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ECE 4060 Fall 2009 Prelim Exam 1 Solution
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•
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deduction in total credit.
•
The time for the exam will be exactly 90 minutes.
Do not be trapped in a question you cannot
answer.
Use your time wisely for distributing your efforts in different problems.
Do not
diverge into irrelevant answers, since this will negatively impact your performance.
1.
Given the potential as
V(x) = cx
2
,
(a)
Write down the timedependent Schrödinger equation in the
x
and
k
space.
(4 pts)
)
,
(
)
,
(
)
,
(
2
)
,
(
ˆ
2
2
2
2
t
x
t
i
t
x
cx
t
x
x
m
t
x
H
x
ψ
∂
∂
=
+
∂
∂
−
=
h
h
)
,
(
)
,
(
)
,
(
2
)
,
(
ˆ
2
2
2
2
t
k
A
t
i
t
k
A
k
c
t
k
A
m
k
t
k
A
H
k
∂
∂
=
∂
∂
−
=
h
h
Notice the symmetric expression in the two spaces, which will lead to the conclusion that the
eigenfunction must be of similar functional form for the parabolic potential.
This will
become the Gaussian function (times the Hermite polynomials for the general cases of (
∞
,
∞
) series).
(b)
Given the solution of the timeindependent Schrödinger equation as
(x)
and
A(k)
from
the previous part, write down the relation between
(x)
and
A(k)
.
(3 pts)
∫
∞
∞
−
=
π
2
)
(
)
(
dk
e
k
A
x
ikx
;
∫
∞
∞
−
−
=
dx
e
x
k
A
ikx
)
(
)
(
(c)
Are the eigenvalues the same for the timeindependent Schrödinger equation in the
x
and
k
space?
Briefly justify your answer. Notice that you cannot use the plane waves as the
eigen functions.
(3 pts)
Yes, they are the same.
This is a general property of Fourier transform on the Hamiltonian as
a Hermitian operator.
∫
∫
∞
∞
−
∞
∞
−
=
=
=
2
)
(
)
(
2
)
(
ˆ
)
(
ˆ
dk
e
k
A
E
x
E
dk
e
k
A
H
x
H
ikx
ikx
x
x
)
(
)
(
ˆ
k
EA
k
A
H
k
=
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2.
We define a new operator with
3
3
2
3
2
ˆ
x
m
i
Q
∂
∂
−
=
h
,
(a)
Write down
Q
ˆ
in the
k
space.
(2 pts)
2
3
3
2
ˆ
m
k
Q
h
=
, which is the energy flux.
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This note was uploaded on 11/26/2010 for the course ECE 3060 at Cornell University (Engineering School).
 '05
 TANG

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