QMGreensite_merged

7.2. OPERATORSANDOBSERVABLES 103

Thishermiticityrelationisquiterestrictive; forexample,itisnotevensatisfied
bythesimpleoperationofdifferentiation:

O ̃ψ(x)= ∂

∂x

ψ(x) (7.32)

Inthatcase

O(x,y)=

∂

∂x

δ(x−y) (7.33)

sothat

O†(x,y)=

∂

∂y

δ(y−x) (7.34)

andtherefore

O ̃†ψ(x) =

∫

dyO†(x,y)ψ(y)

=

∫

dy

[

∂

∂y

δ(y−x)

]

ψ(y)

=

∫

dyδ(x−y)

[

−

∂ψ

∂y

]

= −

∂

∂x

ψ(x)

= −O ̃ψ(x) (7.35)

SinceO ̃isnothermitianinthiscase,itispossiblefor<ψ|O|ψ>tobeimaginary.
Infact,forthegausianwavepacket(6.1)

<φ|Oφ> =

1

√

πa^2

∫

dxexp[−

x^2

2 a^2

−i

p 0 x

̄h

]

∂

∂x

exp[−

x^2

2 a^2

+i

p 0 x

̄h

]

= i

p 0

̄h

(7.36)

So thederivativeoperatorcannot correspond toa physical observable, because it
wouldleadtoimaginaryexpectationvalues.
Theoperatorsofthethreeexamplesabove,correspondingtotheobservablesof
position,momentum,andenergy,hadthereforebetterbeHermitian.Letscheckthis.

HermiticityofthePositionOperator

Fromthematrixrepresentation

X(x,y)=xδ(x−y) (7.37)

wehave,fromthepropertiesofdeltafunctions,

X†(x,y) = X∗(y,x)

= yδ(y−x)

= xδ(x−y)

= X(x,y) (7.38)