7.2. OPERATORSANDOBSERVABLES 101
AnimportantsubsetoflinearoperatorsaretheHermitianoperators.AnHermitian
operatorisalinearoperatorwiththepropertythat
<ψ|O|ψ> =
∫
dxdyψ∗(x)O(x,y)ψ(y)
= arealnumber,foranyψ(x)whatever (7.22)
TheRelationBetweenOperatorsandObservables:
Inquantummechanics,toeveryobservableOthereexistsacorrespond-
inghermitianoperatorO ̃.Knowledgeofthephysicalstateψ(x,t)atsome
time timpliesaknowledgeoftheexpectationvalueofeveryobservable,
accordingtotherule
<O> = <ψ|O|ψ>
=
∫
dx
∫
dyψ∗(x)O(x,y)ψ(y)
=
∫
dxψ∗(x)O ̃ψ(x) (7.23)
Thisisaprincipleofquantummechanicswhichsimplygeneralizestheexamples
wehavealreadyseen.
O=position
<x> = <ψ|x|ψ>
x ̃ψ(x) = xψ(x)
X(x,y) = xδ(x−y) (7.24)
O=momentum
<p> = <ψ|p|ψ>
p ̃ψ(x) = −i ̄h
∂
∂x
ψ(x)
P(x,y) = −i ̄h
∂
∂x
δ(x−y) (7.25)
O=energy