7.2 The Coulomb Wave Functions
105
7.2.2 Coulomb Wave Function: Parabolic Coordinates
The solution of Eq. (7.6) with appropriate scaling factor is given by
ψ(r, z) = exp(−
π
2
γ ) Γ (1 + iγ ) 1 F 1 (−iγ ; 1; ik(r − z)) ,
(7.16)
and therefore, an incident wave in z-direction and an outgoing scattered wave as
ψ k (r, z) = exp(ikz) exp(−
π
2
γ ) Γ (1 + iγ ) 1 F 1 (−iγ ; 1; ik(r − z)) ,
(7.17a)
which can be generalized to arbitrary directions
ψ k (r) = exp(ik · r) exp(−
π
2
γ ) Γ (1 + iγ ) 1 F 1 (−iγ ; 1; i(kr − k · r)) ,
(7.17b)
with |k| = k and |r| = r. An example is plotted in Fig. 7.1. Additional visualizations
are shown in Figs. 7.3 and 7.4.
The partial wave expansion [2] can be constructed by factorization in spherical
coordinates and is given by
ψ(r, z) =
∞
l=0
(2l + 1) i
l exp(iσ l (γ ))
1
ρ
F l (γ , ρ) P l (cos(θ )),
(7.18a)
respectively,
ψ k (r, z) = exp(ikz)
∞
l=0
(2l + 1) i
l exp(iσ l (γ ))
1
ρ
F l (γ , ρ) P l (cos(θ )),
(7.18b)
-40
-30
-20
-10
0
10
20
30
40
x
-20
-10
0
10
20
30
40
z
Fig. 7.1 Coulomb scattering; k = 2, γ = 1