Mathematical Methods for Physics and Engineering - Matematica.NET

APPLICATIONS OF COMPLEX VARIABLESThe result just proved gives∫ an√ 2mV0−a n(a 2sn − x 2s ) 1/2 dx =(n + 1 )π. 2Writing x = va n shows that the integral is proportional to a s+1n I s ,whereI s is the integralbetween −1 and +1 of (1 − v 2s ) 1/2 and does not depend upon n. Thus E n ∝ a 2sn anda s+1n ∝ (n + 1 ), implying that E 2 n ∝ (n + 1 2 )2s/s+1 .Although not asked for, we note that the above result indicates that, for a simpleharmonic oscillator, for which s = 1, the energy levels [ E n ∼ (n + 1 ) ] are equally spaced,2whilst for very large s, corresponding to a square well, the energy levels vary as n 2 .Bothof these results agree with what is found from detailed analyses of the individual cases. ◭25.7.3 Accuracy of the WKB solutionsWe may also ask when we can expect the WKB solutions to the Stokes’ equationto be reasonable approximations. Although our final form for the WKB solutionsis not exactly that used when the condition |n ′ k0 −1|≪|n2| was derived, it shouldgive the same order of magnitude restriction as a more careful analysis. For thederivation of (25.51), k0 2 = −1, n(z) =[f(z)]1/2 = z 1/2 , and the criterion becomes12 |z−1/2 |≪|z|, or, in round terms, |z| 3 ≫ 1.For the more general equation, typified by (25.42), the condition for the validityof the WKB solutions can usually be satisfied by making some quantity, often |z|,sufficiently large. Alternatively, a parameter such as k 0 can be made large enoughthat the validity criterion is satisfied to any pre-specified level. However, from apractical point of view, natural physical parameters cannot be varied at will, andrequiring z to be large may well reduce the value of the method to virtually zero.It is normally more useful to try to obtain an improvement on a WKB solutionby multiplying it by a series whose terms contain increasing inverse powers ofthe variable, so that the result can be applied successfully for moderate, and notjust excessively large, values of the variable.We do not have the space to discuss the properties and pitfalls of suchasymptotic expansions in any detail, but exercise 25.18 will provide the readerwith a model of the general procedure. A few particular points that should benoted are given as follows.(i) If the multiplier is analytic as z →∞, then it will be represented by aseries that is convergent for |z| greater than some radius of convergenceR.(ii) If the multiplier is not analytic as z →∞, as is usually the case, then themultiplier series eventually diverges and there is a z-dependent optimalnumber of terms that the series should contain in order to give the bestaccuracy.(iii) For a fixed value of arg z, the asymptotic expansion of the multiplier isunique. However, the same asymptotic expansion can represent more than902