I need help with HW

1) (20 points) Single walled carbon nanotubes can be approximated by a particle-on-a-cylindrical-surface model. Suppose the cylinder has a length () and radius
a, with the z-axis along the cylinder.
a) Sketch an example of the cylinder with the appropriate labels.
b) Write the kinetic energy of the electron in terms of the length z, the radius a,
and the radial angle of the cylinder.
c) Combining ideas from the particle-in-a-box and 2-D rigid-rotor models, show
that the wavefunction can be written as Y = A sin (172) eimo. What are the
allowed values of the quantum numbers n and m?
d) Write the energy expression in terms of m, n, 1, a, and fundamental constants.
For a nanotube of radius 6 Å and 110 Å, compute the energy spacing between
the two lowest energy levels.
e) If there are 300 n electrons in the tube, calculate the spacing between the
highest occupied and lowest empty orbitals. (Hint: It may be useful to use a
computer program.) What are their (m, n) quantum numbers?
2) (10 points) Show mathematically (derive) that for a rotational mass on a string,
the rotational energy is,
– h2 d2
Erot =
8121 do?
3) (10 points) Using a mathematical program of your choice, plot the following:
a) The l = 1, m = 0 wavefunction of the 3D rigid rotor.
b) The l = 0, m = 0 probability density of the 3D rigid rotor.
c) The l = 2, m = 0 probability density of the 3D rigid rotor.
d) Explain how each differ from one another
4) (10 points) If the H2 molecule rotates in the plane of a crystalline surface (in a
chemisorption situation), it can be approximated as a two-dimensional rigid rotor.
Calculate in cm-“, the lowest energy transition for such a system.
5) (10 points) Define the following terms and explain how each play a role in
rotational spectroscopy:
a) Polarizability
b) Perturbation Theory
c) Dipole Moment
d) Symmetry
e) Selection Rules for rotational and far infrared