QMGreensite_merged

7.3. EIGENSTATESASSTATESOFZEROUNCERTAINTY 107

Ameasurementprocessthereforehasthepropertythat,whateverthephysicalstate
oftheobjectmaybeatthetimejustpriortothemeasurement,thesystemisleftin
oneoftheeigenstatesofO ̃atatimejustafterthemeasurement.
Wecangofurther.Giventhattheexpectationvalueoftheobservablein
theeigenstateφistheeigenvalueλ,andgiventhattheuncertainty∆O= 0 vanishes,
itfollowsthatameasurementofOcouldonlyresultinthevalueO==λ.
Anymeasurementthat wasdifferentfromthemeanvaluewould implyanon-zero
uncertainty. Butsincetheresultofameasurementistoleavethesysteminsucha
zero-uncertaintystate,itfollowsthat

II)AmeasurementoftheobservableOcanonlyresultinavaluewhichis
equaltooneoftheeigenvaluesofthecorrespondingoperatorO ̃.

Ingeneraltherearemanysolutionsφ(x),λtotheeigenvalueequation(7.53),soit
isusefultodenoteasetoflinearlyindependentsolutionsas{φα(x),λα},withdifferent
solutionsdistinguishedbythevalueofthesubscriptα. Theeigenvalueequationis
written
O ̃φα=λαφα (7.54)

withφαtheeigenstateandλα thecorresponding eigenvalue. Ifalltheeigenvalues
aredifferent,theeigenvaluesaresaidtobenon-degenerate. Iftherearenlinearly-
independenteigenstatesφα 1 ,φα 2 ,...,φαnwhoseeigenvaluesarethesame,i.e. λα 1 =
λα 2 =...=λαn,thentheeigenvalueissaidtoben-folddegenerate. Forexample,
theenergyeigenvaluesof afreeparticlearetwo-folddegenergate,becauseforeach
eigenvalueEtherearetwolinearlyindependenteigenstates,e.g.

ei

√ 2 mE/ ̄h

and e−i

√ 2 mE/ ̄h

(7.55)

whichsatisfy

H ̃φ=− ̄h

2

2 m

∂^2

∂x^2

φ=Eφ (7.56)

TheoremH1

TheeigenvaluesofanHermitianoperatorarereal.

Proof:ThisfollowsdirectlyfromthedefinitionofanHermitianoperator

<ψ|O|ψ> isreal (7.57)

Chooseψtobeaneigenstateφα

<φα|O|φα>=λa<φα|φα> (7.58)

Sincetheinnerproductofanyvectorwithitselfisarealnumber,itfollowsthatthe
lhsisrealifandonlyifλαisreal.