QMGreensite_merged

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)