Homework 4. aka “The calm before the storm”
Due Friday Oct 20 at 5pm in Prof. Ginger’s mailbox.
CIRCLE YOUR ANSWERS AND KEY INTERMEDIATE RESULTS
USE MAPLE WHENEVER POSSIBLE
STAPLE YOUR PAPERS TOGETHER
INCLUDE ALL COMPUTER PRINTOUTS (with commentary)
You will note that this problem set is (much) shorter than usual because of the reduced time this
week. I know this will disappoint many of you and I promise I’ll try to make it up to you next
week!
Levine Problems
4.27 vibrations in LiH and ICl
Additional Problems
1) In lecture we introduced the lowering operator as
)(
2
1xipa
µω
µω
++=
−
h
By arguing that energy can’t be negative, we reasoned that there must be some state |
ψ
0
> for
which a
-
|
ψ
0
>=0 Use this relationship to generate a first-order differential equation for
ψ
0
and
solve the differential equation (it should be a “simple” first-order equation) to verify that
ψ
0
is
the same as obtained with the power series solution.
2) Find
ψ
1
, the first excited state of the SHO, by explicitly applying the raising operator, a
+
to
ψ
0
3) The frequencies of the three normal modes of H
2
O are ω
1
=3833 cm
-1
ω
2
=1649 cm
-1
and
ω
3
=3943 cm
-1
. If we describe a vibrationally excited state by the notation (n
1
n
2
n
3
) where n
i
is the
quantum number associated with the i
th
normal mode, what is the energy of the (121) state?
What is the energy difference between the (112) and (010) state? Side note: the anharmonicity in
real bonds tends to mix the normal modes over time.
4) The spatial Parity operator P satisfies the eigenvalue equation: P
ψ
= p
ψ
where the
eigenvalues of P are p=+1 (if
ψ
is even) and p=-1 (if
ψ
is odd) . Only even and odd functions
are eigenfunctions of P. The symmetry of the Hamiltonian has important consequences for the
symmetry of the allowed wave functions that we will examine below.
4a) Show that if the Hamiltonian is a symmetric (even) function, then [P,H]=0 (hint: what is the
parity of an even function times an even function, or an odd function times an odd function, i.e.
P(f
1
f
2
)=?)
4b) Two operators will commute if and only if they have a simultaneous set of eigenfunctions.
Use this fact, and your result from a) to justify that statement that “for a symmetric Hamiltonian,
the only allowed energy eigenfunctions will be even and odd functions.” The analysis of the
spatial symmetry of a wavefunction is a very powerful tool that allows you to predict properties
(i.e. IR and Raman activity of vibrational modes).