path: root/source/know/concept/fermi-gas/index.md

---
title: "Fermi gas"
sort_title: "Fermi gas"
date: 2026-09-02
categories:
- Physics
- Quantum mechanics
layout: "concept"
---

A **Fermi gas** is a system of many fermions
that do not interact directly, only indirectly through
the [Pauli exclusion principle](/know/concept/pauli-exclusion-principle/),
and hence obey [Fermi-Dirac statistics](/know/concept/fermi-dirac-distribution/).

There are several real-life systems for which this model is relevant,
but most notably it serves as the foundation of the quantum-mechanical study
of electrons (or electron holes) in materials.
Obviously, electrons *do* interact strongly via the Coulomb force,
but it is nevertheless a useful starting point to neglect that fact,
and to then add the interactions later (see e.g. [jellium](/know/concept/jellium)).

Consider a collection of infinitely many non-interacting fermions.
For mathematical convenience, we restrict ourselves to a cube with side $$L$$,
and impose periodic boundary conditions.
Then, at the end of our calculation,
we should in theory take the limit $$L \to \infty$$
to recover the "true" system.

In the absence of any potentials, all the fermions' wavefunctions
are simply plane waves $$\ket{\psi_\vb{k}}$$ with wavevector $$\vb{k}$$.
Due to the cube's finite size and its periodic boundary conditions,
those waves have a discrete spectrum of allowed wavevectors $$\vb{k}$$,
meaning that each particle's wavefunction $$\ket{\psi_\vb{k}}$$
is as follows in $$\vb{r}$$-space (modulo a constant phase):

$$\begin{aligned}
    \psi_{\vb{k}}(\vb{r})
    = \frac{1}{\sqrt{L^3}} \exp(i \vb{k} \cdot \vb{r})
    \qquad \qquad
    \vb{k} = \frac{2 \pi}{L} (n_x, n_y, n_z)
\end{aligned}$$

Where $$n_x, n_y, n_z \in \mathbb{Z}$$.
This is a discrete (but infinite) set of independent orbitals,
so it is natural to use the
[second quantization](/know/concept/second-quantization/)'s
operators $$\hat{c}^\dagger$$ and $$\hat{c}$$ in our analysis.

Let the temperature $$T = 0$$,
then the $$N$$ fermions inside our cube
fill the $$N$$ lowest-energy orbitals.
The resulting $$N$$-particle ground state
is known as the **Fermi sea** or **Fermi sphere** $$\ket{\mathrm{FS}}$$,
and can be written as follows, where $$S$$ is the spin degeneracy,
i.e. for each $$\vb{k}$$ there are $$S$$ orbitals
with the same energy but different spin $$s$$
(for most relevant fermions $$S = 2$$):

$$\begin{aligned}
    \ket{\mathrm{FS}}
    = \prod_{s} \prod_{j = 1}^{N/S} \hat{c}_{s,\vb{k}_j}^\dagger \ket{0}
\end{aligned}$$

The energy and wavenumber $$|\vb{k}|$$ of the highest filled orbital
are called the **Fermi energy** $$\varepsilon_F$$ and **Fermi wavenumber** $$k_F$$,
and obey the expected kinetic energy relation:

$$\begin{aligned}
    \boxed{
        \varepsilon_F
        = \frac{\hbar^2}{2 m} k_F^2
    }
\end{aligned}$$

The Fermi sphere can be visualized in $$\vb{k}$$-space
as a sphere with radius $$k_F$$.
Because $$\vb{k}$$ is discrete, the sphere's surface is not smooth,
but in the limit $$L \to \infty$$ that "roughness" disappears.

Now, we would like a relation between the system's parameters,
e.g. $$N$$ and $$L$$, and the resulting values of $$\varepsilon_F$$ or $$k_F$$.
The total number $$N$$ of fermions in our cube is given by:

$$\begin{aligned}
    N
    = \sum_{s} \sum_{\vb{k}} \matrixel{\mathrm{FS}}{\hat{c}_{s,\vb{k}}^\dagger \hat{c}_{s,\vb{k}}}{\mathrm{FS}}
    = \sum_{s} \frac{L^3}{(2 \pi)^3} \int_{-\infty}^\infty \matrixel{\mathrm{FS}}{\hat{c}_{s,\vb{k}}^\dagger \hat{c}_{s,\vb{k}}}{\mathrm{FS}} \dd{\vb{k}}
\end{aligned}$$

