A gas of non-interacting particles has energy-momentum relationship $E=A(\hbar k)^{\alpha}$ for some constants $A$ and $\alpha$. Determine the density of states $g(E) d E$ in a threedimensional volume $V$.

Explain why the chemical potential $\mu$ satisfies $\mu<0$ for the Bose-Einstein distribution.

Show that an ideal quantum Bose gas with the energy-momentum relationship above has

$$p V=\frac{\alpha E}{3} .$$

If the particles are bosons at fixed temperature $T$ and chemical potential $\mu$, 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.

Discuss whether a gas of photons can undergo Bose-Einstein condensation.