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Semiclassical Mechanics with Molecular Applications$
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M. S. Child

Print publication date: 2014

Print ISBN-13: 9780199672981

Published to Oxford Scholarship Online: October 2014

DOI: 10.1093/acprof:oso/9780199672981.001.0001

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The classical S matrix

The classical S matrix

Chapter:
(p.242) 10 The classical S matrix
Source:
Semiclassical Mechanics with Molecular Applications
Author(s):

M. S. Child

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780199672981.003.0010

The classical S matrix is a natural multidimensional extension of the semiclassical phase shift. It is most naturally interpreted as a contraction of the Feynman path integral propagator, between initial and final scattering states. An equivalent derivation in the text employs a multidimensional JWKB wavefunction in place of the propagator. The primitive semiclassical S matrix appears as a sum over classical trajectories, which interfere according to differences in their classical action integrals, while classically forbidden events are handled by the use of complex time trajectories. Appropriate uniform approximations to remove different types of singularity at the classical boundary are described. Symmetry selection rules are seen to arise from the physical equivalence of initial starting geometries. Applications to rotational rainbows and Condon reflection principles are given.

Keywords:   S matrix, Feynman path integral, JWKB, primitive semiclassical, uniform, caustic, complex time, rotational rainbow

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