QMGreensite_merged

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)