Statistical methods and quantum chaology

Chaotic behaviour can arise in any system whose motion is described by a nonlinear differential equation. Whether or not it is prevalent depends on the details of the problem, but it is a general theorem that any system described by a nonlinear differential equation possesses some chaotic regime. 

In quantum mechanics, by contrast, chaos does not occur. We may see this in several ways. First, note that we cannot magnify ad infinitum the volume to be analysed in phase space eventually, we reach the elementary volume h3 within which trajectories lose their meaning. Another way of reaching the same conclusion is to note that any Schrodinger type equation is linear its solutions obey the superposition theorem. Under these circumstances genuine chaos is excluded by fundamental principles. 

Nonetheless, traces of chaotic behaviour persist in some quantum systems. They are found when calculations are extended into the semiclassi-cal limit and the underlying classical dynamics of the corresponding classical system is chaotic. This is anyway an interesting situation, because it relates to the correspondence principle. To some degree, it challenges 

Nevertheless, until relatively recently, chaos has not figured prominently in the preoccupations of atomic physicists. The independent electron, Hartree-Fock model, replaces the jV-electron equation by N one-electron equations, each one of which contains well-ordered Rydberg series, which seems to preclude the possibility of quantum chaos , at least until many-electron correlations are introduced. This suggests that a transition to quantum chaos might appear in the presence of strong configuration mixing. 

In the present chapter, we examine some of these issues, drawing examples from current research. We begin by considering strongly interacting, many-electron systems, which are not necessarily in the semiclassical limit and therefore not true examples of quantum chaos , but which share many of its statistical properties, and we move on to highly-excited atoms in external fields, which can be followed to the semiclassical limit and are therefore very good systems for the study of quantum chaos . 

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