Scattering theory, numerical results

There have been numerous tests since the KPS study of the QCT approach, mostly based on comparisons of QCT results with those from quantum scattering theory (QST) calculations for the same potential-energy surface. These scattering calculations involve solving the... 

Figure 5. Normalized Hght scattering intensity (Pn element of the MueUer scattering matrix) distribution with scattering angle - comparison of FDTD simulation and exact analytical (Mie theory) results. There is a very good agreement between exact and numerical results with a relative error of approximately 5%.

Lord Rayleigh (1871) recognized that the blue of the sky is the result of light scattering. It is obvious from the numerous pictures sent back from outer space that the blackness observed by astronauts is quite different from our earthly observations of a blue sky. Rayleigh developed the theory of light... 

Theories on ThFFF and light scattering have been discussed in numerous publications. Equations that are needed for the discussion of results are briefly reviewed here. In ThFFF, the retention ratio, R, is given by (6)... 

Several aspects of this formulation require emphasis. First, defining the SCR yields a particular statistical theory hence, a wide variety of theories are possible. Second, one should anticipate that a typical scattering event will have both direct and statistical components. Third, an analysis of the degree of statistical behavior in an exact dynamical calculation requires three sequential steps (1) the SCR must first be defined (2) the component of the scattering that does not pass through the SCR is then eliminated from consideration and (3) the product distribution associated with the component passing through the SCR is compared with the results predicted from Eq. (2.17) the latter often requires additional numerical computations. 

This paper reviews this classical S-matrix theory, i.e. the semiclassical theory of inelastic and reactive scattering which combines exact classical mechanics (i.e. numerically computed trajectories) with the quantum principle of superposition. It is always possible, and in some applications may even be desirable, to apply the basic semiclassical model with approximate dynamics Cross7 has discussed the simplifications that result in classical S-matrix theory if one treats the dynamics within the sudden approximation, for example, and shown how this relates to some of his earlier work8 on inelastic scattering. For the most part, however, this review will emphasize the use of exact classical dynamics and avoid discussion of various dynamical models and approximations, the reason being to focus on the nature and validity of the basic semiclassical idea itself, i.e., classical dynamics plus quantum superposition. Actually, all quantum effects—being a direct result of the superposition of probability amplitudes—are contained (at least qualitatively) within the semiclassical model, and the primary question to be answered regards the quantitative accuracy of the description. 

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