ECE 3060 Fall 2008 Prelim Exam 1 Solution
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1.
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3.
The time for the exam will be exactly 90 minutes.
Do not be trapped in a question you
cannot answer, and 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.
Determine if the following operators are Hermitian.
You need to clearly state your reasons.
(a)
x
i
p
∂
∂

=
(3 pts)
Hermitian, it is observable.
You can use definition as well.
(b)
x
i
p
k
∂
∂

=
=
(3 pts)
A real constant times a Hermitian operator is always Hermitian.
(c)
k
∂
∂
(4 pts)
Since we know
k
i
x
∂
∂
=
ˆ
and is Hermitian, hence
k
∂
∂
cannot be Hermitian.
(d)
log
(x/x
0
)
(4 pts)
Any real functions of
x
will be Hermitian.
However, there is a singularity at 0 for this
function.
2.
What are the eigenfunctions and the corresponding eigenvalues for the following operators?
(a)
The momentum operator
x
i
p
∂
∂

=
in
x
space.
(4 pts)
The eigenfunction is
e
ikx
, and the corresponding eigenvalue is
k
i
.
This can be directly
solved from the eigenvalue equation.
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 '09
 KAN
 pts, wave function, ground state, Hermitian

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