QMGreensite_merged

104 CHAPTER7. OPERATORSANDOBSERVATIONS

ThereforethepositionoperatorisHermitian,andisalwaysreal.

HermiticityoftheMomentumOperator

Thematrixrepresentationofthemomentumoperatoris

P(x,y) = p ̃δ(x−y)

= −i ̄h

∂

∂x

δ(x−y) (7.39)

Thenhermitianconjugateofthemomentumoperatorhasmatrixelements

P†(x,y) = P∗(y,x)

= i ̄h

∂

∂y

δ(x−y) (7.40)

andwehave

p ̃†ψ(x) =

∫

dyP†(x,y)ψ(y)

=

∫

dy

[

i ̄h

∂

∂y

δ(x−y)

]

ψ(y)

=

∫

dyδ(x−y)

[

−i ̄h

∂

∂y

]

ψ(y)

= −i ̄h

∂

∂x

ψ(x)

= p ̃ψ(x) (7.41)

ThereforethemomentumoperatorisHermitian,and

isalwaysreal.

HermiticityoftheHamiltonianOperator

Inthiscase

H ̃†ψ(x) =

∫

dy

[{

−

̄h^2

2 m

∂^2

∂y^2

+V(y)

}

δ(x−y)

]

ψ(y)

= −

̄h^2

2 m

∫

dyψ(y)

∂^2

∂y^2

δ(x−y) +

∫

dyV(y)δ(x−y)ψ(y)

=

̄h^2

2 m

∫

dy

∂ψ

∂y

∂

∂y

δ(x−y) + V(x)ψ(x)

= −

̄h^2

2 m

∫

dy

∂^2 ψ

∂y^2

δ(x−y) + V(x)ψ(x)

=

[

−

̄h^2

2 m

∂^2

∂x^2

+V(x)

]

ψ(x)

= H ̃ψ(x) (7.42)