158 CHAPTER10. SYMMETRYANDDEGENERACY
thereforetheHamiltoniansofthefreeparticle,theHarmonicoscillator,andthefinite
squarewellareinvariantundertheParitytransformation.DefinetheParityoperator
asthelinearoperatorwhichhastheproperty
PF(x)=F(−x) (10.30)Onceagain,weseethat
PH ̃[∂x,x]ψ(x) = H ̃[−∂x,−x]ψ(−x)
= H ̃[∂x,x]ψ(−x)
= H ̃[∂x,x]Pψ(x) (10.31)sotheParityoperatorcommuteswiththeHamiltonian
[P,H ̃]= 0 (10.32)andisconserved
d
dt
= 0 (10.33)
ItiseasytoshowthatParityisanHermitianoperator(exercise). Supposeφβisan
eigenstateoftheParityoperatorwitheigenvalueβ
Pφβ(x)=βφβ (10.34)Then
PPφβ(x)=βPφβ(x)=β^2 φβ(x) (10.35)
Ontheotherhand,
PPφβ(x)=Pφβ(−x)=φβ(x) (10.36)
Comparingthelasttwoequations,weseethat
β^2 = 1 =⇒β=± 1 (10.37)Eigenstateswithβ=+1areknownas”even-parity”eigenstates,denotedφ+(x),and
havetheproperty
φ+(−x)=+φ+(x) (10.38)
whileeigenstateswithβ=− 1 areknownas”odd-parity”eigenstates,denotedφ−(x),
withtheproperty
φ−(−x)=−φ−(x) (10.39)
BytheCommutatorTheorem,theHamiltonianoperatorandtheParityoperator
haveacommonset of eigenstates. We havealready seeninLecture 9 that every
energyeigenstateofthefinitesquarewellwaseitheranevenparitystate,oranodd
paritystate.ThesameistruefortheenergyeigenstatesoftheHarmonicoscillator.
Theharmonicoscillatorgroundstateφ 0 (x)isagaussianfunctionwhichisinvariant