Semiclassical collision theory. Application of multidimensional uniform approximations to the atom–rigid-rotor system

Abstract

The multidimensional Bessel and Airy uniform approximations developed earlier in this series for the semiclassical S matrix are applied to the atom rigid-rotor system. The need is shown for (a) using a geoemetrical criterion for determining whether a stationary phase point (s.p.pt) is a maximum, minimum, or saddle point; (b) choosing a proper quadrilateral configuration of the s.p. pts. with the phases as nearly equal as possible; and (c) choosing a unit cell to favor near-separation of variables. (a) and (b) apply both to the Airy and to the Bessel uniform approximations, and (c) to the Bessel. The use of a contour plot both to understand and to facilitate the search in new cases is noted. The case of real and complex-valued stationary phase points is also considered, and the Bessel uniform-in-pairs approximation is applied. Comparison is made with exact quantum results. As in the one-dimensional case, the Bessel is an improvment over the Airy for "k = 0" transitions, while for other transitions they give similar results. Comparison in accuracy with the results of the integral method is also given. As a whole, the agreement can be considered to be reasonable. The improvement of the present over various more approximate results is shown.

Classical-limit S-matrix for heavy ion scattering. [S matrix]

“…x P1(COSX) P 3 (cOSX CA ) exp(i~l/h) 1=1 ,5"" (36) which is similar to Eq. (II ) except for the form factor P3(cOS X CA )'…”

An asymptotic expansion of the hyberbolic umbilic catastrophe integral

“…19], [7,20]. It is used in the scattering theory: the two-dimensional problem of atom-rigid rotor rotationally inelastic scattering [16] and atom-crystal scattering [1]. The hyperbolic umbilic also represents the optical pattern in the neighborhood of the focus in the limit of vanishing wavelength [21, Eq.…”