(Solved): Let us examine a relativistic electron gas, in which the single particle energy reads as a functio ...

Let us examine a relativistic electron gas, in which the single particle energy reads as a function of its momentum e(p) = (mc2)2 + (cp), where m is the mass of the particle and c is the speed of light. a. Determine the single particle density of states and the ultra- and non-relativistic limits of this quantity. (3p) b. Write down (integral) equations for the number and energy densities of the electron gas, parametrized by the temperature and chemical potential. Solve the average energy per particle in the limit of zero temperature. (3p) c. Taking the ultrarelativistic (massless) limit, determine the leading nonvanishing behavior of the heat capacity of the system, Cy = (E/T)lv, in the limit of small temperature. You may assume the particle density to be a predetermined constant here. (3p)