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.

David M. Hanson, Erica Harvey, Robert Sweeney, Theresa Julia Zielinski ("Quantum States of Atoms and Molecules")