Higher-Order Stationary-Phase Approximations in Semiclassical Scattering

Abstract

Higher‐order stationary‐phase approximations are used to calculate corrections to the classical expression for the differential cross section for elastic scattering. An expansion for the cross section as a series in ℏ² is obtained, whose first term is the classical result. An oscillating term is also present, whose ``wavelength'' is approximately Δθ≈2π/kb. Both the ``wavelength'' and the amplitude of this term vary as ℏ. It is shown that the classical differential cross section is valid for angles as small as the critical angle defined by Massey and Mohr. A similar technique is used to obtain corrections to Ford and Wheeler's semiclassical expression for the differential cross section at a rainbow angle. An expansion as a series in ℏ^⅔ is derived, whose first term varies as ℏ^—⅓ and hence diverges in the classical limit. It is shown that the leading term agrees with Ford and Wheeler's result and that the first correction term is very small for a 12–6 potential.