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Branch of physics seeking to explain chaotic dynamical systems in terms of quantum theory

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.

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.

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 orbits. Other phenomena show up in the time evolution of a quantum system, or in its response to various types of external forces. In some contexts, such as acoustics or microwaves, wave patterns are directly observable and exhibit irregular amplitude distributions. Quantum chaos typically deals with systems whose properties need to be calculated using either numerical techniques or approximation schemes (see e. g. Dyson series). Simple and exact solutions are precluded by the fact that the system's constituents either influence each other in a complex way, or depend on temporally varying external forces.

Start with the semiclassical approximation of the time dependent Green's function (the Van Vleck propagator). Realize that for caustics the description diverges and use the insight by Maslov (approximately Fourier transforming to momentum space (stationary phase approximation with h a small parameter) to avoid such points and afterwards transforming back to position space can cure such a divergence, however gives a phase factor). Transform the Greens function to energy space to get the energy dependent Greens function (again approximate Fourier transform using the stationary phase approximation). New divergences might pop up that need to be cured using the same method as step 3
Use (tracing over positions) and calculate it again in stationary phase approximation to get an approximation for the density of states Note: Taking the trace tells you that only closed orbits contribute, the stationary phase approximation gives you restrictive conditions each time you make it. In step 4 it restricts you to orbits where initial and final momentum are the same i. e. periodic orbits. Often it is nice to choose a coordinate system parallel to the direction of movement, as it is done in many books.

For the case of one dimensional system with the boundary condition the density of states obtained from the Gutzwiller formula is related to the inverse of the potential of the classical system by here is the density of states and V(x) is the classical potential of the particle, the half derivative of the inverse of the potential is related to the density of states as in the Wu Sprung potential.

One open question remains understanding quantum chaos in systems that have finite dimensional local Hilbert spaces for which standard semiclassical limits do not apply. Recent works allowed for studying analytically such quantum many body systems. The traditional topics in quantum chaos concerns spectral statistics (universal and non universal features), and the study of eigenfunctions of various chaotic Hamiltonian. For example, before the existence of scars was reported, eigenstates of a classically chaotic system were conjectured to fill the available phase space evenly, up to random fluctuations and energy conservation (Quantum ergodicity). However, a quantum eigenstate of a classically chaotic system can be scarred: the probability density of the eigenstate is enhanced in the neighborhood of a periodic orbit, above the classical, statistically expected density along the orbit (scars). In particular, scars are both a striking visual example of classical quantum correspondence away from the usual classical limit, and a useful example of a quantum suppression of chaos. For example, this is evident in the perturbation induced quantum scarring: More specifically, in quantum dots perturbed by local potential bumps (impurities), some of the eigenstates are strongly scarred along periodic orbits of unperturbed classical counterpart. Further studies concern the parametric dependence of the Hamiltonian, as reflected in e. g. the statistics of avoided crossings, and the associated mixing as reflected in the (parametric) local density of states (LDOS). There is vast literature on wavepacket dynamics, including the study of fluctuations, recurrences, quantum irreversibility issues etc. Special place is reserved to the study of the dynamics of quantized maps: the standard map and the kicked rotator are considered to be prototype problems. Works are also focused in the study of driven chaotic systems, where the Hamiltonian is time dependent, in particular in the adiabatic and in the linear response regimes. There is also significant effort focused on formulating ideas of quantum chaos for strongly interacting many body quantum systems far from semi classical regimes as well as a large effort in quantum chaotic scattering.

In 1977, Berry and Tabor made a still open "generic" mathematical conjecture which, stated roughly, is: In the "generic" case for the quantum dynamics of a geodesic flow on a compact Riemann surface, the quantum energy eigenvalues behave like a sequence of independent random variables provided that the underlying classical dynamics is completely integrable.