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)