quantum chaos

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Quantum chaos is a branch of physics where it is studied how the chaotic classical systems (see dynamical systems and chaos theory) can be shown to be limits of quantum-mechanical systems. The phenomena covered by quantum chaos so far are mainly related to wave theory. An alternative name, proposed by Sir Michael Berry, is quantum chaology.

History

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 exhibits chaos.

In the 1950s, E. P. Wigner introduced the idea that the complex Hamiltonians used to find the energy levels of heavy atom nuclei could be approximated by a random Hamiltonian representing the probability distribution of individual Hamiltonians. This idea was then further developed with advances in random matrix theory and statistics.

This was the first demonstration of the emergence of useful information from a randomized model based on quantum mechanics, contributing to the name quantum chaos. Its emergence in the second half of the twentieth century was aided to a large extent by renewed interest in classical nonlinear dynamics (chaos theory), and by quantum experiments bordering on the macroscopic size regime where laws of classical mechanics are expected to emerge.

A recent notable researcher is Martin Gutzwiller, who pioneered periodic-orbit theory.

Approaches

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. 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.

Examples of recent work

In a 2002 paper, Sudhir R. Jain et al. suggested a relation between the many-body problem and a random matrix theory for pseudo-integrable systems from which they derived a solution to the Schroedinger equations of the systems.

External links

Brian Hayes, "The Spectrum of Riemannium"; American Scientist. Discusses relation to the Riemann zeta function.

References

• Martin C. Gutzwiller (1971). "". Journal of Mathematical Physics 12: 343. 
• Martin C. Gutzwiller, Chaos in Classical and Quantum Mechanics, (1990) Springer-Verlag, New York ISBN=0-387-97173-4.
• Eugene Paul Wigner (1951). "On the statistical distribution of the widths and spacings of nuclear resonance levels". Proc. Cambr. Philos. Soc. 47: 790. 
• Fritz Haake, Quantum Signatures of Chaos 2nd ed., (2001) Springer-Verlag, New York ISBN=3-540-67723-2.
• Sudhir R. Jain, B. Gremaud, A. Khare (2002). "Quantum modes built on chaotic motion : analytically exact results". Physical Review E 66: 016216. 

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