Chapter 8 Scattering Theory - Particle Physics Group

CHAPTER 8. SCATTERING THEORY 142
where the radial part is given by the spherical Bessel functions with constants c lm that have
to be determined. Choosing ⃗ k in the direction of the z-axis the wave function exp(i ⃗ k · ⃗r) =
exp(ikr cosθ) is independent of ϕ so that only the Y lm with m = 0, which are proportional to
the Legendre polynomials P l (θ), can contribute to the expansion
e ikr cos θ =
∞∑
a l j l (kr)P l (cos θ). (8.42)
l=0
With ρ = kr and u = cos θ this becomes
e iρu =
∞∑
a l j l (ρ)P l (u). (8.43)
l=0
One way of determining the coefficients a l is to differentiate this ansatz with respect to ρ,
iue iρu = ∑ l
a l
dj l
dρ P l. (8.44)
The left hand side of (8.44) can now be evaluated by inserting the series (8.43) and using the
recursion relation
(2l + 1)uP m
l
of the Legendre polynomials for m = 0. This yields
i
∞∑
l=0
= (l + 1 − m)P m
l+1 + (l + m)P m
l−1 (8.45)
( l + 1
a l j l
2l + 1 P l+1 +
l )
2l + 1 P l−1 =
∞∑
a l j lP ′ l (8.46)
and, since the Legendre polynomials are linearly independent, for the coefficient of P l
( l
a l j l ′ = i
2l − 1 a l−1j l−1 + l + 1 )
2l + 3 a l+1j l+1 . (8.47)
The derivative j l ′ can now be expressed in terms of j l±1 by using the recursion relations
( d
j l−1 =
dρ + l + 1 )
j l = 1 d
ρ ρ l+1 dρ (ρl+1 j l ) (8.48)
l=0
and
(2l + 1)j l = ρ[j l+1 + j l−1 ], (8.49)
which imply
j ′ l = j l−1 − l + 1
ρ j l = j l−1 − l + 1
2l + 1 (j l+1 + j l−1 ) =
l
2l + 1 j l−1 − l + 1
2l + 1 j l+1 (8.50)
[the equations (8.48-8.50) also holds for the spherical Neumann functions n l ]. Substituting this
expression for j ′ l
into eq. (8.47) we obtain the two equivalent recursion relations
a l
2l + 1 = i a l−1
2l − 1
and
a l
2l + 1 = −i a l+1
2l + 3
(8.51)