Classical chaotic scattering

The classical equations of motion of the Csl molecule can now be derived from (9.1.5). With r = t/to they are given by 

The set of equations (9.1.7) is investigated in detail in the following section. It is shown that, despite its simple appearance, the solutions of the set (9.1.7) contain the full complexity of classical chaotic scattering, reminiscent of the complexity we encountered in Section 1.1 for the trajectories of box C. 

The lifetimes T 9q) displayed in Fig. 9.5 were calculated for fixed Iq. In order to obtain a better impression where the firactal features of T are located in the 9o, lo plane, T 9o,lo) was calculated as a function of the two initial conditions 9o and Iq for e — 0.5 and p = 1. The result is shown in Fig. 9.6 in the form of a grey-scale plot. The darker the shades in Fig. 9.6 the longer lived the molecule. Again there are apparently unresolved regions in Fig. 9.6(a). A magnification of the framed detail of Fig. 9.6(a) is shown in Fig. 9.6(b). Again there are apparently unresolved structures in Fig. 9.6(b). As before in the one-dimensional case one will never be able to resolve the two-dimensional features in T 9q, Iq) as more and more structme appears on smaller and smaller scales. Thus, T 9q, Iq) is a fractal function embedded in the two-dimensional 9q — Iq) space. 

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Other classically chaotic scattering systems have been shown to have repellers described by a symbolic dynamics similar to (4.10). One of them is the three-disk scatterer in which a point particle undergoes elastic collisions on three hard disks located at the vertices of an equilateral triangle. In this case, the symbolic dynamics is dyadic (M = 2) after reduction according to C)V symmetry. Another example is the four-disk scatterer in which the four disks form a square. The C4 symmetry can be used to reduce the symbolic dynamics to a triadic one based on the symbols 0,1,2), which correspond to the three fundamental periodic orbits described above [14]. 

In summary, classically chaotic scattering manifests itself quantum mechanically in the following way ... 

In fact, with the help of Krein s trace formula, the quantum field theory calculation is mapped onto a quantum mechanical billiard problem of a point-particle scattered off a finite number of non-overlapping spheres or disks i.e. classically hyperbolic (or even chaotic) scattering systems. 

In the following section we discuss the classical and quantum dynamics of a Csl molecule scattered off a reaction zone consisting of an arrangement of inhomogenous fields. This system shows classical and quantum chaotic scattering. It can, at least in principle, be built as a laboratory experiment, which would enable the experimenter to check the theoretical predictions advanced in the following sections. 

Fig. 9.8. Distribution of lifetimes (crosses) and scattering angles (squares) in the classically chaotic regime of the three-wire Csl scattering system. (Adapted from Bliimel and Smilansky (1988).)...

The Hamiltonian (9.3.1) contains the control parameter Vq. On the classical level it was found that for small values of Vq there is no chaotic scattering. There exists a critical value which marks the first appearance of a nontrivial set of scattering singularities. Well developed chaotic scattering characterized by a large set of scattering singularities is observed for Vq > Thus, the control parameter Vb allows us to... 

Gaspard, P. and Rice, S.A. (1989b). Semiclassical quantization of the scattering from a classically chaotic repellor, J. Chem. Phys. 90, 2242-2254. 

One-electron atoms subjected to a time-dependent external field provide physically realistic examples of scattering systems with chaotic classical dynamics. Recent work on atoms subjected to a sinusoidal external field or to a periodic sequence of instantaneous kicks is reviewed with the aim of exposing similarities and differences to frequently studied abstract model systems. Particular attention is paid to the fractal structure of the set of trapped unstable trajectories and to the long time behavior of survival probabilities which determine the ionization rates of the atoms. Corresponding results for unperturbed two-electron atoms are discussed. 

A general necessary requirement for the applicability of Wigner-Dyson statistics to physical systems is that there should be no constants of motion other than the energy itself, to eliminate crossing of levels. In classical mechanics such systems are known as non-integrable or chaotic systems (no stable periodic orbits in phase space). Besides impurity scattering, scattering at boundaries can make the system... 

Gaspard and Rice have studied the classical, semiclassical and full quantum mechanical dynamics of the scattering of a point particle from three hard discs fixed in a plane (see Fig. 11). We note that the classical motion (which is chaotic) consists of trajectories which are trapped between the discs. 

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