QMGreensite_merged

21.3. HILBERTSPACE 325

asamatrixequationintheX-representation:

<x|[X,P]|y> = <x|XP|y>−<x|PX|y>

= <x|XIP|y>−<x|PIX|y>

= <x|X

{∫

dx|z><z|

}

P|y>−<x|P

{∫

dx|z><z|

}

X|y>

=

∫

dz [<x|X|z><z|P|y>−<x|P|z><z|X|y>]

=

∫

dz

[

xδ(x−z)

(

−i ̄h

∂

∂z

δ(z−y)

)

−

(

−i ̄h

∂

∂x

δ(x−z)

)

zδ(z−y)

]

= −i ̄h

[

x

∂

∂x

δ(x−y)−y

∂

∂x

δ(x−y)

]

= −i ̄hδ(x−y)

[

−

∂

∂x

x+y

∂

∂x

]

= i ̄hδ(x−y) (21.137)

(In thenext tolast line, the derivative isunderstoodto beacting onsome func-
tionofxstandingtototheright). Thisexampleisanillustrationofthefactthat
thematrixrepresentationofaproductofoperatorsisequaltotheproductofmatrix
representationsofeachoperator,i.e.

<φm|AB|φn> =

∑

k

<φm|A|φk><φk|B|φn>

[AB]mn =

∑

k

AmkBkn (21.138)

• TheP-representation

TakingtheinnerproductoftheeigenvalueequationP|p 0 >=p 0 |p 0 >withthe
bra<p|,wehavethemomentumeignfunctionsinthemomentum-representation

ψp 0 (p)=<p|p 0 >=δ(p−p 0 ) (21.139)

Theeigenstatesofpositionaregivenby

ψx 0 (p) = <p|x 0 >

= <x 0 |p>∗

= ψp(x)∗

=

1

√

2 π ̄h

e−ip^0 x/ ̄h (21.140)