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, and in particular 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.
This question defines the field of quantum chaos, which has emerged in the second half of the twentieth century, 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 intepreted 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 external forces. In some contexts, such as acoustics or microwaves, wave patterns are directly observable and exhibit irregular patterns.
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).
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). For a more detailed overview, see for example [the perspective] offered by J. Delos.
An alternative name for quantum chaos, proposed by Sir Michael Berry, is quantum chaology.