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ECE 4060: Introduction to Quantum and Statistical Mechanics
Quiz 1 Solution
Prob. 1.
Given a wave function of a free particle at
t = 0
as:
(
29
(
29
≤
≤

=
otherwise
x
x
ik
A
x
0
1
1
exp
0
,
0
ψ
(a)
Find
A
.
(5 pts)
By normalization requirement of
∫
=
1
*
dx
ψ
ψ
,
2
1
=
A
.
(b)
Graphically illustrate
(
29
(
29
2
0
,
0
,
x
x
P
=
and Re
(
(x,0)),
assuming 0 <
k
0
< 1.
(10 pts)
(c)
Assuming a dispersion relation of
m
k
2
2
G
=
ϖ
, graphically illustrate
P(x, t)
and Re
(
(x,t))
(10 pts)
(*** Due to a critical error in Homework 2, this problem will not count. ***)
Prob. 2.
Given a wave function of a free particle at
t = 0
as:
(
29
(
29
(
29
2
1
2
1
exp
2
1
exp
2
3
0
,
k
k
x
ik
x
ik
x
≠
+
=
1
P(x,0)
Re
(
(x,0))
x
x
P(x,t)
Re
(
(x,t))
x
x
1/2
1

1
1
1
2
/
1
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View Full Document(a)
What is the expectation value of momentum
p
ˆ
at
t = 0
?
(10 pts)
2
1
4
1
4
3

ˆ

k
k
p
+
=
ψ
(b)
If we measure the momentum at
t = 0
, what are the possible outcome?
What are the
probabilities of obtaining those outcome?
Briefly explain.
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