The WKB and Related Methods
In the method of matched asymptotic expansions studied in Chap. 2, the dependence of the solution on the boundary-layer coordinate was determined by solving the boundary-layer problem. In a similar way, when using multiple scales the dependence on the fas
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The WKB and Related Methods
4.1 Introduction In the method of matched asymptotic expansions studied in Chap. 2, the dependence of the solution on the boundary-layer coordinate was determined by solving the boundary-layer problem. In a similar way, when using multiple scales the dependence on the fast time scale was found by solving a differential equation. This does not happen with the WKB method because one begins with the assumption that the dependence is exponential. This is a reasonable expectation since many of the problems we studied in Chap. 2 ended up having an exponential dependence on the boundary-layer coordinate. Also, with this assumption, the work necessary to find an asymptotic approximation of the solution can be reduced significantly. The popularity of the WKB method can be traced back to the 1920s and the development of quantum mechanics. In particular, it was used to find approximate solutions of Schr¨ odinger’s equation. The name of the method is derived from three individuals who were part of this development, namely, Wentzel, Kramers, and Brillouin. However, as often happens in mathematics, the method was actually derived much earlier. Some refer to it as the method of Liouville and Green since they both published papers on the procedure in 1837. It appears that this too is historically incorrect since Carlini, in 1817, used a version of the approximation in studying elliptical orbits of planets. Given the multiple parenthood of the method it should not be unexpected that there are other names it goes by; these include the phase integral method, the WKBJ method (“J” standing for Jefferys), the geometrical acoustics approximation, and the geometrical optics approximation. The history of the method is surveyed very nicely by Heading (1962) and in somewhat more mathematical detail by Schlissel (1977b). A good, but dated, introduction to its application to quantum mechanics can be found in the book by Borowitz (1967) and to solid mechanics in the review article by Steele (1976). What
M.H. Holmes, Introduction to Perturbation Methods, Texts in Applied Mathematics 20, DOI 10.1007/978-1-4614-5477-9 4, © Springer Science+Business Media New York 2013
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4 The WKB and Related Methods
this all means is that the WKB approximation is probably a very good idea since so many have rediscovered it in a wide variety of disciplines.
4.2 Introductory Example In the same manner as was done for boundary layers and multiple scales, the ideas underlying the WKB method will be developed by using it to solve an example problem. The one we begin with is the equation ε2 y − q(x)y = 0.
(4.1)
For the moment we will not restrict the function q(x) other than to assume that it is smooth. Our intention is to construct an approximation of the general solution of this equation. To motivate the approach that will be used, suppose that the coefficient q is constant. In this case the general solution of (4.1) is √ √ (4.2) y(x) = a0 e−x q/ε + b0 ex q/ε . The hypothesis made in the WKB method is that the exponential soluti
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