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Cambridge University Press eBooks

Semiclassical expansion; beyond WKB

Pierre Cartier, Cécile DeWitt-Morette

Published Nov 30, 2006• 0 citations
Publisher

Abstract

When “WKB breaks down... ” then rainbows and glory scattering may appear. In other words, and with the notation of chapter 4, when the hessian S ′ ′ (q) of the functional S is degenerate S ′ ′ (q) · ξξ = 0 (5.1)
DOI
10.1017/cbo9780511535062.005

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A procedure for the evaluation of one-dimensional uniform semiclassical wave functions is explored. This form of the approximate wave function is tested for the case of an exponential potential and is compared to the exact quantum mechanical and WKB wave functions. It is found that the uniform approximation provides a very accurate representation of the exact wave function, even at the turning points of the classical motion, where the WKB approximation diverges. It is demonstrated how correction terms can be constructed, leading to a formal expansion for the exact wave function. The individual terms of this expansion each involve a finite number of reflection points with uniform semiclassical propagation between these reflections. The analysis is extended to multisurface (nonadiabatic) problems, removing the turning point singularities which are generally present in semiclassical formulations of nonadiabatic problems.