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). Study of probability distribution of individual eigenstates (see scars and Quantum ergodicity). Semiclassical methods such as periodic-orbit theory connecting the classical trajectories of the dynamical system with quantum features. Direct application of the correspondence principle. 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 exhibit chaos. 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. However, classical-quantum correspondence in chaos theory is not always possible. Thus, some versions of the classical butterfly effect do not have counterparts in quantum mechanics. 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 and by connecting spectral periodicities with classical