Free Electrons 3

exercise 3_4932

Show that the kinetic energy of all the electrons in a three dimensional gas of \(N\) free electrons at \(T=0\) is. \(U_0 = \frac{3}{5}NE_F\)

You can use the density of states for a gas in a box: \(g(\varepsilon)=\frac{V}{4\pi^2}{(\frac{2m}{\hbar^2})}^{3/2}\varepsilon^{1/2}\)

Averaged kinetic energy per electron could be achived by dividing the total energy by the number of particles:

\(\bar\varepsilon = \frac{\int\limits_0^\infty\varepsilon \cdot g(\varepsilon)f_{FD}(\varepsilon )d \varepsilon}{\int\limits_0^\infty g(\varepsilon)f_{FD}(\varepsilon )d \varepsilon}\)