184 CHAPTER11. ANGULARMOMENTUM
or
d
dθ
A(θ)=lcotθA(θ) (11.81)
whichissolvedby
A(θ)=const.×sinlθ (11.82)
Then
Yll(θ,φ)=Nsinlθeilφ (11.83)
whereN isanormalization constant, whichisdeterminedfromthe normalization
condition
1 =
∫π
0
dθ sinθ
∫ 2 π
0
dφYl∗lYll
= 2 πN^2
∫π
0
dθ sin(2l+1)θ
=
2 π^3 /^2 l!
Γ(l+^32 )
N^2 (11.84)
wheretheΓ-functionisaspecialfunctionwiththeproperties
Γ(x+1)=xΓ(x) Γ(
1
2
)=
√
π (11.85)
Ofcourse,thenormalizationconditiononlydeterminesN,evenassumingN isreal,
uptoanoverallsign. Theconventionistochoosethissigntobe(−1)l,sothatfinally
Yll(θ,φ)=(−1)l
[
Γ(l+^32 )
2 π^3 /^2 l!
] 1 / 2
sinlθeilφ (11.86)
FromherewecangettheotherYlmusingtheloweringoperator
L ̃−Yl,m(θ,φ)=Cl−mYl,m− 1 (θ,φ) (11.87)
giventheconstantsCl−m.CorrespondingconstantsCl+maredefinedfrom
L ̃+Yl,m(θ,φ)=Cl+mYl,m+1(θ,φ) (11.88)
TogettheCl−mconstants,weagainresorttosomecleveralgebra. Wehave
L ̃−L ̃+ = L ̃^2 −L ̃^2 z− ̄hL ̃z
L ̃+L ̃− = L ̃^2 −L ̃^2 z+ ̄hL ̃z (11.89)
Then
<φlm|L ̃−L ̃+|φlm> = <φlm|(L ̃^2 −L ̃^2 z− ̄hL ̃z)|φlm>
<L ̃+φlm|L ̃+φlm> = h ̄^2 [l(l+1)−m^2 −m)<φlm|φlm>
(Cl+m)∗Cl+m = ̄h^2 (l−m)(l+m+1) (11.90)