Tutorial: Density of states and Landau potential

Background: Sections 6.4-6.6 from Gould and Tobochnik.

1. Show that the grand partition sum (6.70) reduces to a product of single-particle sums (6.74) in the case of an ideal gas (3 pts).
2. Show that Eq. (6.90) is consistent with the definition of the density of states as \( g(\epsilon) = \sum_\mathbf{k}\delta(\epsilon-\epsilon_\mathbf{k}) \) (2 pts).
3. Problem 6.16 (10 pts).
4. Problem 6.17 (10 pts).
5. Problem 6.19 (10 pts).
6. Problem 6.20 (8 pts).
7. Problem 6.22 (8 pts).

Total: 51 points.