Mathematical Methods for Physics and Engineering : A Comprehensive Guide

25.7 WKB methods

there exist many representations, both as linear combinations and as indefinite

integrals, of one in terms of the other.§

25.7 WKB methods

Throughout this book we have had many occasions on which it has been necessary

to solve the equation

d^2 y

dx^2

+k^20 f(x)y= 0 (25.42)

when the notionally general functionf(x) has been, in fact, a constant, usually

the unit functionf(x) = 1. Then the solutions have been elementary and of the

formAsink 0 xorAcosk 0 xwith arbitrary but constant amplitudeA.

Explicit solutions of (25.42) for a non-constantf(x) are only possible in a

limited number of cases, but, as we will show, some progress can be made iff(x)

is a slowly varying function ofx, in the sense that it does not change much in a

range ofxof the order ofk− 01.

We will also see that it is possible to handle situations in which f(x)is

complex; this enables us to deal with, for example, the passage of waves through

an absorbing medium. Developing such solutions will involve us in finding the

integrals of some complex quantities, integrals that will behave differently in the

various parts of the complex plane – hence their inclusion in this chapter.

25.7.1 Phase memory

Before moving on to the formal development of WKB methods¶we discuss the

concept ofphase memorywhich is the underlying idea behind them.

Let us first suppose thatf(x) is real, positive and essentially constant over

arangeofxand definen(x) as the positive square root off(x);n(x)isthen

also real, positive and essentially constant over the same range ofx. We adopt

this notation so that the connection can be made with the description of an

electromagnetic wave travelling through a medium of dielectric constantf(x)

and, consequently, refractive indexn(x). The quantityy(x) would be the electric

or magnetic field of the wave. For this simplified case, in which we can omit the

§These relationships and many other properties of the Airy functions can be found in, for example,

M. Abramowitz and I. A. Stegun (eds),Handbook of Mathematical Functions(New York: Dover,

1965) pp. 446–50.

¶So called because they were used, independently, by Wentzel, Kramers and Brillouin to tackle

certain wave-mechanical problems in 1926, though they had earlier been studied in some depth by

Jeffreys and used as far back as the first half of the nineteenth century by Green.