1
Practice Self Evaluation #3
(Chem113A W’07, Thu. 03/22/07 11:30a2:30p, 100+5pts, max=100pts)
Name:
____
[1] Rotational and vibrational spectroscopy
[1](a) Diatomic vibrational spectroscopy.
[1](b) Diatomic rotational spectroscopy.
[1](c) Rovibrational coupling.
[1](d) Relative population between rotational states by Boltzmann distribution
The relative population between two states i and j are:
Population,i/Population,j = (g
i
/g
j
) exp[(E
i
E
j
)/(k
B
T
)]
where g
i
and g
j
are the degeneracy of these two states and E
i
and E
j
are the energies of these two states.
Please solve the following question.
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[2] Orbital angular momentum: from classical mechanics to quantum mechanics
[2](a) Orbital angular momentum in CM.
QM originates from CM. So let’s see what CM says about orbital
angular momentum. In classical mechanics (CM), angular momentum
L
(a vector) is defined as
L
=
r
×
p
where
vector
L
=
L
x
i
+
L
y
j
+
L
z
k
, vector
r
=
x
i
+
y
j
+
z
k
, and vector
p
=
p
x
i
+
p
y
j
+
p
z
k
. Please show by algebra
that the
x, y, z
components of
L
are:
L
x
=
yp
z

zp
y
;
L
y
=
zp
x

xp
z
;
L
z
=
xp
y
–
yp
x
.
[2](b) Orbital angular momentum in QM.
After promoting all the dynamical variables into operators, one
reaches the corresponding expression in QM. But there is a catch: all QM operators correspond to
experimental observables must be Hermitian. Please
show by adjoint operator algebra
that
L
z
=
xp
y
–
yp
x
(bold means QM operator) is indeed a Hermitian operator.
[2](c) Real eigenvalues theorem.
One of the reasons that all QM operators correspond to experimental
observables must be Hermitian is that the eigenvalues of a Hermitian operator are all real. That is: for
eigenvalue equation
O

Ψ
> =
λ

Ψ
>, if
O
+
=
O
then
λ
is real. Please prove this theorem by algebra (Hint: dual
space correspondence)
[2](d) Commutator relations.
Given the formula in [2](a) for
L
x
,
L
y
and
L
z
and the commutator [
x
,
p
x
] = i (in
reduced units), please show
by commutator algebra
that [
L
x
,
L
y
] = i
L
z
(in reduced units).
[2](e) Common eigenkets.
Similarly, one can show that, in reduced units, [
L
y
,
L
z
] = i
L
x
and [
L
z
,
L
x
] = i
L
y
(no
need to prove). Based on these commutators, please calculate the commutator [
L
2
,
L
z
]
by commutator
algebra.
(Remark: this result suggests that QM operators
L
2
and
L
z
can share a set of common eigenkets)
[2](f) Expectation values.
Denote the common eigenkets of
L
2
and
L
z
by 
l,m
> where
l
and
m
are two quantum
numbers used to label the eigenkets (in reduced units):
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 Winter '07
 Lin
 Atom, Quantum Chemistry, Angular Momentum, orbital angular momentum

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