QMGreensite_merged

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