Please print this sheet, sign the statement at the bottom, use it as cover page when you submit your homework, and write your solutions on separate sheets of paper, leaving some blank space between problems. Remember to always include a complete explanation in full sentences of your reasoning, and to show all calculations.

• 1: Quantum Harmonic Oscillator [adapted from Kennett, Problem 4.13]
  Calculate the entropy S, mean energy \(\bar E\) and heat capacity C for a 1D harmonic oscillator of mass m and spring constant k at temperature T. Show that in the high-temperature limit the value of C agrees with what you expect from the equipartition principle, and that in the \(T \to 0\) limit it goes to zero.

• 2: Hydrogen Gas [adapted from Kennett, Problem 4.19]
  Molecular hydrogen gas (\(H_2\)) can exist in two slightly different forms, ortho and para hydrogen, depending on the state of the spins of the hydrogen nuclei. Treating the (\(H_2\)) molecule as a linear 3D rotor, the rotational energy is

\[ \epsilon_J^{~} = {\hbar^2\,J(J+1)\over2I}\;. \]

For ortho hydrogen the spins are in a triplet state, for which the degeneracy is 3 and J can only take odd integer values 1, 3, 5, ..., while for para hydrogen the spins are in a singlet state, for which the degeneracy is 1 and J can only take even integer values 0, 2, 4, .... Noting that the proton-proton bond distance in hydrogen is 0.747 Å, find the ratio of ortho to para hydrogen at (i) a temperature of 25 K and (ii) high temperatures. You may ignore non-rotational degrees of freedom since these are the same for both species. 

• 3: Ideal Boson Gas
  Show that the heat capacity of an ideal gas of massive bosons vanishes in the \(T \to 0\) limit. Start from the mean energy \(\bar E\) derived in class and, using the information on the second page of the course lecture notes #17, write down and explain in detail all the steps leading to the result.

You are allowed, in fact encouraged, to discuss these problems and how to solve them with other students in the class, but you are not allowed to copy or use in any way anyone else's written solution to any of these problems. This includes solutions that may be provided to you by a student who took the course previously, or solutions you may find online or in print. This policy will hold for all assignments in this class.

Aside from oral discussions I may have had with other students in the class, the solutions to this homework set I am submitting are entirely my own.

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