Consider a non-interacting Fermi gas of electrons. Assume the electrons are nonrelativistic.
(a) Find the density of states $N(E)$ as a function of energy $(N(E)$ is the number of states per unit energy interval) for the following cases:
1) The particles are constrained to move only along a line of length L.
2) The particles move only on a two dimensional area $A$.
3) The particles move in a three dimensional volume $V$.
(b) In a Fermi electron gas in a solid when $T \ll T_{\mathrm{F}}$ (the gas temperature is much less than the Fermi temperature), scattering by phonons and impurities limits electrical conduction. In this case, the conductivity $\sigma$ can be written as
$$
\sigma=e^2 N\left(E_{\mathrm{F}}\right) D,
$$
where $e$ is the electron charge, $N\left(E_{\mathrm{F}}\right)$ is the density of states, defined above, evaluated at the Fermi energy and $D$ is the electron diffusivity. $D$ is proportional to the product of the square of the Fermi velocity and the mean time, $\tau_\epsilon$, between scattering events ( $D \sim v_{\mathrm{F}}^2 \tau_e$ ).
1) Give a physical argument for the dependence of the diffusivity on $N\left(E_{\mathrm{F}}\right)$.
2) Calculate the dependence of $\sigma$ on the total electron density in each of the three cases listed in part (a). The electron density is the total number of electrons per unit volume, or per unit area, or per unit length, as appropriate.