254
Chapter 4. Particle scattering
The difference between these outgoing and incoming wave vectors
is q = k − k 0 . For elastic scattering the magnitudes k 0 and k are
equal.
We postulate a point source at P , radiating an outgoing spherical wave, represented by the complex wave function
1
G(R) = exp(ikR).
(4.59)
R
This point source does not exist physically, but provides a mathematical aid to solve the problem. The amplitude G(R) must satisfy
the Helmholtz equation,
v
2 G(R) + k
2 G(R) = 0
(4.60)
R
everywhere except at P , where G(R) has a singularity. To verify
that this is the case, we express the Laplacian operator v
2 in
spherical coordinates, leading to
d
2
1
[ R G(R) ] + k
2 G(R) = 0.
(4.61)
R dR 2
The solution is immediately recognizable as
R G(R) = exp (± ikR),
(4.62)
in agreement with (4.59) as required. We evaluate G(R) at the
field point x 1 in Figure 4.5, where
R = | x − x 1 |.
(4.63)
For notational purposes, we denote G(R) = G(x, x 1 ) in the following, where
1
G(x, x 1 ) =
exp ( ik| x − x 1 | )
(4.64)
| x − x 1 |
represents the outgoing spherical wave. Multiplying (4.57) by
G(x, x 1 ), multiplying (4.60) by u(x 1 ), and subtracting the two
equations, we obtain
2m
G(x, x 1 ) v 1
2 u(x 1 ) − u(x 1 ) v 1
2 G(x, x 1 ) =
U (x 1 ) G(x, x 1 ) u(x 1 ).
h
2
¯
(4.65)