We can model a solid crystal of N atoms as a set of N three-dimensional simple harmonic oscillators, where every atom feels a restoring force with potential energy kfa pulling them back to their proper positions in the crystal. Since the energy of the atoms depends both on their momentum and on their position, the set of accessible states for a given total energy E is different from that for an ideal gas. The entropy in this case is ~r i , T E 5 ~ (3&ng [1n (:SNhf) + 11 with f : id being the frequency of simple harmonic motion, N the number of atoms, and h:{ is the arbitrary size of our chunks of phase space. Show that if the crystal is put in contact with a heat reservoir at temperature T, we expect the mean energy per atom to be 3.34251". (This will include ngT for the kinetic energy of each atom and 5'11\" 31" for the potential energy.) When translated into a heat capacity, this gives the Dulong-Petit law