10.4. THEQUANTUMCORRAL 165
Inthreedimensions,angularmomentumaroundthez-axisisgivenby
Lz=xpy−ypx (10.86)
Intwodimensions,thisistheonlypossiblecomponentofangularmomentum. The
correspondingoperatorisLz=xp ̃y−yp ̃x,andweseethattherotationoperatoris
justtheexponentiatedangularmomentumoperator
Rδθ=eiδθLz/ ̄h (10.87)
andLzcommuteswiththeHamiltonian
[Lz,H ̃]= 0 (10.88)
Itfollowsthat
d
dt
<Lz>= 0 (10.89)
Inotherwords,symmetryofthepotentialwithrespecttorotationsaroundsomeaxis
impliestheConservationofAngularMomentumaroundthataxis.
Bythe CommutatorTheorem,the operators Lz andH ̃ have acommonsetof
eigenstates. Since
Lz=−i ̄h
∂
∂θ
(10.90)
theHamiltonianoperatorcontainstheangularmomentumoperator
H ̃=− ̄h
2
2 m
(
∂^2
∂r^2
+
1
r
∂
∂r
−
1
̄h^2 r^2
L^2 z
)
+V(r) (10.91)
sowebeginbysolvingfortheeigenstatesofangularmomentum
Lzφ(r,θ) = αφ(r,θ)
−ih ̄
∂
∂θ
φ(r,θ) = αφ(r,θ) (10.92)
whichissimilartothemomentumeigenvalueequation,andhasasimilarsolution
φ(r,θ)=φ(r)eiαθ/ ̄h (10.93)
Thereisonedifference,however,betweentheeigenstatesofmomentumandan-
gularmomentum. Whereasthepositionx+aisdifferentfrompositionx,forany
non-zeroconstanta,theangleθ+ 2 πisidenticaltoθ. Thisimposesanadditional
constraint
φ(r,θ+ 2 π)=φ(r,θ) (10.94)
onthewavefunctionofanyphysicalstate.Inparticular,fortheangularmomentum
eigenstates(10.93),theconstraintcanonlybesatisfiediftheeigenvaluesαofangular
momentumtakeononlydiscretevalues
α=nh, ̄ n= 0 ,± 1 ,± 2 ,± 3 ,.... (10.95)