QMGreensite_merged

11.4. EIGENFUNCTIONSOFANGULARMOMENTUM 183

andwesolvethefirst-orderdifferentialequations

L ̃+Yll= 0 L ̃zYll=lhY ̄ lm (11.70)

bythemethodofseparationofvariables

Ylm(θ,φ)=A(θ)B(φ) (11.71)

TheLzeigenvalueequation(foranym)becomes

−i ̄h

dB

dφ

=m ̄hB (11.72)

whichhasthesolution
B(φ)=eimφ (11.73)
Aswas pointed outinthecaseof the”quantumcorral”, discussedinthelast
lecture,theangleφ= 0 isthesameastheangleφ= 2 π,sothewavefunctionsmust
satisfytheperiodicitycondition

Ylm(θ,φ+ 2 π)=Ylm(θ,φ) (11.74)

Butthismeans,sinceB(φ)=eimφ,thatmmustberestrictedtotheintegervalues

m= 0 ,± 1 ,± 2 ,± 3 ,...,±n,... (11.75)

Asaresult,since−l≤m≤l,thepossiblevaluesforlare

l= 0 , 1 , 2 , 3 ,...,n,... (11.76)

So,althoughwehavefoundalgebraicallythat mcouldhavebothintegerandhalf-
integervalues

m= 0 ,±

1

2

,± 1 ,±

3

2

,...,±n/ 2 ,... (11.77)

itturnsoutthattheperiodicityconditionhasruledoutthehalf-integerpossibilities
fororbitalangularmomentum. Aswe willseenextsemester,thehalf-integerval-
uesarestillpossibleforintrinsicspinangularmomentum,wheretherequirementof
angularperiodicityofthewavefunctiondoesnotapply.
Sofarwehave
Ylm(θ,φ)=A(θ)eimφ (11.78)

andinparticular
Yll(θ,φ)=A(θ)eilφ (11.79)

Applyingtheraisingoperatortothisstate,wemusthave

0 = L ̃+Yll

= ̄heiφ

(

icotθ

∂

∂φ

+

∂

∂θ

)

A(θ)eilφ

= ̄hei(l+1)φ

(

−lcotθ+

∂

∂θ

)

A(θ) (11.80)