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