Unitary Quantum Maps

Abstract

After the crash course on classical maps and classical chaos in the preceding chapter, let us now have a look at the corresponding concepts in quantum mechanics. I shall introduce in this chapter about unitary quantum maps the object of choice for studying chaos in ordinary, i.e. nondissipative quantum mechanics. A standard example, namely a kicked top, will serve as a useful model, not only in this chapter, but for the rest of this book. We shall see how a classical map emerges from the quantum map, and identify signatures of chaos in the quantum world. With the Van Vleck propagator and Gutzwiller’s trace formula, we shall also encounter for the first time semiclassical theories that try to bridge the gap between chaos in the classical realm and in the quantum world. The generalization of these semiclassical theories to dissipative dynamics will be the main topic of later chapters.