In the grand canonical ensemble, at temperature $T$ and chemical potential $\mu$, what is the probability of finding a system in a state with energy $E$ and particle number $N$ ?

A particle with spin degeneracy $g_{s}$ and mass $m$ moves in $d \geqslant 2$ spatial dimensions with dispersion relation $E=\hbar^{2} k^{2} / 2 m$. Compute the density of states $g(E)$. [You may denote the area of a unit $(d-1)$-dimensional sphere as $S_{d-1}$.]

Treating the particles as non-interacting fermions, determine the energy $E$ of a gas in terms of the pressure $P$ and volume $V$.

Derive an expression for the Fermi energy in terms of the number density of particles. Compute the degeneracy pressure at zero temperature in terms of the number of particles and the Fermi energy.

Show that at high temperatures the gas obeys the ideal gas law (up to small corrections which you need not compute).