7
Coulomb Wave Functions
Abstract
Topic of this chapter are the regular and irregular Coulomb functions and
scattering states. The regular and irregular Coulomb functions, based on a
spherical representation of the Schrödinger equation with different scaling techniques, will be discussed and evaluated, as well as the Coulomb wave function
(scattering states) based on the Schrödinger equation in parabolic coordinates.
The corresponding programming codes are available for download.
Topic of this chapter are the regular and irregular Coulomb wave functions.
The Schrödinger equation for two point particles [2] with charge ±qZ 1 and
±qZ 2 and distance r reads
−
¯
h 2
2m
Δ + ±
q 2 Z 1 Z 2
r
< r|ϕ >= E < r|ϕ > ,
(7.1)
where +-sign stands for a repulsive and −-sign for an attractive Coulomb interaction, e.g., the hydrogen atom. With the abbreviations
k
2
=
2mE
¯
h 2
and β = ±mq
2 Z 1 Z 2
¯
h 2 ,
(7.2)
we get
Δ + k
2
−
2β
r
< r|ϕ >= 0.
(7.3)
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
W. Schweizer, Special Functions in Physics with MATLAB,
https://doi.org/10.1007/978-3-030-64232-7_7
101