102
7 Coulomb Wave Functions
For Z 1 = Z 2 = 1 and m the mass of the hydrogen atom, the scaling corresponds
to atomic units with length measured in units of the Bohr radius and the energy in
units of the Rydberg energy.
Spherical Coordinates: Parameterization (a)
In spherical coordinates (23.5) plus additional scaling of the radial coordinate, kr =
ρ and
1
k β = γ 1 lead to the radial Schrödinger equation [1, 3]
d 2
dρ 2 + 1 −
2γ
ρ
−
l(l + 1)
ρ 2
Φ l (γ , ρ) = 0 ,
(7.4)
with γ negative for attractive potentials and positive for repulsive potentials. In
units of the Bohr radius (a B = ¯
h 2
mq 2 ), the Sommerfeld parameter γ becomes
γ =
Z 1 Z 2
k . This second order equation has two linear independent solutions: the
regular Coulomb function F l (γ , ρ) and the irregular Coulomb functions H
±
l (γ , ρ)
and G l (γ , ρ).
Spherical Coordinates: Parameterization (b)
An alternative parameterization is given by Olver et al. [3] ζ = −βr and =
k 2
β 2 ,
and hence, the radial Schrödinger equation becomes
d 2
dζ 2 + −
2
ζ
−
l(l + 1)
ζ 2
˜
Φ l ((, ζ ) = 0 ,
(7.5)
with ζ > 0 for attractive potentials and for repulsive potentials ζ < 0. Again, this
second order equation has, to independent solutions, the regular Coulomb function
f l ((, ζ ) and the irregular Coulomb function h l ((, ζ ).
Parabolic Coordinates
Due to the rotational symmetry around the beam direction, we select parabolic
coordinates, Eq. (23.16c), such that the beam direction is parallel to the z-axis, k||z,
with energy of the relative motion E = ¯
hk 2
2m . From Eqs. (23.17b) and (7.1), we get
−
¯
h 2
2m
4
η + ζ
∂
∂η
η
∂
∂η
+
∂
∂ζ
ζ
∂
∂ζ
+
1
ηζ
∂ 2
∂φ 2
+ ±
q 2 Z 1 Z 2
η + ζ
− E
· < η, ζ, φ|ϕ >= 0,
1 Alternative notation for γ is η, e.g., in [1].