11.4. EIGENFUNCTIONSOFANGULARMOMENTUM 181
whilevaluesofLxandLyareindefinite;althoughtheirsquaredexpectationvalues
mustsatisy
<L^2 x>+<L^2 y>+<L^2 z>=<L^2 > (11.58)
Wecanpicturetheangularmomentumassociatedwithφlmastheconeofallvectors
%Lsatisfying(11.57).Theconesforl=2,m=− 2 ,− 1 , 0 , 1 ,2,areshowninFig.[11.2].
Inclassicalmechanicsitiseasytoaddtwoormoreangularmomenta;itisjusta
matterofvectoraddition. Inquantummechanics,asonemightimagine,theprocess
ismorecomplicated. Forexample,supposeonehastwoelectronswhichareeachin
angularmomentumeigenstates,andweask: ”whatisthetotalangularmomentum
ofthesystem?”(Howdoesonegoaboutadding”cones”ofangularmomentum?)We
willlearnthe quantum-mechanicalrulesforaddition ofangularmomentuminthe
secondsemesterofthiscourse.
Problem:Showthatinaφlmeigenstate,that
<Lx>=<Ly>= 0 (11.59)andthat
<L^2 x>=<L^2 y>=1
2
̄h^2[
l(l+1)−m^2]
(11.60)Fromthisinformation,verifythegeneralizeduncertaintyprinciple(eq. (7.116))for
∆Lx∆Ly.
11.4 Eigenfunctions of Angular Momentum
Inthecaseof theharmonicoscillator, wefoundthegroundstatewavefunctionby
solvingthefirst-orderdifferentialequation
aφ 0 (x)= 0 (11.61)andthenallothereigenstatescanbeobtainedbyapplyingtheraisingoperatora†.
Inthecaseofangularmomentum,thestrategyisverysimilar.Foragivenl,wefirst
solvethefirst-orderdifferentialequations
L+φll(x,y,z) = 0
Lzφll(x,y,z) = l ̄hφll (11.62)andthenobtainallotherφlmwavefunctionsinthemultipletbyapplyingsuccessively
theloweringoperatorL−.
Itis mucheasier to solve these differential equations in sphericalcoordinates
r, θ, φ
z = rcosθ
x = rsinθcosφ
y = rsinθsinφ (11.63)