Homework Set # 4 - Due September 25

1. (a) Derive Eq.3.2.35 using the definition of Γ given in 3.2.13.

   (b) Derive Eq.3.2.36 from Eq.3.2.14 and 3.2.35.

   (c) Derive Eq.3.2.39 from Eq.3.2.38 (Hint: calculate ∂U/∂ω_r=0).

   (d) Derive Eq. 3.2.40 from Eq. 3.2.39 and 3.2.37

2. Problem 3.4

3. Problem 3.6

4. Problem 3.15

5. Consider an isolated system consisting of a large number N of very weakly interacting localized particles of spin ½. Each particle has a magnetic moment μ which can point either parallel or antiparallel to an applied field H. The energy E of the system is then E=-(n₁-n₂)μH, where n₁ is the number of spins aligned parallel to H and n₂ the number of spins aligned antiparallel to H.

(a) Consider the energy range between E and E+δE where δE is very small compared to E but is microscopically large so that δE>> μH. What is the total number of states Ω(E) lying in this energy range?

(b) Write down an expression for lnΩ(E) as a function of E. Simplify this expression by applying Stirling's formula (lnN!≈NlnN-N).

(c) Assume that the energy E is in a region where Ω(E) is appreciable, i.e., that is not close to the extreme values +/- NμH which it can assume. In this case apply a Gaussian approximation to part (a) to obtain a simple expression for Ω(E) as a function of E.