Please print this sheet, sign the statement at the bottom, and use it as cover page when you submit your homework. Remember to always include a complete explanation of your reasoning, and to show all calculations.

• 1: Dependence of the entropy on the number of states
  Assuming that the entropy S of a physical system is given by an arbitrary function S = f(Ω) of the number of states Ω, show that 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 entropy for a classical ideal gas in a microcanonical state 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
  Calculate the entropy of a system of N uncoupled 1-dimensional simple harmonic oscillators of mass m and spring constant k if the energy of the system is known to be in an interval of width Δ around E, with \(\Delta \ll E\).

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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