1
ECE 4060 Fall 2009 Prelim Exam 1 Solution
Rules of the Exam (Please read carefully before start)
•
This is an open-book, open-note exam.
You are allowed to use your computer as a browser for
downloaded course files, but you are NOT allowed to connect to Internet in any form.
Connection to Internet during the exam period is a violation of academic integrity.
•
Grading will ONLY consider what you legibly put down on the exam paper.
References to
textbook or class notes will NOT count for credit.
Irrelevant answers, even though the content
is correct, will NOT receive any partial credit.
Wrong information will always cause a
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 time-dependent 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
ψ
ψ
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 time-independent Schrödinger equation as
ψ
(x)
and
A(k)
from
the previous part, write down the relation between
ψ
(x)
and
A(k)
.
(3 pts)
∫
∞
∞
−
=
π
ψ
2
)
(
)
(
dk
e
k
A
x
ikx
;
∫
∞
∞
−
−
=
dx
e
x
k
A
ikx
)
(
)
(
ψ
(c)
Are the eigenvalues the same for the time-independent 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.
∫
∫
∞
∞
−
∞
∞
−
=
=
=
π
ψ
π
ψ
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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