QMGreensite_merged

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