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Self Evaluation #2
(Chem113A W’07, Fri. 03/02/07 5:007:30pm+20mins, 100+5pts, max=100pts)
Name:
_______
By writing down my name, I confirm that I strictly
obey the academic ethic code when taking this exam.
(i)
Five questions + one extra credit question. Please budget your time. You may want to start with the parts you
are more familiar with. Formula sheets are attached at the end. Need more formula? Please raise your hand.
(ii) It is very important to show calculation or algebraic details.
(iii) Theme and coverage of SE#1 is shown below.
(iv) Academic ethics need to be strictly obeyed. No exceptions and no kidding.
Theme of SE#2.
In 1930 Dirac published “The Principles of Quantum Mechanics” and
for this work he was awarded the Nobel Prize for Physics in 1933. Some reviewers
comment that this book ".
.. reflects Dirac's very characteristic approach: abstract but
simple, always selecting the important points and arguing with unbeatable logic."
Questions in the SE#2 are mainly from material in this book.
[1] Applicaions of Simple Quantum Models (total 16 pts)
[1](a) (4 pts) Octatetraene
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[1](b) (4 pts) Benzene
What is the energy of a delocalized
π
electron in benzene in the following state? Assume the radius a=0.1nm.
[1] (c) (8 pts) Diatomic vibrational spectroscopy
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[2] From CM to QM (total 40pts)
[2](a) (3pts) Dirac quantum condition.
The Dirac quantum condition i
ħ
{
f
,
g
}
Æ
[
f
,
g
] is a mapping between
the classical Poisson bracket of dynamical variables
f
,
g
and the commutator of the corresponding quantum
operators
f, g
. First, please calculate commutator [
x
,
p
] based on the Dirac quantum condition
. (Hint: classical
Poisson bracket is defined as
)
The choice of the representation for quantum operators
x
,
p
must
satisfy the commutator [
x
,
p
] that you just calculated in [2](a). Someone chose the representation as
p
Æ
p
and
x
Æ
i
ħ
∂
/
∂
p
. Is this a valid choice? Please prove your answer by algebra. (Hint: calculate the
commutator [
x
,
p
] by acting [
x
,
p
] on an arbitrary function
f(p)
and then compare this result with your answer
in [2](a))
Dirac was the first one to recognize the connection
between the commutator and the Uncertainty Principle: for QM operators
A
and
B
, if [
A
,
B
] = i
C
, then
∆
A
∆
B
>= ½ <
C
> , where (
∆
A
)
2
= <
A
2
>  <
A
>
2
(definition). Based on your results in [2](a), please calculate
∆
x
∆
p
.
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This note was uploaded on 10/12/2009 for the course CHEM 113A taught by Professor Lin during the Winter '07 term at UCLA.
 Winter '07
 Lin
 Quantum Chemistry

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