11.4. EIGENFUNCTIONSOFANGULARMOMENTUM 185
sothat
Cl+m= ̄h
√
(l−m)(l+m+1)eiω (11.91)
Likewise
<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) (11.92)
sothat
Cl−m= ̄h
√
(l+m)(l−m+1)eiω (11.93)
whereωisanarbitraryphase. Thiscanalwaysbeabsorbedintoarescalingofthe
wavefunction;i.e. φlm→eiωφlm,whichdoesnotaffectthephysicalstateatall.So
wecanalwayschoosetheCl−mtobereal.Itisnothardtoshowthat
Cl−m=(Cl+,m− 1 )∗ (11.94)
Exercise: Provethisrelation.
Therefore,iftheC−coefficientsarereal,thentheC+coefficientsarealsoreal,
andwehave
Cl−m= ̄h
√
(l+m)(l−m+1)
Cl+m= ̄h
√
(l−m)(l+m+1) (11.95)
andwecannowcomputealltheYlmusing
Yll(θ,φ) = (−1)l
[
Γ(l+^32 )
2 π^3 /^2 l!
] 1 / 2
sinlθeilφ
L ̃−Ylm = ̄h
√
(l+m)(l−m+1)Yl,m− 1
L ̃+Ylm = ̄h
√
(l−m)(l+m+1)Yl,m+1 (11.96)
Sincethe Yll wasnormalized to 1 inequation(11.84), all ofthe Ylm obtained by
applyingtheloweringoperatorwillalsobenormalizedto1. Also,sinceL ̃^2 andL ̃z
areHermitianoperators,sphericalharmonicscorresponding todifferentvalues ofl
and/ormwillbeorthogonal:
∫π
0
dθsinθ
∫ 2 π
0
dφYlm∗(θ,φ)Yl′m′(θ,φ)=δll′δmm′ (11.97)
Example:the l= 1 multiplet