Due December 2
1. This problem concnerns the momentum representation.
a. Show that
Hint:Look at the ovelap between two momentum operators, insert the identity operator in an
appropriate place and use the following facts:
b. Show that the explict x operator im momentum on the momentum space wave functionφ(p)
is given by
c. Show that momentum space Schrodinger equation is given by
2. The purpose of this problem is to show that the momentum operator is the generator of
translations. We will do this in two ways: acting on a wavefunction and on an abstract eigentste
of x.
a. Show that
Hint: Differentiate both sides with respect to a. demonstrate the validity and then integrate.
b. Show that
p
pd
d
ipx −=
ˆ
)(')(
ˆpipx φφ =
xxdx
∫
=1
ˆ
π2
xpi
e
px =
( )
t
tp
itpiV
m
p
x∂
∂
=
∂+ ),(
),(
2
2φ
φ
)()(
/
ˆaxxe api +=ψψ
axxe api −=
/
ˆ