7.E: Rotational States (Exercises)

Q7.1

Consider a homonuclear diatomic molecule described by the rotational wavefunction \(Y^0_1 (\theta , \varphi )\).

1. Sketch graphical representations of this function by plotting the amplitude of the function vs. some coordinate with all other coordinates held constant.
2. Sketch a three-dimensional polar plot of this function where the three dimensions are x, y, and z.
3. Sketch a picture to show how this molecule is rotating in space.

Q7.2

Consider a homonuclear diatomic molecule of mass M and bond length D described by the rotational wavefunction \(Y^{-1}_2 (\theta , \varphi )\).

1. What is the rotational energy of this molecule?
2. What is the rotational angular momentum?
3. What is the z-component of the angular momentum?
4. What angle does the angular momentum vector make with respect to the z-axis?
5. If the molecule is oxygen, what are the numerical answers to 1) – 4)?

Q7.3

Develop an equation for the stimulated emission of a photon. Compare your result to Equation (7-58).

Q7.4

When centrifugal stretching is included in the energy for the states of the rigid rotor, equation has an extra term \(v_{allowed} = 2B (J_i + 1) - 4D(J_i + 1)^3\), Equation (7-67), where J is the quantum number for the initial rotational state, B is the rotational constant and D is the centrifugal distortion constant. Use the data in Table 7.2 to determine both B and D graphically. Be careful how you use units. Compare the magnitudes of B and D. What is the percent difference between B determined without centrifugal stretching and that found here including centrifugal stretching? What would be the corresponding percent error in the bond length computed from B?

Q7.5

Write a paragraph explaining why you might expect the same functions involving spherical coordinates to describe both the rigid rotor and the hydrogen atom.

Contributors

• Adapted from "Quantum States of Atoms and Molecules" by David M. Hanson, Erica Harvey, Robert Sweeney, Theresa Julia Zielinski