assignment Homework
Fermi gas Relativity
Thermal and Statistical Physics 2020
For electrons with an energy \(\varepsilon\gg mc^2\), where
\(m\) is the mass of the electron, the energy is given by
\(\varepsilon\approx pc\) where \(p\) is the momentum. For electrons
in a cube of volume \(V=L^3\) the momentum takes the same values as
for a non-relativistic particle in a box.
Show that in this extreme relativistic limit the Fermi energy of a
gas of \(N\) electrons is given by \begin{align}
\varepsilon_F &= \hbar\pi c\left(\frac{3n}{\pi}\right)^{\frac13}
\end{align} where \(n\equiv \frac{N}{V}\) is the number density.
Show that the total energy of the ground state of the gas is
\begin{align}
U_0 &= \frac34 N\varepsilon_F
\end{align}