CHEM 550/475 2005
Homework 2. Due Monday Oct 10 at 5pm in Prof. Ginger’s mailbox. Page 1/3
CIRCLE YOUR ANSWERS AND KEY RESULTS
STAPLE YOUR PAPERS TOGETHER
Levine
2.17 – time dependent wave function for a free particle
2.29 – probability concept review
3.5 – basic operator practice
3.15 – commutator identity
3.42 – operator concept review – provide counter-examples for false cases!
Additional Problems:
1) Why is simple harmonic motion so common in classical mechanics?
We solved the ODE for the simple harmonic oscillator (mass on a spring) in class last
week, we will solve the quantum version next week. In the meantime, why do you
think so many classical systems exhibit simple harmonic motion? We derived the
equation of motion based on a Hooke’s law restoring force (i.e. F=-k*x), however, we
can also see this results from any harmonic potential.
a) Since VF
⋅
−∇
=
for any conservative force, what kind of potential function leads to a
Hooke’s law like restoring force and resulting simple harmonic motion?
b) Take any arbitrary potential that we will call U(x). Write out the Taylor expansion for
U(x) about a local minimum x0.
c) Thus, from b) what kind of motion can be expected for sufficiently small
displacements from the local potential minimum? What do we mean by “sufficiently
small displacements”?
2) Do Levine problem 2.15 for a longer molecule and compare with the butadiene result
from Levine 2.15 (you don’t need to work the butadiene problem if you are familiar
with it):
a) γ-Carotene, one of the precursors of vitamin A, is a conjugated system containing 11-
double bonds (you will need to calculate the total bond length). Calculate the
wavelength of light needed to excite the HOMO-LUMO transition of the pi electrons
for carotene using the simple free electron model. Compare the calculated value with
the observed transition at 460 nm.
b) Explain why the ‘free electron model’ gives an error for butadiene?
Hint: do electrons attract or repel one another?
