# 3.6] What is quantum chaos? (nonlinear science)

## Description

This article is from the Nonlinear Science FAQ, by James D. Meiss jdm@boulder.colorado.edu with numerous contributions by
others.

# 3.6] What is quantum chaos? (nonlinear science)

(Thanks to Leon Poon for contributing to this answer)

According to the correspondence principle, there is a limit where classical

behavior as described by Hamilton's equations becomes similar, in some

suitable sense, to quantum behavior as described by the appropriate wave

equation. Formally, one can take this limit to be h -> 0, where h is Planck's

constant; alternatively, one can look at successively higher energy levels.

Such limits are referred to as "semiclassical". It has been found that the

semiclassical limit can be highly nontrivial when the classical problem is

chaotic. The study of how quantum systems, whose classical counterparts are

chaotic, behave in the semiclassical limit has been called quantum chaos. More

generally, these considerations also apply to elliptic partial differential

equations that are physically unrelated to quantum considerations. For

example, the same questions arise in relating classical waves to their

corresponding ray equations. Among recent results in quantum chaos is a

prediction relating the chaos in the classical problem to the statistics of

energy-level spacings in the semiclassical quantum regime.

Classical chaos can be used to analyze such ostensibly quantum systems as the

hydrogen atom, where classical predictions of microwave ionization thresholds

agree with experiments. See Koch, P. M. and K. A. H. van Leeuwen (1995).

"Importance of Resonances in Microwave Ionization of Excited Hydrogen Atoms."

Physics Reports 255: 289-403.

See also:

http://sagar.physics.neu.edu/qchaos/qc.html Quantum Chaos

http://www.mpipks-dresden.mpg.de/~noeckel/microlasers.html Microlaser

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