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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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The WKB(J) Approximation Revisited

The WKB(J) Approximation Revisited

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

John A. Adam

Princeton University Press

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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