Mathematical Methods for Physics and Engineering - Matematica.NET

25.7 WKB METHODSthere exist many representations, both as linear combinations and as indefiniteintegrals, of one in terms of the other. §25.7 WKB methodsThroughout this book we have had many occasions on which it has been necessaryto solve the equationd 2 ydx 2 + k2 0f(x)y = 0 (25.42)when the notionally general function f(x) has been, in fact, a constant, usuallythe unit function f(x) = 1. Then the solutions have been elementary and of theform A sin k 0 x or A cos k 0 x with arbitrary but constant amplitude A.Explicit solutions of (25.42) for a non-constant f(x) are only possible in alimited number of cases, but, as we will show, some progress can be made if f(x)is a slowly varying function of x, in the sense that it does not change much in arange of x of the order of k0 −1.We will also see that it is possible to handle situations in which f(x) iscomplex; this enables us to deal with, for example, the passage of waves throughan absorbing medium. Developing such solutions will involve us in finding theintegrals of some complex quantities, integrals that will behave differently in thevarious parts of the complex plane – hence their inclusion in this chapter.25.7.1 Phase memoryBefore moving on to the formal development of WKB methods we discuss theconcept of phase memory which is the underlying idea behind them.Let us first suppose that f(x) is real, positive and essentially constant overarangeofx and define n(x) as the positive square root of f(x); n(x) isthenalso real, positive and essentially constant over the same range of x. We adoptthis notation so that the connection can be made with the description of anelectromagnetic wave travelling through a medium of dielectric constant f(x)and, consequently, refractive index n(x). The quantity y(x) would be the electricor 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 tacklecertain wave-mechanical problems in 1926, though they had earlier been studied in some depth byJeffreys and used as far back as the first half of the nineteenth century by Green.895