Problem 1. For the 3D simple harmonic oscillator (SHO), described by the Hamiltonian,
$$H = \frac{\vec{p}^2}{2m_0} + \frac{1}{2}m_0\omega^2\vec{x}^2,$$ (1)
with a mass $m_0$ and spring constant $k \equiv m_0\omega^2$,
(a) Calculate the (classical) partition function,
$$Z_{cl} \equiv \int \frac{d^3p}{(2\pi h)^3} \int d^3xe^{-\beta H(\vec{p},\vec{x})}.$$ (2)
(b) Then, using the partition function, derive the expression for the energy $U$ and determine the heat capacity, $C \equiv \frac{\partial U}{\partial T}$.
(c) What then, is the result if you have $N$ such atoms, whose vibrations are independently described by this SHO?
