Chemistry 256

Problem set No. 5; due Monday, Oct. 18

Reading: MS - Ch. 5

The potential kx2/2 gives Hooke's law, F = -kx, and leads to harmonic oscillations at frequency  = (k/m)1/2. The quantum mechanical oscillator can be solved exactly and has even and odd eigenfunctions fn(x) with energy (n + 1/2), n = 0,1,2...; n = 0 is a Gaussian, the bell-shaped curve. The vibrations or infrared spectra of diatomic molecules are accurately approximated by a harmonic oscillator; the 3N-6 normal coordinates of polyatomic molecules are 3N-6 oscillators. Rotational spectra of diatomics, typically at microwave frequencies, illustrate motion on the unit sphere and introduce angular functions that also appear in atomic orbitals. Resolved microwave spectra give the spectacularly accurate bond lengths seen in textbooks, since only knowledge of nuclear masses is needed. These simple models have many spectroscopic applications and more elaborate versions are widely used in current research.

1. MS - 5.4 / 4. MS - 5.22
2. MS - 5.13 / 5. MS - 5.34
3. MS - 5.15 / 6. See below.

NOTE: The first test (120 min) is on Tuesday, Oct. 19. The material covered is through Chapter 5 and including the material from the lecture notes, but excluding math chapter D. Monday, Oct. 18, is a review and a chance for you to ask questions.

6. The following question is an extended question that covers a lot of the formal material thus far. It may help you digest this material, and pieces of this question may well show up in the midterm exam. Consider once more the perpetual particle in the box with eigenfunctions and orthonormal eigenfunctions . Consider the time-dependent wavefunction . For simplicity you can use that , but this is not essential. The results obtained below are quite general. Questions:

  1. Show that is normalized at all times.
  2. Is an eigenfunction of the Hamiltonian? What about ?
  3. What is the expectation value for the energy?
  4. What is the expectation value for ?
  5. What is the variance ?
  6. Does any of the quantities in c, d, or e depend on time?
  7. Is your answer under e consistent with your answer of question b? Do you know what consistency I am referring to?
  8. What are the values for the energy that one can obtain if one would actually measure the energy on an ensemble described by this state?
  9. What are the probabilities to find the possible outcomes of the energy measurement, listed under h?
  10. Does any of the quantities obtained under h and i depend on time?