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:

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