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33 Coulomb FunctionsVariables ρ,η

Β§33.2 Definitions and Basic Properties

  1. Β§33.2(i) Coulomb Wave Equation
  2. Β§33.2(ii) Regular Solution Fℓ⁑(Ξ·,ρ)
  3. Β§33.2(iii) Irregular Solutions Gℓ⁑(Ξ·,ρ),Hℓ±⁑(Ξ·,ρ)
  4. Β§33.2(iv) Wronskians and Cross-Product

Β§33.2(i) Coulomb Wave Equation

33.2.1 d2wdρ2+(1βˆ’2β’Ξ·Οβˆ’β„“β’(β„“+1)ρ2)⁒w=0,

This differential equation has a regular singularity at ρ=0 with indices β„“+1 and βˆ’β„“, and an irregular singularity of rank 1 at ρ=∞ (§§2.7(i), 2.7(ii)). There are two turning points, that is, points at which d2w/dρ2=0 (Β§2.8(i)). The outer one is given by

33.2.2 ρtp⁑(Ξ·,β„“)=Ξ·+(Ξ·2+ℓ⁒(β„“+1))1/2.

Β§33.2(ii) Regular Solution Fℓ⁑(Ξ·,ρ)

The function Fℓ⁑(Ξ·,ρ) is recessive (Β§2.7(iii)) at ρ=0, and is defined by

33.2.3 Fℓ⁑(Ξ·,ρ)=Cℓ⁑(Ξ·)⁒2βˆ’β„“βˆ’1⁒(βˆ“i)β„“+1⁒MΒ±i⁒η,β„“+12⁑(Β±2⁒i⁒ρ),

or equivalently

33.2.4 Fℓ⁑(Ξ·,ρ)=Cℓ⁑(Ξ·)⁒ρℓ+1⁒eβˆ“i⁒ρ⁒M⁑(β„“+1βˆ“i⁒η,2⁒ℓ+2,Β±2⁒i⁒ρ),

where Mκ,μ⁑(z) and M⁑(a,b,z) are defined in §§13.14(i) and 13.2(i), and

33.2.5 Cℓ⁑(Ξ·)=2ℓ⁒eβˆ’Ο€β’Ξ·/2⁒|Γ⁑(β„“+1+i⁒η)|(2⁒ℓ+1)!.

The choice of ambiguous signs in (33.2.3) and (33.2.4) is immaterial, provided that either all upper signs are taken, or all lower signs are taken. This is a consequence of Kummer’s transformation (Β§13.2(vii)).

Fℓ⁑(Ξ·,ρ) is a real and analytic function of ρ on the open interval 0<ρ<∞, and also an analytic function of Ξ· when βˆ’βˆž<Ξ·<∞.

The normalizing constant Cℓ⁑(Ξ·) is always positive, and has the alternative form

33.2.6 Cℓ⁑(Ξ·)=2ℓ⁒((2⁒π⁒η/(e2β’Ο€β’Ξ·βˆ’1))⁒∏k=1β„“(Ξ·2+k2))1/2(2⁒ℓ+1)!.

Β§33.2(iii) Irregular Solutions Gℓ⁑(Ξ·,ρ),Hℓ±⁑(Ξ·,ρ)

The functions Hℓ±⁑(Ξ·,ρ) are defined by

33.2.7 Hℓ±⁑(Ξ·,ρ)=(βˆ“i)ℓ⁒e(π⁒η/2)Β±i⁒σℓ⁑(Ξ·)⁒Wβˆ“i⁒η,β„“+12⁑(βˆ“2⁒i⁒ρ),

or equivalently

33.2.8 Hℓ±⁑(Ξ·,ρ)=eΒ±i⁒θℓ⁑(Ξ·,ρ)⁒(βˆ“2⁒i⁒ρ)β„“+1Β±i⁒η⁒U⁑(β„“+1Β±i⁒η,2⁒ℓ+2,βˆ“2⁒i⁒ρ),

where Wκ,μ⁑(z), U⁑(a,b,z) are defined in §§13.14(i) and 13.2(i),

33.2.9 θℓ⁑(Ξ·,ρ)=Οβˆ’Ξ·β’ln⁑(2⁒ρ)βˆ’12⁒ℓ⁒π+σℓ⁑(Ξ·),


33.2.10 σℓ⁑(Ξ·)=ph⁑Γ⁑(β„“+1+i⁒η),

the branch of the phase in (33.2.10) being zero when Ξ·=0 and continuous elsewhere. σℓ⁑(Ξ·) is the Coulomb phase shift.

Hβ„“+⁑(Ξ·,ρ) and Hβ„“βˆ’β‘(Ξ·,ρ) are complex conjugates, and their real and imaginary parts are given by

33.2.11 Hβ„“+⁑(Ξ·,ρ) =Gℓ⁑(Ξ·,ρ)+i⁒Fℓ⁑(Ξ·,ρ),
Hβ„“βˆ’β‘(Ξ·,ρ) =Gℓ⁑(Ξ·,ρ)βˆ’i⁒Fℓ⁑(Ξ·,ρ).

As in the case of Fℓ⁑(Ξ·,ρ), the solutions Hℓ±⁑(Ξ·,ρ) and Gℓ⁑(Ξ·,ρ) are analytic functions of ρ when 0<ρ<∞. Also, eβˆ“i⁒σℓ⁑(Ξ·)⁒Hℓ±⁑(Ξ·,ρ) are analytic functions of Ξ· when βˆ’βˆž<Ξ·<∞.

Β§33.2(iv) Wronskians and Cross-Product

With arguments η,ρ suppressed,

33.2.12 𝒲⁑{Gβ„“,Fβ„“}=𝒲⁑{Hβ„“Β±,Fβ„“}=1.
33.2.13 Fβ„“βˆ’1⁒Gβ„“βˆ’Fℓ⁒Gβ„“βˆ’1=β„“/(β„“2+Ξ·2)1/2,