174 CHAPTER11. ANGULARMOMENTUM
eigenstatesofangularmomentum,byanalgebraicmethodswhichcloselyresembles
thetreatmentoftheharmonicoscillator.
ThefactthattheangularmomentumoperatorscommutewiththeHamiltonian,
butnotwitheachother,meansthattheenergyeigenvaluesoftheHamiltonianare
degenerate.Italsomeansthatallthreecomponentsofangularmomentumcannotbe
measuredsimultaneously,whichtellsusthattheredoesnotexistanyphysicalstate
inwhichthedirectionofangularmomentumisdefinite. However,despitethefact
thatthedirectionofangularmomentumisindefinite,theredoexiststatesinwhich
themagnitudeofangularmomentumisdefinite;theseareeigenstatesofthesquared
angularmomentumoperator
L ̃^2 =L ̃^2 x+L ̃^2 y+L ̃^2 z (11.15)EachofthecomponentsofangularmomentumcommuteswithL ̃^2 .Forexample,
[L ̃z,L ̃^2 ] = [L ̃z,L ̃x^2 +L ̃^2 y+L ̃^2 z]
= [L ̃z,L ̃^2 x]+[L ̃z,L ̃^2 y] (11.16)Weevaluatethefirstcommutatorwiththehelpoftherelationsineq. (11.14)
[L ̃z,L ̃^2 x] = L ̃zL ̃xL ̃x−L ̃xL ̃xL ̃z
= (L ̃xL ̃z+[L ̃z,L ̃x])L ̃x−L ̃^2 xL ̃z
= L ̃xL ̃zL ̃x+i ̄hL ̃yL ̃x−L ̃^2 xL ̃z
= L ̃x(L ̃xL ̃z+[L ̃z,L ̃x])+i ̄hL ̃yL ̃x−L ̃^2 xL ̃z
= i ̄h(L ̃xL ̃y+L ̃yL ̃x) (11.17)Thesecondcommutatorgivesasimilarresult:
[L ̃z,L ̃^2 y] = L ̃zL ̃yL ̃y−L ̃yL ̃yL ̃z
= (L ̃yL ̃z+[L ̃z,L ̃y])L ̃y−L ̃^2 yL ̃z
= L ̃yL ̃zL ̃y−i ̄hL ̃xL ̃y−L ̃^2 xL ̃z
= L ̃y(L ̃yL ̃z+[L ̃z,L ̃y])−i ̄hL ̃xL ̃y−L ̃^2 yL ̃z
= −i ̄h(L ̃yL ̃x+L ̃xL ̃y)
= −[L ̃z,L ̃^2 x] (11.18)Addingthetwocommutatorstogethergiveszero,andthereforeL ̃zcommuteswith
L ̃^2 .Thesameresultisobtainedfortheothercomponentsofangularmomentum:
[L ̃x,L ̃^2 ]=[L ̃y,L ̃^2 ]=[L ̃z,L ̃^2 ]= 0 (11.19)