Subatomic Physics

6.11. More Details on Scattering and Structure 177
is built up from delta functions, δ(r ′ ), each with a weight (2m/� 2 )V (r ′ )ψ(r ′ )so
that
ψs(r) = 2m
� 2
�
d 3 r ′ G(r,r ′ )V (r ′ )ψ(r ′ ), (6.85)
where G(r,r ′ ) is the Green’s function for a delta function potential, Eq. (6.80). The
appropriate solution of the homogeneous Schrödinger equation describes a particle
that impinges on the target along the z axis; the general solution is therefore
ψ(r) =e ikz + 2m
� 2
�
d 3 r ′ G(r,r ′ )V (r ′ )ψ(r ′ ). (6.86)
The original Schrödinger differential equation for the wave function ψ has been
transformed into an integral equation, called the scattering integral equation. For
many problems, it is more convenient to start from such an integral equation rather
than from the differential equation.
In scattering experiments, the incident beam is prepared far outside the scattering
potential, and the scattered particles are also analyzed and detected far away.
The detailed form of the wave function inside the scattering region is consequently
not investigated, and what is needed is the asymptotic form of the scattered wave,
ψs(x). With ˆr = r/r and k = kˆr, asindicatedinFig.6.26,|r − r ′ | becomes
|r − r ′ �
2r · r′
| = r 1 −
r2 r′2
+
r2 �1/2
and the Green’s function takes on the asymptotic value
G(r,r ′ ) ∼
r→∞
−→ r − ˆr · r′
r→∞
(6.87)
−1 exp(ikr)
exp(−ik · r
4π r
′ ). (6.88)
Inserting G(r,r ′ ) into Eq. (6.85) and comparing with Eq. (6.76) yields the expression
for the scattering amplitude,
f(θ, ϕ) = −m
2π�2 �
d 3 r ′ e ik·r′
V (r ′ )ψ(r ′ ).• (6.89)
The First Born Approximation The first Born approximation corresponds to
the case of a weak interaction. If the interaction were negligible, the scattering
amplitude would vanish and ψ(r ′ ) would be given by exp(ikz ′ ) ≡ exp(ik0 · r ′ ). As
a first approximation, this value of the wave function is inserted in Eq. (6.89), with
the result
f(θ, ϕ) = −m
2π�2 �
d 3 r ′ V (r ′ )exp(iq · r ′ /�), (6.90)