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**Unformatted text preview: **Homework #4 Solutions Question 1a) K+K Chapter 3, Problem 1. The free energy can be obtained from the partition function using the formula F =- τ log Z where Z = e + e- ε/τ . Hence F =- τ log(1 + e- ε/τ ) 1b) Entropy is related to the free energy by σ =- ∂F ∂τ V . Note that the energy levels are a property of the system and cannot depend on temperature. Since volume is held fixed we conclude that in this differentiation ε is fixed. σ = log(1 + e- ε/τ ) + τε τ 2 (1 + e- ε/τ ) e- ε/τ Energy is given by U = F + τσ =- τ log(1+ e- ε/τ )+ τ log(1+ e- ε/τ )+ εe- ε/τ 1+ e- ε/τ U = ε 1 + e ε/τ which agrees with Eq.(14) on page 62. Question2a) K+K Chapter 3, Problem 3. We calculate the partition func- tion first and use the standard relation to get the free energy. Z = ∞ X s =0 e- ε s /τ = ∞ X s =0 ( e- ¯ hω/τ ) s where we have used the fact that ε s = s ¯ hω . This sum can be performed exactly using 1 1- x = ∞ X s =0 x s This formula is only valid for...

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