1)(Ultra-relativistic degenerate fermion gas). In the lectures, we have derived the equation of state
for the non-relativistic degenerate fermion gas and showed that it behaves like P ～ ρ5/3. In this problem, we
will derive the equation of state for the ultra-relativistic case. Assume that fermion has degeneracy
(i) The dispersion relation for a relativistic particle is given by
In Homework 3, you have shown that in ultra-relativistic case, p>>mc such that E = pc. Show that the
density of states for this case is then
i.e. it’s the same as the massless case.
(ii) Assuming that the hNi = 1 for E < EF and hNi = 0 for E > EF (i.e. fully degenerate gas) where
EF is the Fermi energy, calculate the mean particle number and mean energy of the system and show
that they are
(iii) Equivalently with the non-relativistic Fermi momentum, the relativistic Fermi momentum is given
by pF = EF =c. Show that, in terms of density, this is
(iv) Calculate the equation of state of the ultra-relativistic degenerate fermion gas and show that it scales
like P ~ (N/V )4/3.