Quantum chaos

Quantum chaos is a branch of physics which studies how chaotic classical dynamical systems can be described in terms of quantum theory. The primary question that quantum chaos seeks to answer is: "What is the relationship between quantum mechanics and classical chaos?" The correspondence principle states that classical mechanics is the classical limit of quantum mechanics, specifically in the limit as the ratio of Planck's constant to the action of the system tends to zero. If this is true, then there must be quantum mechanisms underlying classical chaos (although this may not be a fruitful way of examining classical chaos). If quantum mechanics does not demonstrate an exponential sensitivity to initial conditions, how can exponential sensitivity to initial conditions arise in classical chaos, which must be the correspondence principle limit of quantum mechanics?In seeking to address the basic question of quantum chaos, several approaches have been employed:

Development of methods for solving quantum problems where the perturbation cannot be considered small in perturbation theory and where quantum numbers are large.
Correlating statistical descriptions of eigenvalues (energy levels) with the classical behavior of the same Hamiltonian (system).
Semiclassical methods such as periodic-orbit theory connecting the classical trajectories of the dynamical system with quantum features.
Direct application of the correspondence principle.

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