Quantum chaos
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Quantum chaos is an interdisciplinary branch of physics, arising from semiclassical models.
Classical mechanics has historically been one of the fundamental theories of physics, and is complete in the sense that all its axioms are mutually consistent and not in need of further incremental refinement. However, many of the most difficult unsolved problems in contemporary physics and applied mathematics in fact originate in classical mechanics, particularly in the field of deterministic chaos. Laws of classical mechanics govern the macroscopic world of everyday experience.
An important question of quantum mechanics is how to obtain the laws of classical mechanics as limiting cases of the more fundamental laws governing the microscopic constituents of matter. The correspondence principle is an expression of this goal, which strongly influenced the early development of quantum mechanical theories and their applications. However, the classical limit of a quantum description may lead to a mechanical system with chaotic dynamics.
During the first half of the twentieth century, chaotic behavior in mechanics was recognized (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 reapproximated 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. This transition regime between classical and quantum systems is also called semiclassical physics.
Similar questions arise in many different branches of physics, ranging from nuclear to atomic, molecular and solid-state physics, and even to acoustics, microwaves and optics. This is what makes quantum chaos an interdisciplinary field, unified by wave phenomena that can be interpreted as fingerprints of classical chaos. Such phenomena can be 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.
Important observations often associated with classically chaotic quantum systems are level repulsion in the spectrum, dynamical localization in the time evolution (e.g. ionization rates of atoms), and enhanced stationary wave intensities in regions of space where classical dynamics exhibits only unstable trajectories (wave function scarring).
An alternative name for quantum chaos, proposed by Sir Michael Berry, is quantum chaology.
[edit] History
Important methods applied in the theoretical study of quantum chaos include random-matrix theory (significant contributions by Oriol Bohigas, see also American Scientist) and periodic-orbit theory (pioneered by Martin Gutzwiller).
[edit] References
- A. Einstein (1917). "Zum Quantensatz von Sommerfeld und Epstein". Verhandlungen der Deutschen Physikalischen Gesellschaft 19: 82-92. Reprinted in The Collected Papers of Albert Einstein, A. Engel translator, (1997) Princeton University Press, Princeton. 6 p.434. (Provides an elegant reformulation of the Bohr-Sommerfeld quantization conditions, as well as an important insight into the quantization of non-integrable (chaotic) dynamical systems.)
- Joeseph B. Keller (1958). "". Annals of Physics (NY) 4: 180. (An independent rediscovery of the A. Einstein quantization conditions.)
- Joeseph B. Keller (1960). "". Annals of Physics (NY) 9: 24.
- 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.
