This article is from the Nonlinear Science FAQ, by James D. Meiss firstname.lastname@example.org with numerous contributions by others.
(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.
http://sagar.physics.neu.edu/qchaos/qc.html Quantum Chaos