168 CHAPTER10. SYMMETRYANDDEGENERACY
10.5 Complete Sets of Observables
Tosummarizethelessonsofthislecture:
- Iftwo(ormore)hermitianoperatorscommutewiththeHamiltonian,butnotwith
eachother,thentheenergyeigenvaluesoftheHamiltonianmustbedegenerate.
Reason:ifHcommuteswithAandB,thenHandAshareacommonsetofeigen-
states,andHandBshareacommonsetofeigenstates. ButifAdoesn’tcommute
withB,thesetwosetsofeigenstatesmustbedifferent. However,iftheeigenvalues
of Hwerenon-degenerate, thenthereisonlyonepossiblechoiceof energy eigen-
states. Sincethereare(atleast)twodifferentsetsofenergyeigenstates,theenergy
eigenvaluesmustbedegenerate.
Example:ThefreeparticleHamiltoniancommuteswithmomentumpandparityP,
but[p,P]+=0.ThecommonsetofeigenstatesofHandpis
{
ei√
2 mEx/ ̄h, e−i
√
2 mEx/ ̄h, E∈[0,∞]} (10.111)andthecommonsetofeigenvaluesofHandPis
{
cos[
√
2 mEx/ ̄h], sin[√
2 mEx/ ̄h], E∈[0,∞]}
(10.112)Eachof these setsis aComplete Set of States, in the sensethat any energy
eigenstatecan bewritten as alinearcombination ofthe eigenstates ofeither set.
Then,bytheoremH3,anywavefunctionφ(x)canbewrittenasasumofeigenstates
ofeitherset.
- SymmetriesoftheHamiltonianimplyconservationlaws.
Reason: Symmetryundercoordinatetransformationscanbeassociatedwithopera-
torswhichproducethosetransformations;theseoperatorscommutewithanyHamil-
tonianwhichisinvariantunderthegiventransformation.Operatorswhichcommute
withtheHamiltonianareconserved,accordingtoeq. (10.20).
Examples:Symmetryofthefree-particleHamiltonianundertranslationsimpliescon-
servationofmomentum; symmetryofthe Hamiltonianunderrotations,e.g. fora
particleboundinsideacircle,impliesconservationofangularmomentum.
- As ageneralrule,symmetryoperations donotcommute. Therefore, asacon-
sequence of item1,the moresymmetricthe Hamiltonian,thegreateris the
degeneracyinitsenergyeigenvalues.