Background: Sections 4.12 and 6.3 from Gould and Tobochnik.
- Review section II of the note on the connection of statistics with thermodynamics, which establishes the connection between \( \ln Z\) and the free energy. Provide a similar derivation of the connection between \( \ln Z_G \) and \( \Omega \), where \( Z_G \) is the grand partition function (4.144) (10 pts).
- Problem 4.53 (7 pts).
- Explain why the use of occupation numbers enables the correct enumeration of the states of a quantum gas, while the listing of states occupied by each particle does not (5 pts).
- Explain why it is easier to use the grand canonical ensemble for a quantum ideal gas compared to the canonical ensemble [with Eq. (6.65) and (6.66)] (3 pts).
- Problem 6.14 (5 pts).
- Problem 6.15 (8 pts).
- Problem 6.54 (12 pts).
Total: 50 pts.