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Rays, Waves, and ScatteringTopics in Classical Mathematical Physics$
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John A. Adam

Print publication date: 2017

Print ISBN-13: 9780691148373

Published to Princeton Scholarship Online: May 2018

DOI: 10.23943/princeton/9780691148373.001.0001

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PRINTED FROM PRINCETON SCHOLARSHIP ONLINE (www.princeton.universitypressscholarship.com). (c) Copyright Princeton University Press, 2019. All Rights Reserved. An individual user may print out a PDF of a single chapter of a monograph in PRSO for personal use.date: 19 October 2019

The WKB(J) Approximation Revisited

The WKB(J) Approximation Revisited

Chapter:
(p.434) Chapter Twenty-Two The WKB(J) Approximation Revisited
Source:
Rays, Waves, and Scattering
Author(s):

John A. Adam

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691148373.003.0022

This chapter reexamines the WKB(J) approximation and applies it to some simple one-dimensional potentials, with a focus on the case of a triangular barrier. It first considers the connection formulas and proposes an an alternative approach before discussing tunneling from a physical standpoint. It then turns to the case of a triangular barrier and goes on to explore the phase shift, offering some comments on convergence and the transition to classical scattering. It also describes the asymptotic behavior of the Coulomb wave function and revisits the spherical coordinate system. Finally, it finds the WKB(J) approximation with respect to Coulomb scattering and the formal WKB(J) solutions for the time-independent radial Schrödinger equation, and justifies the Langer transformation by showing how the asymptotic phase of the radial WKB(J) wave function is recovered.

Keywords:   triangular barrier, WKB(J) approximation, tunneling, phase shift, convergence, classical scattering, spherical coordinate system, Coulomb scattering, Schrödinger equation, Langer transformation

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