QMGreensite_merged

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)