Quantum chaotic scattering

The quantization of the dipole model defined in Sections 9.1 and 9.2 is not difficult. The Hamiltonian for the three-wire version of the Csl system is given by 

The stationary scattering function for total energy E can be expanded according to 

The boundary conditions satisfied by the expansion amplitudes 7 (x) are given by 

The incident wave is prepared in the state n) as a plane wave in x. The channel wave numbers are given by 

We denote by N the number of open channels, i.e. N is the largest integer n for which E h n /2I. The scattered wave is a superposition 

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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. 

In this section we define the concept of Ericson fluctuations and discuss their relevance in quantum chaotic scattering systems of atomic and molecular physics. Our discussion follows the excellent introduction to Ericson fluctuations in Ericson s own paper of 1963. A nuclear compound reaction is taken as an illustrative example. 

Baranger, H.U., Jalabert, R.A. and Stone, A.D. (1993a). Quantum chaotic scattering effects in semiconductor microstructures. Chaos 3, 665-682. 

Bliimel, R. (1993b). Quantum chaotic scattering with Csl molecules, Chaos 3, 683-690. 

Marcus, C.M., Westervelt, R.M., Hopkings, P.F. and Gossard, A.C. (1993). Conductuance fluctuations as quantum chaotic scattering, inQuantum Dynamics of Chaotic Systems, eds. J.-M. Yuan, D. H. Feng and G. M. Zaslavsky (Gordon and Breach, Amsterdam). 

The complex scattering wave function can be specified by nodal points at which u = 0,v = 0. They have great physical significance since they are responsible for current vortices. We have calculated distribution functions for nearest distances between nodal points and found that there is a universal form for open chaotic billiards. The form coincides with the distribution for the Berry function and hence, it may be used as a signature of quantum chaos in open systems. All distributions agree well with numerically computed results for transmission through quantum chaotic billiards. 

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

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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