cycle, when a system is subjected to cyclic adiabatic processes, which results from the geometrical properties of the parameter space of the Ha ...
-like term stemming from spin or other internal degrees of freedom. The index
distinguishes the primitive
periodic orbit In mathematics, in the study of iterated functions and dynamical systems, a periodic point of a function is a point which the system returns to after a certain number of function iterations or a certain amount of time.
Iterated functions
Given a ...
s: the shortest period orbits of a given set of initial conditions.
is the period of the primitive periodic orbit and
is its classical action. Each primitive orbit retraces itself, leading to a new orbit with action
and a period which is an integral multiple
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
.
is the orbit's
Maslov index In mathematics, the Lagrangian Grassmannian is the smooth manifold of Lagrangian subspaces of a real symplectic vector space ''V''. Its dimension is ''n''(''n'' + 1) (where the dimension of ''V'' is ''2n''). It may be identified with the homogene ...
.
The amplitude factor,
, 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
characterizes the instability of the orbit. A stable orbit moves on a
torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does not tou ...
in phase space, and neighboring trajectories wind around it. For stable orbits,
becomes
, where
is the winding
number of the periodic orbit.
, where
is the number of times that neighboring orbits intersect the periodic orbit in one period. This presents a difficulty because
at a classical
bifurcation
Bifurcation or bifurcated may refer to:
Science and technology
* Bifurcation theory, the study of sudden changes in dynamical systems
** Bifurcation, of an incompressible flow, modeled by squeeze mapping the fluid flow
* River bifurcation, the for ...
. This causes that orbit's contribution to the energy density to diverge. This also occurs in the context of photo-
absorption spectrum
Absorption spectroscopy refers to spectroscopic techniques that measure the absorption of radiation, as a function of frequency or wavelength, due to its interaction with a sample. The sample absorbs energy, i.e., photons, from the radiating f ...
.
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
Anisotropy () is the property of a material which allows it to change or assume different properties in different directions, as opposed to isotropy. It can be defined as a difference, when measured along different axes, in a material's physic ...
Kepler
Johannes Kepler (; ; 27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best known for his laws o ...
problem (a single particle in a
potential with an anisotropic mass
tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...
)
semiclassically. He found agreement with quantum computations for low lying (up to
) 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
(tracing over positions) and calculate it again in stationary phase approximation to get an approximation for the density of states
.
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
![Cotcomp](https://upload.wikimedia.org/wikipedia/commons/c/ce/Cotcomp.jpg)
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
A Rydberg atom is an excited atom with one or more electrons that have a very high principal quantum number, ''n''. The higher the value of ''n'', the farther the electron is from the nucleus, on average. Rydberg atoms have a number of peculia ...
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
is given by a smooth background plus an oscillatory sum of the form
:
is a phase that depends on the Maslov index and other details of the orbits.
is the recurrence amplitude of a closed orbit for a given initial state (labeled
). 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
A Rydberg atom is an excited atom with one or more electrons that have a very high principal quantum number, ''n''. The higher the value of ''n'', the farther the electron is from the nucleus, on average. Rydberg atoms have a number of peculia ...
in strong fields, the
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
of an oscillator strength spectrum computed at fixed
as a function of
is called a recurrence spectrum, because it gives peaks which correspond to the scaled action of closed orbits and whose heights correspond to
.
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
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
For the case of one-dimensional system with the boundary condition
the density of states obtained from the Gutzwiller formula is related to the inverse of the potential of the classical system by
here
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
One open question remains understanding quantum chaos in systems that have finite-dimensional local
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
s for which standard semiclassical limits do not apply. Recent works allowed for studying analytically such
quantum many-body systems.
The traditional topics in quantum chaos concerns spectral statistics (universal and non-universal features), and the study of eigenfunctions of various chaotic Hamiltonian. For example, before the existence of scars was reported, eigenstates of a classically chaotic system were conjectured to fill the available phase space evenly, up to random fluctuations and energy conservation (
Quantum ergodicity
In quantum chaos, a branch of mathematical physics, quantum ergodicity is a property of the Quantization (physics), quantization of classical mechanics, classical mechanical systems that are chaos theory, chaotic in the sense of exponential sensit ...
