Subatomic Physics

176 Structure of Subatomic Particles
The Scattering Integral Equation • To find the general solution of the
Schrödinger equation, Eq. (6.74), we recall that it can be written as the sum of a
special solution and of the appropriate solution of the corresponding homogeneous
equation, where V = 0. To find a special solution of Eq. (6.74), it is convenient to
consider the term (2m/� 2 )Vψ on the right-hand side as the given inhomogeneity,
even though it contains the unknown wave function ψ. As a first step, then, we solve
the scattering problem for a point source for which the inhomogeneity becomes a
three-dimensional Dirac delta function and Eq. (6.74) takes on the form
(∇ 2 + k 2 )G(r, r ′ )=δ(r − r ′ ). (6.79)
The solution of this equation that corresponds to an outgoing wave is
G(r,r ′ )= −1
4π
eik|r−r′ |
|r − r ′ . (6.80)
|
To verify that this Green’s function indeed satisfies Eq. (6.79), we set, for simplicity,
r ′ =0, |r| = r, and use the relations (62)
∇ 2
� �
1
= −4πδ(r) (6.81)
r
∇ 2 (polar coord.) = 1
r 2
∇ 2 (FG)=(∇ 2 F )G
+
After some calculations we obtain
+2(∇F ) · (∇G)+F ∇ 2 G (6.82)
�
∂ 2 ∂
r
∂r ∂r
1
r 2 sin θ
∂
∂θ
(∇ 2 + k 2 ) eikr
r
�
�
sin θ ∂
�
+
∂θ
= −4πδ(r)eikr
1
r 2 sin 2 θ
∂2 . (6.83)
∂φ2 = −4πδ(r). (6.84)
The second step in this identity follows from the fact that
�
d 3 �
rδ(r)f(r) and d 3 rδ(r)exp(ikr)f(r)
give the same result, f(0), for any continuous function f. The solution of Eq. (6.55)
for a potential V (r) is found by assuming that the inhomogeneity (2m/�2 )V (r)ψ(r)
62For a derivation of Eq. (6.81) see, for instance, Jackson, Section 1.7.