A semiclassical study of quantum maps

The study of the behavior of quantum systems whose classical limit exhibits chaos defines the problem of quantum chaos. Because of some strongly contrasting properties exhibited by classical and quantum theories, one would naturally ask how quantum mechanics approaches the classical limit ;For studying the quantum mechanics corresponding to generic classical motion it is desirable to use the simplest possible model. The model system we use is the kicked rotator, which is one of the most studied model systems due to its simplicity and generic features. Detailed computations of both classical and quantum mechanics are feasible for this system.;The relationship between invariant classical phase space structures and quantum eigenfunctions has been the focus of recent semiclassical studies. We study the eigenstates of the quantum standard map associated with both integrable and non-integrable regions in classical phase space. The coherent-state representation is used to make the correspondence between the quantum eigenstates and the classical phase space structure.;The importance of periodic orbits in the quantum eigenstates of classically chaotic Hamiltonians has become a popular topic in study of semiclassical limits of the systems. It is well known that periodic orbits arise without any assumption in the trace formula developed by Gutzwiller. We calculate the semiclassical coherent-state propagator. Since computing all the complex stationary orbits is not practical, we make a further assumption which we call the periodic point dominance (PPD). We present arguments and evidence to show that the PPD approximation works well in hard chaos regions where the full semiclassical approximation is not practical to use. The method fails in some boundary regions where both stable and unstable points are present, but the full semiclassical approximation is not a much better method than the PPD in many of this kind of situations.