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 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 not equivalent to a pure state but in which those probabilities are still 1/2. (b) Calculate the mean value and fluctuation of the x component of the electron spin, S_x, in both states.

• 2: Corrections to Ideal Gas Thermodynamics
  Evaluate the degeneracy parameter \(\delta = \rho\lambda^3\) for He and N gas at STP conditions. Discuss the physical significance of the values you obtain by explaining (a) The physical interpretation of the parameter \(\delta\) for a general system, and (b) What you can conclude based on those values about which theory should be used to describe the two gases.
   
• 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.
   

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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