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: Quantum Pure and Mixed States for an Electron
  (a) Write down a pure quantum state for the spin of an electron in which the probability of its spin component in the z direction being found to be +ħ/2 and −ħ/2 is 1/2, represented both as a vector ψ in a 2-dimensional Hilbert space and as a density matrix ρ_ψ, and a mixed quantum state ρ for the spin of the electron that is inequivalent to a pure state but in which the probabilities are still 1/2. (b) Calculate the mean value and fluctuation of the x component of the electron spin, S_x.

• 2: Quantum Correction to Ideal Gas Thermodynamics
  Complete the calculation of the quantum partition function for a monatomic ideal gas we did in class, starting from the point where one expands the sum over all permutations of the N particles into an identity term, terms where only two particles are switched, etc (see the second page of the lecture notes #15), but stop after those two terms. Then evaluate the degeneracy parameter δ = ρλ³ for helium gas at STP (standard T and P) conditions. What can you conclude about whether quantum corrections to ideal gas thermodynamics for this gas are important in a normal lab setting?

• 3: Massive Bosons in 2D: Calculate the density of states for massive bosonic particles in two spatial dimensions. Then show that in 2D Bose-Einstein condensation does not occur.
   

Aside from discussions I may have had with other students in the class about how to solve these problems, the solutions to this homework set I am submitting are entirely my own. While I was writing these solutions I did not look at anyone else's solutions, and I was not told by anyone what to write.

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