10.5. COMPLETESETSOFOBSERVABLES 169
Example:Itwillbeseeninthenextlecturethattheenergydegeneracyofaspherically
symmetricHamiltonianinthreedimensionsisgreaterthantheenergydegeneracyof
arotationallyinvariantHamiltonianintwodimensions.
Items 1 and 3 actuallyholdtruenotjustfortheHamiltonian,butforalloperators.
IfAcommuteswithBandC,butBandCdon’tcommutewitheachother,thenthe
eigenvaluesofAaredegenerate. Themoresymmetriesanoperatorhas,thegreater
isthedegeneracyinitseigenvalues.
Supposethe operatorsA,B,C commute, and that theset of eigenstates of A
whicharealsoeigenstatesofBandCisunique. Thentheset{A,B,C}isknownas
aCompleteSetofObservables. Measurementofacompletesetofobservablesis
sufficienttouniquelydeterminethequantumstate(whichmustbeaneigenstateof
alloperatorsintheset).Examples:
a. Momentumpisacompletesetof observables. Infact,anyoperatorwithnon-
degenerateeigenvaluesisacompleteset.
b. EnergyEisacompleteset,fortheHarmonicOscillatorandSquareWellpoten-
tials.
c. EnergyEandparityPisacompleteset,forthefreeparticle.
d. EnergyEandreflectionRxisacompletesetfortheparticleinasquare. SoisE
andtheobservablecorrespondingtox-yinterchangeI.
e. EnergyEandangularmomentumLzisacompletesetforthe”quantumcorral”.