CHM 3411, Dr.
Chatfield, Fall 2009
Problem Set 5
Due Friday, Feb. 20
Suggested (b) Exercises from
This problem set explores rotational motion and perturbation theory. The first three problems are purely mathematical. They are designed to help you understand the particle-on-a-ring and particle-on-a-sphere problems. The fourth question applies the solutions for the energy levels of the particle-on-a-sphere to the problem of rotation of a diatomic molecule, a problem of significance for chemistry. The fifth problem is theoretical and explores the concept of space quantization. The last problem is a simple application of perturbation theory; we will use perturbation theory later in the course, so doing this problem will help you understand the method.
1. We stated in
class that the quantum mechanical operator 
for the z-component of the angular momentum expressed
in Cartesian coordinates, is:
Show that when we convert to polar
coordites, the operator becomes:
2. By integration, show explicitly that the spherical harmonics Y0,0 and Y1,0 (given in Table 9.3) are orthonormal (i.e., that each is normalized and that they are orthogonal to each other). Be careful with the two-dimensional volume element in the integration; it is dt = sin q d q df (see Comment 9.6 on p. 302 of Atkins).
3. In spherical polar coordinates, the operators
for the square of the total angular momentum, 
,
the z-component of the angular momentum, , and the x-component of the angular momentum,
, are
[Note that Atkins uses a small l for the angular momentum operator: 
, 
etc. but I use capital
letters instead because that is more common.]
(a) Show that
the spherical harmonic Y1,1 given in Table
9.3 is an eigenfunction of both and ,
but not of .
(b) Calculate
the corresponding eigenvalues of and and show that they are equivalent to the
values given by eqs. 9.54a and
9.54b in Atkins.
(c) Explain what the result in (a) has to do with the cone model of angular momentum that we discussed in class (see Fig. 9.40b in Atkins).
4. This problem explores the energy levels for the particle-on-a-sphere problem. As we will see later in the course, the rotation of a diatomic molecule about its center of mass is equivalent to the rotation of a single particle having a “reduced mass” of m (see Atkins Problem 9.4 for the definition of m) about a fixed point, at a distance r from the point. [This particular problem concerns 1H127I; the H atom is much lighter than the I atom, so the I atom hardly moves. Thus this is almost identical to rotation of the H about a fixed I, which is obviously identical to the particle-on-a-sphere problem.]
(a) Atkins Problem 9.4
(b) It is possible to excite a molecule from the ground rotational level (l=0) to the first excited rotational level (l=1). Calculate the wavelength of light needed to do this for the the 1H127I molecule. [We will explore this in more detail when we cover spectroscopy later in the course.]
5. The quantization of angular momentum is sometimes hard to grasp, particularly since its spatial representation is so counterintuitive. This problem attempts to make the concept more concrete.
a) For a
particle on a sphere, calculate the angle that the angular momentum vector
makes with the z-axis when the system is in the state represented by the
spherical harmonic 
. The angle should be
expressed in terms of l and ml.
b) Show that the minimum angle approaches zero as l approaches infinity.
c) Calculate the allowed angles when l is 1, 2, and 3.
d) Draw a visual representation of the angular momentum vector with the allowed angles for l equal to 1, 2, and 3.
6. Atkins 9.6. This problem applies perturbation theory to a simple situation.