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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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Introduction to the WKB(J) Approximation: All Things Airy

Introduction to the WKB(J) Approximation: All Things Airy

(p.137) Chapter Seven Introduction to the WKB(J) Approximation: All Things Airy
Rays, Waves, and Scattering

John A. Adam

Princeton University Press

This chapter deals with the WKB(J) approximation, commonly used in applied mathematics and mathematical physics to find approximate solutions of linear ordinary differential equations (of any order in principle) with spatially varying coefficients. The WKB(J) approximation is closely related to the semiclassical approach in quantum mechanics in which the wavefunction is characterized by a slowly varying amplitude and/or phase. The chapter first introduces an inhomogeneous differential equation, from which the first derivative term is eliminated, before discussing the Liouville transformation and the one-dimensional Schrödinger equation. It then presents a physical interpretation of the WKB(J) approximation and its application to a potential well. It also considers the “patching region” in which the Airy function solution (the local turning point) is valid, the relation between Airy functions and Bessel functions, Airy integral and related topics, and related integrals.

Keywords:   differential equations, WKB(J) approximation, Liouville transformation, Schrödinger equation, potential well, Airy functions, Bessel functions, Airy integral, integrals

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