Mathematical Methods for Physics and Engineering : A Comprehensive Guide

APPLICATIONS OF COMPLEX VARIABLES

Apply the WKB method to the problem of finding the quantum energy levelsEof a

particle of massmbound in a symmetrical one-dimensional potential wellV(x)that has

only a single minimum. The relevant Schr ̈odinger equation is

−

^2

2 m

d^2 ψ

dx^2

+V(x)ψ=Eψ.

Relate the problem close to each of the classical ‘turning points’,x=±aat whichE−

V(x)=0, to Stokes’ equation and assume that it is appropriate to use the solutionAi(x)

given in equations (25.52) and (25.53) atx=a. Show that if the general WKB solution in

the ‘classically allowed’ region−a<x<ais to match such Airy solutions atbothturning

points, then

∫a

−a

k(x)dx=(n+^12 )π,

wherek^2 (x)=2m[E−V(x)]/
^2 andn=0, 1 , 2 ,....

For a symmetric potentialV(x)=V 0 x^2 s,wheresis a positive integer, show that in this

approximation the energy of thenth level is given byEn=cs(n+^12 )^2 s/(s+1),wherecsis a

constant depending onsbut not uponn.

We start by multiplying the equation through by 2m/
^2 ,writing2m[E−V(x)]/
^2 ask^2 (x),
and rearranging the equation to read

d^2 ψ

dx^2

+k^2 (x)ψ=0, (25.54)

noting that, withEandV(x) given, the equationE=V(a) determines the value ofaand
thatk(a)=0.
For−a<x<a,wherek^2 (x) is positive, the form of the WKB solutions are given
directly by (25.49) as

ψ±=

C

√

k(x)

exp

[

±i

∫x

k(u)du

]

.

Just beyond the turning pointx=a,where

E−V(x)=0−V′(a)(x−a)+O[(x−a)^2 ],

equation (25.54) can be approximated by

d^2 ψ

dx^2

−

2 mV′(a)


^2

(x−a)ψ=0. (25.55)

This, in turn, can be reduced to Stokes’ equation by first settingx−a=μzandψ(x)≡y(z),
so converting it into

1

μ^2

d^2 y

dz^2

−

2 μmV′(a)


^2

zy=0,

and then choosingμ=[
^2 / 2 mV′(a)]^1 /^3. The equation then reads

d^2 y

dz^2

=zy.

Since the solution must be evanescent forx>a,i.e.forz>0, we assume that the
appropriate solution there is Ai(z); this implies that, forzsmall and negative (just inside
the classically allowed region), the solution has the form given by (25.53), namely

A

(−z)^1 /^4

sin

[

2

3

(−z)^3 /^2 +

π

4

]

,