108 CHAPTER7. OPERATORSANDOBSERVATIONS
TheoremH2
If two eigenstates of an Hermitian operator correspond to different
eigenvalues,thenthetwoeigenstatesareorthogonal.
Proof:Supposeφα′ andφα′′areeigenstatesofanHermitianoperatorO ̃,andλα′+=
λα′′.Fromeq. (7.45),wehave
<φα′|O|φα′′> = <Oφα′|φα′′>
<φα′|λα′′|φα′′> = <λα′φα′|φα′′>
λa′′<φα′|φα′′> = λ∗α′<φα′|φα′′>
λa′′<φα′|φα′′> = λa′<φα′|φα′′> (7.59)Giventhattheeigenvaluesaredifferent,theonlywaythisequationcanholdtrueis
that
<φα′|φα′′>= 0 (7.60)
TheoremH3
Foranysquare-integrablefunctionψ(x)andanyHermitianoperatorO ̃,
thefunctionψcanalwaysbeexpressedassomelinearcombinationofthe
eigenstatesofO ̃
ψ(x)=
∑
αcαφα(x) (7.61)wherethesumisreplacedbyanintegraliftheeigenvaluesspanacontin-
uousrange.
(Wewillnottrytoprovethishere.)
Withthehelpofthesetheorems,wearereadytosaysomethingaboutphysics:TheGeneralizedBornInterpretation(I)
SupposethatanobservableOcorrespondstoaHermitianoperatorO ̃whoseeigen-
valuesarediscreteandnon-degenerate,andwhoseeigenstatesarenormalizedto
<φα′|φα′′>=δα′α′′ (7.62)Denotethequantumstateofthesystematthetimeofmeasurementby|ψ>. Then:
(I)Theoutcomeofanymeasurementoftheobservablewillbeoneofthe
eigenvalues{λα}.