Mathematical Methods for Physicists: A concise introduction - Site Map

SPHERICAL BESSEL FUNCTIONS
We can express j l …x† in terms of j 0 …x†. To do this, let us go back to J n …x† and we
®nd that
d
dx fxn J n …x†g ˆ x n J n‡1 …x†; or J n‡1 …x† ˆx n d
dx fxn J n …x†g:
The proof is simple and straightforward:
d
dx fxn J n …x†g ˆ d X 1
dx
kˆ0
X1
n
ˆ x
kˆ0
X1
n
ˆ x
kˆ0
…1† k x 2k
2 n‡2k k!…n ‡ k ‡ 1†
…1† k x n‡2k1
2 n‡2k1 …k 1†!…n ‡ k ‡ 1†
…1† k‡1 x n‡2k‡1
2 n‡2k‡1 k!‰…n ‡ k ‡ 2g ˆxn J n‡1 …x†:
Now if we set n ˆ l ‡ 1 2 and divide by xl‡3=2 , we obtain
J l‡3=2 …x†
ˆ 1
d J l‡1=2 …x† j
or l‡1 …x†
x dx
x l‡1 ˆ 1 x
x l‡3=2
x l‡1=2
d
dx
j l …x†
x l :
Starting with l ˆ 0 and applying this formula l times, we obtain
j l …x† ˆx l 1
d l
j
x dx 0 …x† …l ˆ 1; 2; 3; ...†: …7:103†
Once j 0 …x† has been chosen, all j l …x† are uniquely determined by Eq. (7.103).
Now
p
let us go back to Eq. (7.102) and see why we chose the constant factor C to
be =2 . If we set l ˆ 0 in Eq. (7.101), the resulting equation is
xy 00 ‡ 2y 0 ‡ xy ˆ 0:
Solving this equation by the power series method, the reader will ®nd that functions
sin (x†=x and cos (x†=x are among the solutions. It is customary to de®ne
Now by using Eq. (7.76), we ®nd
J 1=2 …x† ˆX1
kˆ0
ˆ …x=2†1=2 p
…1=2†
…1† k …x=2† 1=2‡2k
k!…k ‡ 3=2†
j 0 …x† ˆsin…x†=x:
1 x2
3! ‡ x4
5! !
ˆ …x=2†1=2 p
…1=2†
sin x
x
r
ˆ 2
sin x:
x
p
Comparing this with j 0 …x† shows that j 0 …x† ˆ
=2xJ 1=2 …x†, and this explains the
factor =2
p
chosen earlier.
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