10.2. PARITY 159
underx→−x;itisthereforeaneven-parityeigenstate.Theraisingoperator,onthe
otherhandtransformsas
a†(−x,−∂x)=−a†(x,∂x) (10.40)Usingthegeneralformfortheharmonicoscillatoreigenstates
φn(x)=1
√
n!(a†)nφ 0 (x) (10.41)itisclearthat
Pφn(x)=φn(−x)=(−1)nφn(x) (10.42)
andtheenergyeigenstatesareeven-parityforneven,andodd-parityfornodd.
Finally,thefreeparticleHamiltonianisinvariantunderParity,sotheremustbe
acompletesetofenergyeigenstateswhicharealsoeigenstatesoftheparityoperator.
Notethattheeigenstatesofmomentumarenoteigenstatesofparity
Peipx/ ̄h=e−ipx/ ̄h+=±eipx/ ̄h (10.43)Ontheotherhand,anenergyeigenstatedoesnothavetobeamomentumeigenstate.
Aswehaveseen,aneigenstateofenergyEhasthegeneralform
φE(x)=aeipx/ ̄h+be−ipx/ ̄h p=√
2 mE (10.44)IfφE(x)isaneven-parityeigenstate,itrequiresthata=b,or
φE+(x)=Acos[px
̄h] (10.45)
whileifφE(x)isanodd-parityeigenstate,itmeansthata=−b,or
φE−(x)=Bsin[px
̄h] (10.46)
Soacompletesetofenergyeigenstatesisgivenbytheset
{φE+(x), φE−(x), E∈[0,∞]} (10.47)Thisiscompletebecauseanyenergyeigenstateoftheform(10.44)canbeexpressed
asalinearcombinationofφE+andφE−.NotethatspecifyingtheenergyE,andthe
parity±1,determinestheenergyeigenstateuniquely.
Itmayseemoddthatan eigenstateof momentumisnotan eigenstateof par-
ity(andviceversa). Afterall,bothareassociatedwithcoordinatetransformations
whicharesymmetriesofthefreeparticleHamiltonian. However,theorderofthese
transformationsisimportant.Supposewehaveaparticleatpointx.Ifwefirstmake
atranslationx′=x+a,andthenatranformationx′′=−x′,theparticlewindsup
atcoordinatesx′′=−x−a.Ontheotherhand,ifwemakethesetransformationsin