11.4. EIGENFUNCTIONSOFANGULARMOMENTUM 187
havejustcomputed,are
Y 00 =
√
1
4 π
Y 11 = −
√
3
8 π
sinθeiφ
Y 10 =
√
3
4 π
cosθ
Y 1 ,− 1 =
√
3
8 π
sinθe−iφ
Y 22 =
√
15
32 π
sin^2 θe^2 iφ
Y 21 = −
√
15
8 π
sinθcosθeiφ
Y 20 =
√
5
16 π
(3cos^2 θ−1)
Y 2 ,− 1 =
√
15
8 π
sinθcosθe−iφ
Y 2 ,− 2 =
√
15
32 π
sin^2 θe−^2 iφ (11.103)
Thereisanicegraphicalwayofrepresentingthesphericalharmonics. Themodulus
ofthesphericalharmonics|Ylm|isafunctiononlyofθ,andisindependentofφ.So,
inpolarcoordinates(r,θ)whereθ isthe angleawayfromthe z-axis,we plotthe
function
r(θ)=|Ylm(θ,φ)| (11.104)
TheresultingdiagramsareshowninFig.[11.3].
- TheRigidRotator Asafirstapplicationofthesphericalharmonics,letus
considerarigidrodwithmomentofinertiaI,whichpivotsfreelyaroundonefixed
endatr=0.Sincethelengthoftherodisfixed,thegeneralizedcoordinatesarethe
angularpositionoftherod(θ,φ).TheclassicalHamiltoniandescribingthedynamics
ofthissimplesystemis
H=
1
2 I
L^2 (11.105)
where%Listheangularmomentum.ThecorrespondingSchrodingerequationisthen
simply
H ̃φlm(θ,φ)=^1
2 I
L ̃^2 φlm(θ,φ)=Elmφlm(θφ) (11.106)