Problem 3.23. Show that the entropy of a two-state paramagnet, expressed as a function of temperature, is S = Nk[ln (2 cosh x) - x tanh x], where x = μB/kT. Check that this formula has the expected behavior as T → 0 and T → ∞o. The following two problems apply the techniques of this section to a different sys- tem, an Einstein solid (or other collection of identical harmonic oscillators) at arbitrary temperature. Both the methods and the results of these problems are extremely important. Be sure to work at least one of them, preferably both.

Problem 3.23. Show that the entropy of a two-state paramagnet, expressed as a function of temperature, is S = Nk[ln (2 cosh x) - x tanh x], where x = μB/kT. Check that this formula has the expected behavior as T → 0 and T → ∞o. The following two problems apply the techniques of this section to a different sys- tem, an Einstein solid (or other collection of identical harmonic oscillators) at arbitrary temperature. Both the methods and the results of these problems are extremely important. Be sure to work at least one of them, preferably both.