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: Boltzmann entropy.
  A physical system in microcanonical equilibrium is assumed to have the same probability of being in any of its microstates, and its entropy S is taken to simply be a function S = f(Ω) of the number of states Ω the system can be in. Show that in that case the additive character of S and the multiplicative character of Ω necessarily require that S be of the form k ln Ω, where k is a constant. Provide a detailed proof, showing all steps!

• 2: Ideal gas in a microcanonical state.
  Starting from the expression for the number of states Ω for a classical monatomic ideal gas of N particles of mass m in a microcanonical state with total energy H(q, p) in the interval [E, E + Δ], with \(\Delta \ll E\), derived in class (and given in the lecture notes), show that the pressure equation of state for such a gas is pV = NkT.
   
• 3: Harmonic oscillators in a microcanonical state.
  Consider a system of two classical, uncoupled 1-dimensional simple harmonic oscillators, both of mass m and spring constant K, in a microcanonical state with energy H(q, p) in the interval [E, E + Δ], with \(\Delta \ll E\). Calculate the entropy of this system.

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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