Problem 4 (15 points): Consider a collection of N identical harmonic oscillators (perhaps an Einstein
solid or the internal vibrations of gas molecules) at temperature T. The allowed energies of each
oscillator are 0, \epsilon, 2\epsilon, and so on. (a) Evaluate the partition function for a single harmonic oscillator.
To simplify your answer as much as possible, use the fact that for a geometric series
$\sum_{n=0}^{\infty} r^n = \frac{1}{1 - r}$
(b) Use the calculated partition function to find an expression for the average energy (E) of a single
oscillator at temperature T. (c) Find an expression for the Helmholtz free energy of a system of N
harmonic oscillators.