Homework 4
Due Mon Oct 22 at 5:00 pm in Prof. Gingerās mailbox
CIRCLE YOUR ANSWERS AND KEY INTERMEDIATE RESULTS
USE MAPLE WHENEVER POSSIBLE
STAPLE YOUR PAPERS TOGETHER
INCLUDE ALL COMPUTER PRINTOUTS (with commentary)
Levine Problems (5
th
Ed. in parenthesis) 6
th
Ed.
(4.16) 4.14 most likely particle position
(4.18) 4.19 3D SHO (note: you donāt need to resolve parts weāve already solved)
(4.27) 4.25 vibrations in LiH and ICl
(4.29) 4.28 Morse Potential ā also: c) Given that D
e
= 7.31E-19 J/molecule, and a=1.82E10 m
-1
for HCl, calculate the force constant k for the HCl bond. Plot the Morse Potential for HCl and
the corresponding Harmonic oscillator potential on the same graph.
d) for what displacements from equilibrium is the harmonic oscillator (HO) potential a good
approximation of the Morse potential (this requires only a qualitative answer)?
e) For what physical situations would the HO potential yield a poor approximation to the actual
potential? (again, requires a qualitative answer)
Additional Problems:
1) In lecture we derived the density of states for particle in a 3D box as:
2/1
2/3
22
2
2
)( E
mV
ED ļ£·
ļ£ø
ļ£¶
ļ£¬
ļ£
ļ£«
=
h
Ļ
, starting with an approximation for the number of states with n
x2
+ n
y2
+n
z2
ā¤
n
2
as a function of n. Test this starting approximation by writing a program or routine in
Maple (or C or Fortran or Basic or Labview or whatever) to calculate the ACTUAL number of
states below the energy corresponding to a given n and compare it to the formula (and calculate
% error of the formula) for values of n from n=1 to n=200 (i.e. n^2=1 to 40000).
2) In lecture we introduced the lowering operator as
)(
2
1xipa
ĀµĻ
ĀµĻ
++=
ā
h
By arguing that energy canāt be negative, we reasoned that there must be some state |
Ļ
0
> for
which a
-
|
Ļ
0
>=0 Use this relationship to generate a first-order differential equation for
Ļ
0
and
solve the resulting differential equation (it should be a āsimpleā first-order equation) to find
Ļ
0
.
The answer is the same as obtained with the power series solution in Levine.
2) Find
Ļ
1
, the first excited state of the SHO, by explicitly applying the raising operator, a
+
to
Ļ
0
3) The frequencies of the three normal modes of H
2
O are Ļ
1
=3833 cm
-1
Ļ
2
=1649 cm
-1
and
Ļ
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 not needed for the
problem: anharmonicity in real bonds tends to mix the normal modes over time so energy will
spread out between the modes.
