Semiclassical dynamics

The properties of the classical motion plays an important role in determining the quantum dynamics of a system. The fixed points of the classical map generated by the classical motion have a deep effect on the quantum dynamics: however, to exploit this one must study not so much the orbits but the evolution of curves which classically represent ensembles corresponding to wavefunctions. These fixed points also generate new quantum mechanical time scales, such that for times larger than these critical times (depending on Planck's constant) the quantum dynamics will profoundly differ from the classical one. Those regions of the classical phase, where the hyperbolic fixed points generate a stochastic motion contain states described by wave-functions with an interesting and unusual structure. The probability density in position exhibits many peaks of the type usually only associated with the classical turning points of the motion, but appearing in the interior of the classically allowed domain. The momentum distribution is equally complicated showing the spectral impurity of the wave-functions