# Quantum chaos

**Berry–Tabor conjecturequantumrelativistic quantum chaotic dynamicssemiclassicalsemiclassical way of quantizing chaotic systemstrace formulaVan Vleck formula**

Quantum chaos is a branch of physics which studies how chaotic classical dynamical systems can be described in terms of quantum theory.wikipedia

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### Classical limit

**classical limit of quantum mechanicspreviously known resultquantum-classical correspondence**

The correspondence principle states that classical mechanics is the classical limit of quantum mechanics.

It is less clear, however, how the classical limit applies to chaotic systems, a field known as quantum chaos.

### Michael Berry (physicist)

**Michael BerryBerryMichael Victor Berry**

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).

He specialises in semiclassical physics (asymptotic physics, quantum chaos), applied to wave phenomena in quantum mechanics and other areas such as optics.

### Random matrix

**random matricesrandom matrix theory random matrix theory**

Random matrix theory was developed in an attempt to characterize spectra of complex nuclei.

In quantum chaos, the Bohigas–Giannoni–Schmit (BGS) conjecture asserts that the spectral statistics of quantum systems whose classical counterparts exhibit chaotic behaviour are described by random matrix theory.

### Semiclassical physics

**semiclassicalsemi-classicalsemiclassical approximations**

3) Semiclassical methods such as periodic-orbit theory connecting the classical trajectories of the dynamical system with quantum features.

quantum chaos; quantization of classical chaotic systems.

### Chaos theory

**chaoticchaoschaotic behavior**

Quantum chaos is a branch of physics which studies how chaotic classical dynamical systems can be described in terms of quantum theory.

Quantum chaos

### Quantum ergodicity

**quantum unique ergodicity**

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).

In quantum chaos, a branch of mathematical physics, quantum ergodicity is a property of the quantization of classical mechanical systems that are chaotic in the sense of exponential sensitivity to initial conditions.

### Einstein–Brillouin–Keller method

**EBK quantizationquantization**

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.

In 1976–1977, Berry and Tabor derived an extension to Gutzwiller trace formula for the density of states of an integrable system starting from EBK quantization.

### Kicked rotator

**kicked rotor**

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.

The kicked rotator, also spelled as kicked rotor, is a prototype model for chaos and quantum chaos studies.

### Martin Gutzwiller

**GutzwillerGutzwiller trace formulaGutzwiller, Martin**

Quantum Chaos by Martin Gutzwiller (1992 and 2008, Scientific American)

Martin Charles Gutzwiller (12 October 1925 – 3 March 2014 ) was a Swiss-American physicist, known for his work on field theory, quantum chaos, and complex systems.

### Hans-Jürgen Stöckmann

**Stöckmann, Hans-Jürgen**

Hans-Jürgen Stöckmann, Quantum Chaos: An Introduction, (1999) Cambridge University Press ISBN: 0-521-59284-4.

Hans-Jürgen Stöckmann (8 January 1945 in Göttingen, Lower Saxony, Germany) is a German physicist who works in the area of quantum chaos.

### Zeev Rudnick

**Rudnick, Zeev**

What is... Quantum Chaos by Ze'ev Rudnick (January 2008, Notices of the American Mathematical Society)

Zeev Rudnick or Ze'ev Rudnick (born 1961 in Haifa, Israel) is a mathematician, specializing in number theory and in mathematical physics, notably quantum chaos.

### Bifurcation theory

**bifurcationbifurcationsbifurcation point**

This presents a difficulty because at a classical bifurcation.

The dominant reason for the link between quantum systems and bifurcations in the classical equations of motion is that at bifurcations, the signature of classical orbits becomes large, as Martin Gutzwiller points out in his classic work on quantum chaos.

### Lagrangian Grassmannian

**Maslov correctionMaslov indexes**

These repetitions are separately classified by the intermediate sum over the indices n. \alpha_{nk} is the orbit's Maslov index.

It appeared originally in the study of the WKB approximation and appears frequently in the study of quantization, quantum chaos trace formulas, and in symplectic geometry and topology.

### Scar (physics)

**scar**

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).

In physics, and especially quantum chaos, a wavefunction scar is an enhancement (i.e. increased norm squared) of an eigenfunction along unstable classical periodic orbits in classically chaotic systems .They were discovered and explained in 1984 by E.J. Heller and are part of the large field of quantum chaos.

### Rydberg atom

**RydbergRydberg states**

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.

Quantum chaos

### Physics

**physicistphysicalphysicists**

Quantum chaos is a branch of physics which studies how chaotic classical dynamical systems can be described in terms of quantum theory.

### Dynamical system

**dynamical systemsdynamic systemdynamic systems**

Quantum chaos is a branch of physics which studies how chaotic classical dynamical systems can be described in terms of quantum theory.

### Correspondence principle

**correspondencecorrespondslarge orbits**

The correspondence principle states that classical mechanics is the classical limit of quantum mechanics.

### Perturbation theory (quantum mechanics)

**perturbation theoryperturbativeperturbation**

1) Development of methods for solving quantum problems where the perturbation cannot be considered small in perturbation theory and where quantum numbers are large.

### Hamiltonian (quantum mechanics)

**HamiltonianHamiltoniansHamiltonian operator**

2) Correlating statistical descriptions of eigenvalues (energy levels) with the classical behavior of the same Hamiltonian (system).

### Three-body problem

**3-body problemrestricted three-body problemthree-body**

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.

### Celestial mechanics

**celestial dynamicscelestial mechaniciancelestial**

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.

### Nuclear physics

**nuclear physicistnuclearnuclear scientist**

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.

### Atomic physics

**atomicatomic physicistatomic scientist**

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.

### Molecular physics

**molecularmolecular scienceMolecular physicist**

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.