APPLICATIONS OF COMPLEX VARIABLES
The result just proved gives
∫an
−an√
2 mV 0
(a^2 ns−x^2 s)^1 /^2 dx=(n+^12 )π.Writingx=vanshows that the integral is proportional toasn+1Is,whereIsis the integral
between−1 and +1 of (1−v^2 s)^1 /^2 and does not depend uponn. ThusEn∝a^2 nsand
asn+1∝(n+^12 ), implying thatEn∝(n+^12 )^2 s/s+1.
Although not asked for, we note that the above result indicates that, for a simple
harmonic oscillator, for whichs= 1, the energy levels [En∼(n+^12 ) ] are equally spaced,
whilst for very larges, corresponding to a square well, the energy levels vary asn^2 .Both
of 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’ equation
to be reasonable approximations. Although our final form for the WKB solutions
is not exactly that used when the condition|n′k− 01 ||n^2 |was derived, it should
give the same order of magnitude restriction as a more careful analysis. For the
derivation of (25.51),k^20 =−1,n(z)=[f(z)]^1 /^2 =z^1 /^2 , and the criterion becomes
1
2 |z
− 1 / (^2) ||z|, or, in round terms,|z| (^3) 1.
For the more general equation, typified by (25.42), the condition for the validity
of the WKB solutions can usually be satisfied by making some quantity, often|z|,
sufficiently large. Alternatively, a parameter such ask 0 can be made large enough
that the validity criterion is satisfied to any pre-specified level. However, from a
practical point of view, natural physical parameters cannot be varied at will, and
requiringzto 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 solution
by multiplying it by a series whose terms contain increasing inverse powers of
the variable, so that the result can be applied successfully for moderate, and not
just excessively large, values of the variable.
We do not have the space to discuss the properties and pitfalls of such
asymptotic expansions in any detail, but exercise 25.18 will provide the reader
with a model of the general procedure. A few particular points that should be
noted are given as follows.
(i) If the multiplier is analytic asz→∞, then it will be represented by a
series that is convergent for|z|greater than some radius of convergence
R.
(ii) If the multiplier is not analytic asz→∞, as is usually the case, then the
multiplier series eventually diverges and there is az-dependent optimal
number of terms that the series should contain in order to give the best
accuracy.
(iii) For a fixed value of argz, the asymptotic expansion of the multiplier is
unique. However, the same asymptotic expansion can represent more than