(a) A gas of non-interacting particles with spin degeneracy $g_{s}$ has the energymomentum relationship $E=A(\hbar k)^{\alpha}$, for constants $A, \alpha>0$. Show that the density of states, $g(E) d E$, in a $d$-dimensional volume $V$ with $d \geqslant 2$ is given by

$$g(E) d E=B V E^{(d-\alpha) / \alpha} d E$$

where $B$ is a constant that you should determine. [You may denote the surface area of a unit $(d-1)$-dimensional sphere by $S_{d-1}$.]

(b) Write down the Bose-Einstein distribution for the average number of identical bosons in a state with energy $E_{r} \geqslant 0$ in terms of $\beta=1 / k_{B} T$ and the chemical potential $\mu$. Explain why $\mu<0$.

(c) Show that an ideal quantum Bose gas in a $d$-dimensional volume $V$, with $E=A(\hbar k)^{\alpha}$, as above, has

$$p V=D E,$$

where $p$ is the pressure and $D$ is a constant that you should determine.

(d) For such a Bose gas, write down an expression for the number of particles that do not occupy the ground state. Use this to determine the values of $\alpha$ for which there exists a Bose-Einstein condensate at sufficiently low temperatures.