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33 Coulomb FunctionsVariables r,Ο΅

Β§33.14 Definitions and Basic Properties

  1. Β§33.14(i) Coulomb Wave Equation
  2. Β§33.14(ii) Regular Solution f⁑(Ο΅,β„“;r)
  3. Β§33.14(iii) Irregular Solution h⁑(Ο΅,β„“;r)
  4. Β§33.14(iv) Solutions s⁑(Ο΅,β„“;r) and c⁑(Ο΅,β„“;r)
  5. Β§33.14(v) Wronskians

Β§33.14(i) Coulomb Wave Equation

Another parametrization of (33.2.1) is given by

33.14.1 d2wdr2+(Ο΅+2rβˆ’β„“β’(β„“+1)r2)⁒w=0,


33.14.2 r =βˆ’Ξ·β’Ο,
Ο΅ =1/Ξ·2.

Again, there is a regular singularity at r=0 with indices β„“+1 and βˆ’β„“, and an irregular singularity of rank 1 at r=∞. When Ο΅>0 the outer turning point is given by

33.14.3 rtp⁑(Ο΅,β„“)=(1+ϡ⁒ℓ⁒(β„“+1)βˆ’1)/Ο΅;

compare (33.2.2).

Β§33.14(ii) Regular Solution f⁑(Ο΅,β„“;r)

The function f⁑(Ο΅,β„“;r) is recessive (Β§2.7(iii)) at r=0, and is defined by

33.14.4 f⁑(Ο΅,β„“;r)=ΞΊβ„“+1⁒MΞΊ,β„“+12⁑(2⁒r/ΞΊ)/(2⁒ℓ+1)!,

or equivalently

33.14.5 f⁑(Ο΅,β„“;r)=(2⁒r)β„“+1⁒eβˆ’r/κ⁒M⁑(β„“+1βˆ’ΞΊ,2⁒ℓ+2,2⁒r/ΞΊ)/(2⁒ℓ+1)!,

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

33.14.6 ΞΊ={(βˆ’Ο΅)βˆ’1/2,Ο΅<0,r>0,βˆ’(βˆ’Ο΅)βˆ’1/2,Ο΅<0,r<0,Β±iβ’Ο΅βˆ’1/2,Ο΅>0.

The choice of sign in the last line of (33.14.6) is immaterial: the same function f⁑(Ο΅,β„“;r) is obtained. This is a consequence of Kummer’s transformation (Β§13.2(vii)).

f⁑(Ο΅,β„“;r) is real and an analytic function of r in the interval βˆ’βˆž<r<∞, and it is also an analytic function of Ο΅ when βˆ’βˆž<Ο΅<∞. This includes Ο΅=0, hence f⁑(Ο΅,β„“;r) can be expanded in a convergent power series in Ο΅ in a neighborhood of Ο΅=0 (Β§33.20(ii)).

Β§33.14(iii) Irregular Solution h⁑(Ο΅,β„“;r)

For nonzero values of Ο΅ and r the function h⁑(Ο΅,β„“;r) is defined by

33.14.7 h⁑(Ο΅,β„“;r)=Γ⁑(β„“+1βˆ’ΞΊ)π⁒κℓ⁒(WΞΊ,β„“+12⁑(2⁒r/ΞΊ)+(βˆ’1)ℓ⁒S⁑(Ο΅,r)⁒Γ⁑(β„“+1+ΞΊ)2⁒(2⁒ℓ+1)!⁒MΞΊ,β„“+12⁑(2⁒r/ΞΊ)),

where ΞΊ is given by (33.14.6) and

33.14.8 S⁑(Ο΅,r)={2⁒cos⁑(π⁒|Ο΅|βˆ’1/2),Ο΅<0,r>0,0,Ο΅<0,r<0,eΟ€β’Ο΅βˆ’1/2,Ο΅>0,r>0,eβˆ’Ο€β’Ο΅βˆ’1/2,Ο΅>0,r<0.

(Again, the choice of the ambiguous sign in the last line of (33.14.6) is immaterial.)

h⁑(Ο΅,β„“;r) is real and an analytic function of each of r and Ο΅ in the intervals βˆ’βˆž<r<∞ and βˆ’βˆž<Ο΅<∞, except when r=0 or Ο΅=0.

Β§33.14(iv) Solutions s⁑(Ο΅,β„“;r) and c⁑(Ο΅,β„“;r)

The functions s⁑(Ο΅,β„“;r) and c⁑(Ο΅,β„“;r) are defined by

33.14.9 s⁑(Ο΅,β„“;r) =(B⁑(Ο΅,β„“)/2)1/2⁒f⁑(Ο΅,β„“;r),
c⁑(Ο΅,β„“;r) =(2⁒B⁑(Ο΅,β„“))βˆ’1/2⁒h⁑(Ο΅,β„“;r),


33.14.10 B⁑(Ο΅,β„“)={A⁑(Ο΅,β„“)⁒(1βˆ’exp⁑(βˆ’2⁒π/Ο΅1/2))βˆ’1,Ο΅>0,A⁑(Ο΅,β„“),ϡ≀0,


33.14.11 A⁑(Ο΅,β„“)=∏k=0β„“(1+ϡ⁒k2).

An alternative formula for A⁑(Ο΅,β„“) is

33.14.12 A⁑(Ο΅,β„“)=Γ⁑(1+β„“+ΞΊ)Γ⁑(ΞΊβˆ’β„“)β’ΞΊβˆ’2β’β„“βˆ’1,

the choice of sign in the last line of (33.14.6) again being immaterial.

When Ο΅<0 and β„“>(βˆ’Ο΅)βˆ’1/2 the quantity A⁑(Ο΅,β„“) may be negative, causing s⁑(Ο΅,β„“;r) and c⁑(Ο΅,β„“;r) to become imaginary.

The function s⁑(Ο΅,β„“;r) has the following properties:

33.14.13 ∫0∞s⁑(Ο΅1,β„“;r)⁒s⁑(Ο΅2,β„“;r)⁒dr=δ⁑(Ο΅1βˆ’Ο΅2),

where the right-hand side is the Dirac delta (Β§1.17). When Ο΅=βˆ’1/n2, n=β„“+1,β„“+2,…, s⁑(Ο΅,β„“;r) is exp⁑(βˆ’r/n) times a polynomial in r/n, and

33.14.14 Ο•n,ℓ⁑(r)=(βˆ’1)β„“+1+n⁒(2/n3)1/2⁒s⁑(βˆ’1/n2,β„“;r)=(βˆ’1)β„“+1+nnβ„“+2⁒((nβˆ’β„“βˆ’1)!(n+β„“)!)1/2⁒(2⁒r)β„“+1⁒eβˆ’r/n⁒Lnβˆ’β„“βˆ’1(2⁒ℓ+1)⁑(2⁒r/n)


33.14.15 ∫0βˆžΟ•m,ℓ⁑(r)⁒ϕn,ℓ⁑(r)⁒dr=Ξ΄m,n.

Note that the functions Ο•n,β„“, n=β„“,β„“+1,…, do not form a complete orthonormal system.

Β§33.14(v) Wronskians