Consider a 3-dimensional gas of $N$ non-interacting particles in a box of size $L$ where the allowed momenta are $\left\{\mathbf{p}_{i}\right\}$. Assuming the particles have an energy $\epsilon(|\mathbf{p}|), \epsilon^{\prime}(p)>0$, calculate the density of states $g(\epsilon) d \epsilon$ as $L \rightarrow \infty$.

Treating the particles as classical explain why the partition function is

$$Z=\frac{z^{N}}{N !}, \quad z=\int_{0}^{\infty} d \epsilon g(\epsilon) e^{-\epsilon / k T}$$

Obtain an expression for the total energy $E$.

Why is $\mathbf{p}_{i} \propto 1 / L ?$ By considering the dependence of the energies on the volume $V$ show that the pressure $P$ is given by

$$P V=\frac{N}{3 z} \int_{0}^{\infty} d \epsilon g(\epsilon) p \epsilon^{\prime}(p) e^{-\epsilon / k T}$$

What are the results for the pressure for non-relativistic particles and also for relativistic particles when their mass can be neglected?

What is the thermal wavelength for non-relativistic particles? Why are the classical results correct if the thermal wavelength is much smaller than the mean particle separation?