Bound-scattering matrix

In Chapter 3 we derived a general expression for the amplitude scattering matrix for an arbitrary particle. An unstated assumption underlying that derivation is that the particle is confined within a bounded region, a condition that is not satisfied by an infinite cylinder. Nevertheless, we can express the field scattered by such a cylinder in a concise form by resolving the incident and scattered fields into components parallel and perpendicular to planes determined by the cylinder axis (ez) and the appropriate wave normals (see Fig. 8.3). That is, we write the incident field... 

Resonances in half and in full collisions have exactly the same origin, namely the temporary excitation of quasi-bound states at short or intermediate distances irrespective of how the complex was created. In full collisions one is essentially interested in the asymptotic behavior of the stationary wavefunction L(.E) in the limit R —> 00, i.e., the scattering matrix S with elements Sif as defined in (2.59). The S-matrix contains all the information necessary to construct scattering cross sections for a transition from state i to state /. In the case of a narrow and isolated resonance with energy Er and width hT the Breit- Wigner expression... 

First, it is necessary to specify the relevant n and i quantum numbers that e the bound-free matrix elements (E, n, q deg Ej). For the continuum n = [k, vj, m) where k is the scattering direction, v and j are the vibrationaf rotational product quantum numbers and m is the space-fixed z projection of/yl... 

The product of the bound-free matrix elements of Eq. (8.31), which enter Eq. (3.79), integrated over scattering angles and averaged over the initial [280] Mi) quantum numbers, is... 

The most accurate method for multilevel curve crossing problems is, of course, to solve the close-coupling differential equations numerically. This is not the subject here, however instead, we discuss the applications of the two-state semiclassical theory and the diagrammatic technique. With these tools we can deal with various problems such as inelastic scattering, elastic scattering with resonance, photon impact process, and perturbed bound state in a unified way. The overall scattering matrix 5, for instance, can be defined as... 

Multichannel variants of the phase equation can also be defined, and this is where supercomputers would become attractive for implementing the method. Calogero defines a first order differential equation for the reactance matrix, R, and the scattering matrix, S. It is more enlightening to consider the radial evolution of S. S is complex, but it is bounded and S(r) shows directly the evolution of the transition probabilities as r is increased. This can display the radial regions where coupling is strong. The differential equation for S(r) is Eqn (7) ... 

In summary, we have shown how the absorption of a photon leads to the formation of a resonant scattering state. Explicit formulas involving quadrature over the system energy spectrum have been presented but not evaluated. When the resonant scattering state may be approximated in terms of a set of quasistationary bound states, an explicit relationship is obtained for the rate of dissociation in terms of the matrix elements coupling zero-order states and the corresponding densities of states. In principle this permits the use of experimental rate data to evaluate the matrix elements vx and v2, if px and p2 can be estimated. 

The two-particle Boltzmann collision term if and the three-particle contribution for k = 0 were considered in Section II. It was possible to express those collision integrals in terms of the two- and three-particle scattering matrices. It is also possible to introduce the T matrix in if for the channels k = 1, 2,3, that is, in those cases where three are asymptotically bound states. Here we use the multichannel scattering theory, as outlined in Refs. 9 and 26. 

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