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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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Quantization

Quantization

Chapter:
(p.33) 3 Quantization
Source:
Semiclassical Mechanics with Molecular Applications
Author(s):

M. S. Child

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780199672981.003.0003

The quantization formula, ∮p(x)dx=(n+δ)h is shown to include a Maslov index, δ , dependent on the distribution of turning points and singularities. A related argument explains why singularities in the angular momentum equation require the Langer substitution ℓ(ℓ+1)→(ℓ+1/2) The subsequent sections show how semiclassical connection formulae may be used to stitch JWKB fragments of the wavefunction together in a variety of tunnelling and curve–crossing situations. Diagrammatic representations of the connection formulae are used to illustrate the quantization of double minimum and restricted rotation problems. Applications are also given to the widths of shape resonance widths and the rates of curve-crossing predissociation. The latter are particularly interesting in showing rapid fluctuations from one quasi-bound level to another.

Keywords:   Maslov, Langer, connection formulae, tunnelling, curve–crossing, double minimum, restricted rotation, shape resonance, predissociation

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