QMGreensite_merged

21.3. HILBERTSPACE 327

where

x ̃=i ̄h

∂

∂p

(21.148)

Ingeneral,onecanshowthatinthep-representation

<p 1 |Xn|φ 2 >=

(

i ̄h

∂

∂p 1

)n

δ(p 1 −p 2 ) (21.149)

ThematrixelementofaHamiltonianwithpotentialV(x)thereforetakestheform

<p|H|p′>=H ̃δ(p−p′) (21.150)

where

H ̃= p

2

2 m

+V

[

i ̄h

∂

∂p

]

(21.151)

TheSchrodingerequationinthemomentumrepresentationisobtainedbystarting
from
i ̄h∂t|ψ>=H|ψ> (21.152)

andtakingtheinnerproductwiththebra<p|

i ̄h∂t<p|ψ> = <p|H|ψ>

i ̄h∂t<p|ψ> = <p|H

{∫

dp′|p′><p′|

}

|ψ>

=

∫

dp′<p|H|p′>ψ(p′)

= H ̃

∫

dp′δ(p−p′)ψ(p′) (21.153)

sofinally

i ̄h∂tψ(p,t)=

{

p^2

2 m

+V

[

i ̄h

∂

∂p

]}

ψ(p,t) (21.154)

istheSchrodingerequationinthisrepresentation.Unlessthepotentialiseitherzero
(thefreeparticle)orquadratic(theharmonicoscillator),thisformoftheSchrodinger
equationisusuallyhigherthan2ndorderinderivatives,andthereforehardertosolve
thanthecorrespondingequationintheX-representation.

Exercise: Verifyeq. (21.149).

Exercise: Usingeq. (21.146),verifythat

<p|[X,P]|p′>=i ̄hδ(p−p′) (21.155)

• TheHarmonicOscillatorRepresentation