PHY4604 Fall 2006
Final Exam
Department of Physics
Page 1 of 6
PHY 4604 Final Exam
Take Home Exam (Total Points = 100)
You may use books, notes, internet, and computers. However, you are not allowed to get help
from other people.
You must do the work yourself!
You can spend as much time on it as you
want, but you must return the exam on Wednesday December 6 (
in class
).
If I do not receive
your exam by 10:30am on Wednesday December 6, you will receive a zero on the exam.
Problem 1 (25 points):
Consider the following onedimensional potential:
⎪
⎩
⎪
⎨
⎧
∞
+
−
=
0
0
)
(
V
x
V
L
x
L
x
L
x
≥
<
≤
<
0
where
V
0
is a positive (real) constant.
(A)
Suppose that particles with mass
m
and energy
E > 0
enter from the left in
region 1 and travel to the right and encounter the potential
V(x)
.
Let
2
2
2
2
ε
mL
E
h
=
and
E
V
r
0
1
+
=
.
(1) (4 points)
Calculate the probability that particles entering from the left in region 1 will be
reflected back (
i.e.
calculate the ratio of the probability flux in region 1 traveling to the left to
the incident flux in region 1 traveling to the right,
1
1
/
j
j
P
R
r
s
=
).
Express your answer in terms of
ε
and r.
(2) (4 points)
Calculate the probability that particles traveling to the right in region 2 will be
reflected back (
i.e.
calculate the ratio of the probability flux in region 2 traveling to the left to
flux in region 2 traveling to the right,
2
2
/
j
j
R
r
s
=
).
Express your answer in terms of
ε
and r.
(3) (6 points)
Calculate the ratio of the probability flux in region 2 traveling to the right to the
incident flux in region 1 traveling to the right,
1
2
/
j
j
P
r
r
=
).
Express your answer in terms of
ε
and r. What are the maximum and minimum values of P(
ε
) (express your answer in terms of r)?
Evaluate the maximum and minimum values of P(
ε
) for the case
V
0
= 3E
. Explain what is going
on.
(B)
Now consider the case
E < 0
and look for possible bound states.
Let
2
2
2
2
ε
mL
E
h
−
=
and
2
2
2
0
2
α
mL
V
h
=
.
(1) (5 points)
How many bound states are there if
α
=
π
/4
?
What is the ground state energy (
i.e.
what is
ε
for the ground state)?
(2) (6 points)
How many bound states are there if
α
= 11
π
/4
?
What is the ground state energy
(
i.e.
what is
ε
for the ground state)?
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 Spring '07
 FIELDS
 mechanics, Work, ground state, op, Department of Physics

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