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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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Matrix elements

Matrix elements

Chapter:
(p.99) 5 Matrix elements
Source:
Semiclassical Mechanics with Molecular Applications
Author(s):

M. S. Child

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780199672981.003.0005

Two types of matrix element approximation are adopted according to whether the wavefunctions are taken in angle–action or normalized JWKB forms. The former gives the Heisenberg correspondence between matrix elements and classical Fourier components. The latter approximation is appropriate to situations for which the dominant contribution to the integral comes from stationary phase or ‘Condon’ points, at which both coordinates and momenta are conserved between the two states. The presence of a single such point leads to a ‘Condon reflection’ pattern such that the energy variation of the matrix element mimics the nodal pattern of the parent wavefunction. Complications arising from multiple Condon points are discussed.

Keywords:   Heisenberg correspondence, Fourier components, matrix elements, stationary phase, Condon point, Condon reflection

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