1
Self Evaluation #3
(Chem113A W’07, Thu. 03/22/07 11:30am-2:30pm+15mins, 100+10pts, max=100pts)
Name:
_______
By writing down my name, I confirm that I strictly
obey the academic ethic code when taking this exam.
(i)
Six 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) Academic ethics need to be strictly obeyed. No exceptions and no kidding.
Theme of SE#3.
We blend in additional advanced, up-to-date materials into Chem113A W’07 to pursue a solid
understanding to some abstract, deep, and striking concepts and to make these concepts useful to our future
careers. Richard Feynman: “I think I can safely say that nobody understands quantum mechanics”. After our 10
weeks of intensive, diligent stud together, let’s see how much we understand quantum chemistry now.
[1] Objective #1 “Basic Material”: Molecular Spectroscopy (total 14 pts)
[1](a) Eigenfunctions and eigenvalues of the Hamiltonian operator of a particle on a sphere (4 pts)
Denote the angular momentum operator by
L
. The eigenfunctions and eigenvalues for the operators
L
2
are:
L
2
Y
lm
(
θ,φ
) =
l
(
l
+1)
ħ
2
Y
lm
(
θ,φ
), where
l
=0,1,2,… and
m
=0,±1,±2,..., ,±
l
. In classical mechanics, the energy for a
rotating particle (e.g., an imagined relative-motion particle representing the relative motion of a rotating diatomic
molecule) is
E
=
L
2
/(2
I
), where moment of inertia
I
= μR
2
. Please derive the eigenfunctions and eigenvalues for the
Hamiltonian operator of a rotating particle with moment of inertia
I
.