Show that for a degenerate gas of fermions at T=0, the...

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. ]

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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Key Concepts

Fermi Energy

The Fermi energy is the highest occupied energy level at absolute zero temperature in a system of fermions. It defines the threshold energy below which all quantum states are filled and plays a crucial role in determining the electronic properties of materials.

Degenerate Fermi Gas

A degenerate Fermi gas describes a collection of fermions at temperatures near absolute zero, where the effects of quantum statistics dominate. In such a system, the Pauli exclusion principle forces the particles to occupy the lowest available energy states, filling them up to the Fermi energy.

Density of States

The density of states is a function that represents the number of available quantum mechanical states at each energy level. In a three-dimensional system, it often varies as the square root of energy, which is important when integrating to calculate physical quantities like total particle number and energy.

Average Energy Calculation

The average energy of particles in a degenerate Fermi gas is calculated by integrating the energy multiplied by the density of states over all occupied states and then dividing by the total number of particles. This method directly relates the mean energy per particle to the Fermi energy, exemplified by the derivation showing that the average energy is three-fifths of the Fermi energy.

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