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Coulomb functions

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1: 33 Coulomb Functions
Chapter 33 Coulomb Functions
2: 33.22 Particle Scattering and Atomic and Molecular Spectra
§33.22(i) Schrödinger Equation
Attractive potentials: Z 1 Z 2 < 0 , η < 0 .
Attractive potentials: Z 1 Z 2 < 0 , κ < 0 .
§33.22(iv) Klein–Gordon and Dirac Equations
3: 33.14 Definitions and Basic Properties
§33.14(ii) Regular Solution f ( ϵ , ; r )
§33.14(iii) Irregular Solution h ( ϵ , ; r )
For nonzero values of ϵ and r the function h ( ϵ , ; r ) is defined by … The function s ( ϵ , ; r ) has the following properties: …
§33.14(v) Wronskians
4: 33.18 Limiting Forms for Large
§33.18 Limiting Forms for Large
5: 33.2 Definitions and Basic Properties
§33.2(i) Coulomb Wave Equation
The function F ( η , ρ ) is recessive (§2.7(iii)) at ρ = 0 , and is defined by … The functions H ± ( η , ρ ) are defined by … …
§33.2(iv) Wronskians and Cross-Product
6: 33.20 Expansions for Small | ϵ |
§33.20(i) Case ϵ = 0
§33.20(ii) Power-Series in ϵ for the Regular Solution
§33.20(iii) Asymptotic Expansion for the Irregular Solution
where A ( ϵ , ) is given by (33.14.11), (33.14.12), and …
§33.20(iv) Uniform Asymptotic Expansions
7: 33.15 Graphics
§33.15 Graphics
§33.15(i) Line Graphs of the Coulomb Functions f ( ϵ , ; r ) and h ( ϵ , ; r )
§33.15(ii) Surfaces of the Coulomb Functions f ( ϵ , ; r ) , h ( ϵ , ; r ) , s ( ϵ , ; r ) , and c ( ϵ , ; r )
See accompanying text
Figure 33.15.9: h ( ϵ , ; r ) with = 1 , - 2 < ϵ < 2 , - 15 < r < 15 . Magnify 3D Help
8: 33.17 Recurrence Relations and Derivatives
§33.17 Recurrence Relations and Derivatives
33.17.1 ( + 1 ) r f ( ϵ , - 1 ; r ) - ( 2 + 1 ) ( ( + 1 ) - r ) f ( ϵ , ; r ) + ( 1 + ( + 1 ) 2 ϵ ) r f ( ϵ , + 1 ; r ) = 0 ,
33.17.2 ( + 1 ) ( 1 + 2 ϵ ) r h ( ϵ , - 1 ; r ) - ( 2 + 1 ) ( ( + 1 ) - r ) h ( ϵ , ; r ) + r h ( ϵ , + 1 ; r ) = 0 ,
33.17.3 ( + 1 ) r f ( ϵ , ; r ) = ( ( + 1 ) 2 - r ) f ( ϵ , ; r ) - ( 1 + ( + 1 ) 2 ϵ ) r f ( ϵ , + 1 ; r ) ,
33.17.4 ( + 1 ) r h ( ϵ , ; r ) = ( ( + 1 ) 2 - r ) h ( ϵ , ; r ) - r h ( ϵ , + 1 ; r ) .
9: 33.24 Tables
§33.24 Tables
10: 33.1 Special Notation
The main functions treated in this chapter are first the Coulomb radial functions F ( η , ρ ) , G ( η , ρ ) , H ± ( η , ρ ) (Sommerfeld (1928)), which are used in the case of repulsive Coulomb interactions, and secondly the functions f ( ϵ , ; r ) , h ( ϵ , ; r ) , s ( ϵ , ; r ) , c ( ϵ , ; r ) (Seaton (1982, 2002a)), which are used in the case of attractive Coulomb interactions. …
  • Greene et al. (1979):

    f ( 0 ) ( ϵ , ; r ) = f ( ϵ , ; r ) , f ( ϵ , ; r ) = s ( ϵ , ; r ) , g ( ϵ , ; r ) = c ( ϵ , ; r ) .