102 CHAPTER7. OPERATORSANDOBSERVATIONS
E = <ψ|H|ψ>H ̃ψ(x) =[
−̄h^2
2 m∂^2
∂x^2+V(x)]
ψ(x)H(x,y) =[
−̄h^2
2 m∂^2
∂x^2+V(x)]
δ(x−y) (7.26)Notethat
H ̃=H[p ̃,x]= p ̃2
2 m+V(x) (7.27)sothat,asnotedinLecture5,theoperatorcorrespondingtotheHamiltonianfunction
isobtainedbyreplacingthemomentumpbythemomentumoperatorp ̃.
ThereasonthatanobservablemustcorrespondtoanHermitianoperatoristhat
expectationvaluesarerealnumbers. Thereality condition(7.22)requires, foran
Hermitianoperator,
<ψ|O|ψ> = <ψ|O|ψ>∗
∫
dxdyψ∗(x)O(x,y)ψ(y) =∫
dxdyψ(x)O∗(x,y)ψ∗(y)=∫
dxdyψ∗(y)O∗(x,y)ψ(x)=∫
dxdyψ∗(x)O∗(y,x)ψ(y)=∫
dxdyψ∗(x)O†(x,y)ψ(y)= <ψ|O†|ψ> (7.28)wherewerenamedvariablesx→yandy→xinthesecondtolastline.Theoperator
O ̃†isknownastheHermitianConjugateoftheoperatorO ̃.Itispronounced“O-
dagger,”andhasmatrixelements
O†(x,y)=O∗(y,x) (7.29)AnoperatorisHermitianiff
O ̃ψ(x)=O ̃†ψ(x) (7.30)or,inmatrixrepresentation
∫
dyO(x,y)ψ(y)=∫
dyO∗(y,x)ψ(y) (7.31)foranyphysicalstate|ψ>.