ECE 306
Homework Set #2
Spring 2005
(Due 12 noon, Wednesday, February 9, 2005.)
1. Suppose an electron is confined to a zero-potential region between two impenetrable
walls at x = 0 and
a
for all times. Its initial wave function is given by:
a
x
x
a
x
a
x
a
x
≥
≤
<
<
=
,
0
0
,
,
0
)
3
sin(
2
{
)
(
π
ψ
a.
Calculate the expectation value of the energy <
^
H
> in this state
b.
Calculate the energy eigen value corresponding to this state.
c.
What is the time dependence of
Ψ
(x,t)?
d.
Calculate the uncertainty in the total energy in this state,
∆
^
H
. Is the answer
what you expect? Explain.
2. Prove the following commutation relationships:
a.
[A + B,C] = [A,C] + [B,C]
b.
[A, BC] = [A,B]C + B[A,C]
3. Prove the following commutation relations:
a.
[
p
x
, x
n
] =
-i
Ñ
nx
n-1
b.
[
x, p
x
2
] =
2i
Ñ
p
x
c.
From b above, what can you say about the possibility of measuring the
position and kinetic energy of a particle in an arbitrary state simultaneously
with zero uncertainty in each measurement?

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