). However, a quantum eigenstate of a classically chaotic system can be scarred:
the probability density of the eigenstate is enhanced in the neighborhood of a periodic orbit, above the classical, statistically expected density along the orbit (
scar
A scar (or scar tissue) is an area of fibrous tissue that replaces normal skin after an injury. Scars result from the biological process of wound repair in the skin, as well as in other organs, and tissues of the body. Thus, scarring is a na ...
s). In particular, scars are both a striking visual example of classical-quantum correspondence away from the usual classical limit, and a useful example of a quantum suppression of chaos. For example, this is evident in the perturbation-induced quantum scarring: More specifically, in quantum dots perturbed by local potential bumps (impurities), some of the eigenstates are strongly scarred along periodic orbits of unperturbed classical counterpart.
Further studies concern the parametric (
) 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
The standard map (also known as the Chirikov–Taylor map or as the Chirikov standard map) is an area-preserving chaotic map from a square with side 2\pi onto itself. It is constructed by a Poincaré's surface of section of the kicked rota ...
and the
kicked rotator are considered to be prototype problems.
Works are also focused in the study of driven chaotic systems, where the Hamiltonian
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 semi-classical regimes as well as a large effort in quantum chaotic scattering.
Berry–Tabor conjecture
In 1977,
Berry
A berry is a small, pulpy, and often edible fruit. Typically, berries are juicy, rounded, brightly colored, sweet, sour or tart, and do not have a stone or pit, although many pips or seeds may be present. Common examples are strawberries, raspb ...
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
In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first i ...
.
See also
*
Scar (physics)
Quantum scarring refers to a phenomenon where the Quantum state, eigenstates of a classically chaotic quantum system have enhanced Density matrix, probability density around the paths of unstable classical periodic orbits. The instability of the p ...
*
Statistical mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...
References
Further resources
*
* Martin C. Gutzwiller, ''Chaos in Classical and Quantum Mechanics'', (1990)
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
Originally founded in 1842 in ...
, New York .
*
Hans-Jürgen Stöckmann
Hans-Jürgen Stöckmann (8 January 1945 in Göttingen, Lower Saxony, Germany) is a German physicist who works in the area of quantum chaos. He is a professor at the University of Marburg, Germany.
He is the son of the physicist Fritz Stöckmann. S ...
, ''Quantum Chaos: An Introduction'', (1999)
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press
A university press is an academic publishing hou ...
.
*
*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
Quantum Chaosby
Martin Gutzwiller Martin may refer to:
Places
* Martin City (disambiguation)
* Martin County (disambiguation)
* Martin Township (disambiguation)
Antarctica
* Martin Peninsula, Marie Byrd Land
* Port Martin, Adelie Land
* Point Martin, South Orkney Islands
Austral ...
(1992 and 2008, ''Scientific American'')
Quantum ChaosMartin Gutzwiller Martin may refer to:
Places
* Martin City (disambiguation)
* Martin County (disambiguation)
* Martin Township (disambiguation)
Antarctica
* Martin Peninsula, Marie Byrd Land
* Port Martin, Adelie Land
* Point Martin, South Orkney Islands
Austral ...
Scholarpedia
''Scholarpedia'' is an English-language wiki-based online encyclopedia with features commonly associated with open-access online academic journals, which aims to have quality content in science and medicine.
''Scholarpedia'' articles are written ...
2(12):3146.
doi:10.4249/scholarpedia.3146Quantum Chaos ScholarpediaWhat is... Quantum Chaosby
Ze'ev Rudnick (January 2008, ''
Notices of the American Mathematical Society
''Notices of the American Mathematical Society'' is the membership journal of the American Mathematical Society (AMS), published monthly except for the combined June/July issue. The first volume appeared in 1953. Each issue of the magazine since ...
'')
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
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
.
Eigenfunctions in chaotic quantum systemsby Arnd Bäcker.
ChaosBook.org
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