Chapter 11
Angular Momentum
Whenthepotentialenergyinthreedimensionsdependsonlyontheradialcoordinate,
thepotentialissaidtobe”sphericallysymmetric”,andtheHamiltonianisinvariant
underarbitraryrotations;inparticular,theHamiltonianisinvariantunderrotations
aroundthex,y,andz-axes. Inthelastlecturewefoundthatforrotationsaround
theorigininthex−yplane,therotationoperatoris
Rz(δθ)=eiLzδθ/ ̄h (11.1)Thereisnothingspecialaboutthez-axis;rotationsaroundthexandy-axesmust
havethecorrespondingforms
Rx(δθ) = eiLxδθ/ ̄h
Ry(δθ) = eiLyδθ/h ̄ (11.2)Inthecaseofrotationsarounddifferentaxes,theorderofrotationsisimportant.
Forexample,consideraparticleatapointonthez-axis,showninFig. [11.1],and
imaginedoingfirstarotationby 90 oaroundthez-axis,andthenarotationby 900
alongthex-axis. Thefirstrotation,aroundthez-axis,leavestheparticlewhereit
is. The secondrotation,aroundthe x-axis, bringstheparticle to afinalposition
alongthey-axis. Now reversethe orderofcoordinatetransformations: Arotation
aroundthex-axisbringstheparticle toapointtothey-axis, andthe subsequent
rotationaroundthez-axisbringstheparticletoapointonthex-axis. Thusthetwo
transformations,performedindifferentorders,resultindifferentfinalpositionsfor
theparticle.
Itfollowsthat,althoughweexpecttheangularmomentumoperatorstocommute
withasymmetric Hamiltonian,we donotexpectthe Lx,Ly,Lz operatorstocom-
mutewitheachother. Theenergyeigenvalueswillthereforebedegenerate,angular
momentumisconserved,andtheangularmomentumoperatorscanbeusedtoselect
amongdifferentcompletesetsofenergyeigenstates. Inthislecture,wewilllearnhow
to constructeigenstates of angularmomentum, whichisanecessarysteptowards
solvingtheSchrodingerequationforsphericallysymmetricpotentials