Blundell Thermal Physics · Problem 20.3

The partition function

Physics — Statistical Mechanics · partition-function-sho-rewrite

Problem

Show that eqn 20.29 can be rewritten as

U = \frac{\hbar\omega}{2} \coth \frac{\beta \hbar\omega}{2}, \tag{20.50}

and eqn 20.32 can be rewritten as

S = k_B \left[\frac{\omega}{2} \coth \frac{\beta \hbar\omega}{2} - \ln \left(2 \sinh \frac{\beta \hbar\omega}{2}\right)\right]. \tag{20.51}