Department of Physics and Astronomy
Arizona State University
Tempe, AZ U.S.A.
Many of you are having trouble taking the limits of a finite system to infinity and converting the sums to the corresponding integrals especially when Bessel functions are involved.
Let's look at examples of taking the limit of two simple systems as their size goes to infinity and converting the corresponding sum of basis functions to an integral. We covered the simplest example in class, the usual coulomb Green's function for free space,
|G(,') = ,||(1)|
In the usual case, we imagine a cubic box of side L , with periodic boundary conditions. The Laplacian separates in cartesian coordinates, and we need to solve the three equations
|+ ki2fn(xi) = 0 .||(2)|
|= (m + n + o) ,||(4)|
The Green's function is
|G(,') = 4||(5)|
In the limit of L for any nonzero q , the integers m , n , o go to infinity, and the spacing between adjacent q values goes to zero. This means we can convert the sum to an integral, and for sums spaced by 1,
|G(,') = dn dm do .||(8)|
|dn||=||dqx = dqx|
|dm||=||dqy = dqy|
|do||=||dqz = dqz||(9)|
|dn dm do = d 3q .||(10)|
|G(,') = d 3q||(11)|
|G(,') = dq .||(12)|
|G(,') = du ,||(13)|
The last integral
|I = du||(14)|
Now that we did it the easy way, let's use a harder method that may help in taking other limits. Let's again calculate the free space Green's function, but let's take the limit using Dirichlet boundary conditions on a sphere of radius R that we will let go to infinity. Separating variables as in class, the eigenfunctions of are
|() = jl(qlor)Ylm(,)||(17)|
The Green's function is
|G(,') = 4jl(qlor)jl(qlor')Ylm(,)Y *lm(',') ,||(18)|
In the limit of R for any nonzero q value, the integer o must go to infinity, and the spacing between adjacent q values goes to zero. This means the the o sum can be converted to an integral. Just as in the cartesian case, we need to calculate do/dql to change to an integration over ql . Since o goes to for any finite ql value, the Bessel function zeroes can be evaluated for large argument, since the small arguments contribute an amount of measure zero as R .
The assymptotic expansion of jl(x) is
|jl(x) sin(x - l/2) ,||(19)|
|qlo = (l + 2o)||(20)|
|G(,') = 8dqljl(qlr)jl(qlr')Ylm(,)Y *lm(',') .||(21)|
|G(,') = 8dqjl(qr)jl(qr')Ylm(,)Y *lm(',') .||(22)|
Rather than perform the integration in Eq. 22, we can verify that this is the correct expression by comparing with our previous result. Jackson Eq. 10.43 gives the expansion of a plane wave in terms of spherical Bessel functions,
|exp(i ) = 4i ljl(kr)Ylm(,)Ylm*(,)||(23)|
We can therefore write
|exp(i [ - ']) = (4)2i ljl(qr)Ylm(,)Ylm*(,)(- i)l'jl'(qr')Y *l'm'(',')Yl'm'(,)||(24)|
It is also possible to do the integrals, but we know the result
that must occur, and that means the integrals will be somewhat ugly.
For example for l = 0 , the integral can be done readily with contour
integration similarly to the contour integral above to give
The other integrals could be done similarly, but this is a hard way of getting the standard result.