A diatomic molecule, free to move in two space dimensions, has classical Hamiltonian

$$H=\frac{1}{2 m}|\mathbf{p}|^{2}+\frac{1}{2 I} J^{2}$$

where $\mathbf{p}=\left(p_{1}, p_{2}\right)$ is the particle's momentum and $J$ is its angular momentum. Write down the classical partition function $Z$ for an ideal gas of $N$ such molecules in thermal equilibrium at temperature $T$. Show that it can be written in the form

$$Z=\left(z_{t} z_{r o t}\right)^{N}$$

where $z_{t}$ and $z_{\text {rot }}$ are the one-molecule partition functions associated with the translational and rotational degrees of freedom, respectively. Compute $z_{t}$ and $z_{r o t}$ and hence show that the energy $E$ of the gas is given by

$$E=\frac{3}{2} N k T$$

where $k$ is Boltzmann's constant. How does this result illustrate the principle of equipartition of energy?

In an improved model of the two-dimensional gas of diatomic molecules, the angular momentum $J$ is quantized in integer multiples of $\hbar$ :

$$J=j \hbar, \quad j=0, \pm 1, \pm 2, \ldots$$

Write down an expression for $z_{\text {rot }}$ in this case. Given that $k T \ll\left(\hbar^{2} / 2 I\right)$, obtain an expression for the energy $E$ in the form

$$E \approx A T+B e^{-\hbar^{2} / 2 I k T}$$

where $A$ and $B$ are constants that should be computed. How is this result compatible with the principle of equipartition of energy? Find $C_{v}$, the specific heat at constant volume, for $k T \ll\left(\hbar^{2} / 2 I\right)$.

Why can the sum over $j$ in $z_{\text {rot }}$ be approximated by an integral when $k T \gg\left(\hbar^{2} / 2 I\right)$ ? Deduce that $E \approx \frac{3}{2} N k T$ in this limit.