Question
Calculate the average energy per particle, $\varepsilon$, for a Fermi gas at $T=0$, given that $\varepsilon_F$ is the Fermi energy.
Step 1
The Fermi energy, $\varepsilon_F$, is the energy of the highest occupied state in a Fermi gas at absolute zero temperature ($T=0$). It represents the energy level up to which all quantum states are filled with particles. Show more…
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Show that for a degenerate gas of fermions at $T=0$, the average energy per particle and the Fermi energy $E_{\mathrm{F}}$ (the energy of the highest occupied states) are related by $\langle E\rangle=(3 / 5) E_{\mathrm{F}} .[$ Hint $:$ Avoid writing any messy constants in your calculation, by noting that we can write $$ \langle E\rangle=\frac{\int_{0}^{E_{\mathrm{F}}} p(E) E d E}{\int_{0}^{E_{\mathrm{F}}} p(E) d E} $$ where $p(E)=C \sqrt{E}$ and $C$ is a constant that you don't need to know because it cancels. ]
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