Where the periodic boundary conditions have
[enabled us](/know/concept/discrete-spectrum-summation/)
to convert the sum over $$\vb{k}$$ to an integral.
For $$T = 0$$, the matrix element
$$\matrixel{\mathrm{FS}}{\hat{c}_{s,\vb{k}}^\dagger \hat{c}_{s,\vb{k}}}{\mathrm{FS}}$$
is either $$0$$ or $$1$$,
depending on whether $$\vb{k}$$ is outside or inside the Fermi sphere.
We can write this using
a [Heaviside step function](/know/concept/heaviside-step-function/):

$$\begin{aligned}
    N
    = \sum_{s} \frac{L^3}{(2 \pi)^3} \int_{-\infty}^\infty \Theta(k_F - |\vb{k}|) \dd{\vb{k}}
    = \frac{S L^3}{(2 \pi)^3} \int_{-\infty}^\infty \Theta(k_F - |\vb{k}|) \dd{\vb{k}}
\end{aligned}$$

Where we realized that spin does not matter,
to replace the sum with a factor $$S$$.
To evaluate this 3D integral, we transition to
[spherical coordinates](/know/concept/spherical-coordinates/)
$$(|\vb{k}|, \theta, \varphi)$$:

$$\begin{aligned}
    N
    &= \frac{S L^3}{8 \pi^3} \int_0^{2 \pi} \int_0^\pi \int_0^\infty \Theta(k_F - |\vb{k}|) |\vb{k}|^2 \sin(\theta) \dd{|\vb{k}|} \dd{\theta} \dd{\varphi}
    \\
    &= \frac{S L^3}{8 \pi^3} 4 \pi \int_0^\infty \Theta(k_F - |\vb{k}|) |\vb{k}|^2 \sin(\theta) \dd{|\vb{k}|}
    \\
    &= \frac{S L^3}{2 \pi^2} \int_0^{k_F} |\vb{k}|^2 \dd{|\vb{k}|}
    \\
    &= \frac{S L^3}{6 \pi^2} k_F^3
\end{aligned}$$

Since the particle density $$n = N / L^3$$,
we can rearrange this result to the following relation:

$$\begin{aligned}
    \boxed{
        k_F^3
        = \frac{6 \pi^2}{S} n
    }
    \qquad
\end{aligned}$$

Consequently, the Fermi energy $$\varepsilon_F$$
and the corresponding orbital's velocity $$v_F = \hbar k_F / m$$
can be expressed as a function of the density $$n$$:

$$\begin{aligned}
    \boxed{
        \varepsilon_F
        = \frac{\hbar^2}{2 m} \bigg( \frac{6 \pi^2}{S} \bigg)^{2/3} n^{2/3}
    }
    \qquad \qquad
    \boxed{
        v_F
        = \frac{\hbar}{m} \bigg( \frac{6 \pi^2}{S} \bigg)^{1/3} n^{1/3}
    }
\end{aligned}$$

This is an important result, especially for electrons in metals.
We know the electron density $$n$$ for many conductors,
and then these relations tell us that $$v_F \ll c$$,
and that the "Fermi temperature" $$T_F = \varepsilon_F / k_B$$
is very large (e.g. $$T_F \approx 8 \cdot 10^4 \: \mathrm{K}$$ for copper).
This justifies our implicit assumptions that relativity
and thermal fluctuations are negligible under normal circumstances.

We now have an expression for $$\varepsilon_F$$ as a function of $$n$$,
which we can control by adding or removing fermions from the system.
But it is also useful to isolate this relation for $$n$$ instead:

$$\begin{aligned}
    n
    &= \frac{S}{6 \pi^2} \bigg( \frac{2 m}{\hbar^2} \bigg)^{3/2} \varepsilon_F^{3/2}
\end{aligned}$$

The total population $$N = L^3 n$$ can therefore be expressed
as a function of $$\varepsilon_F$$:

$$\begin{aligned}
    N(\varepsilon_F)
    &= \frac{S L^3}{6 \pi^2} \bigg( \frac{2 m}{\hbar^2} \bigg)^{3/2} \varepsilon_F^{3/2}
\end{aligned}$$

And from this we obtain a formula for the
[density of states](/know/concept/density-of-states/)
$$g$$ of a 3D Fermi gas:

$$\begin{aligned}
    \boxed{
        g(\varepsilon_F)
        = \dv{N}{\varepsilon_F}
        = \frac{S L^3}{4 \pi^2} \bigg( \frac{2 m}{\hbar^2} \bigg)^{3/2} \varepsilon_F^{1/2}
    }
\end{aligned}$$

Now, $$\varepsilon_F$$ is the highest energy of a single fermion,
but what about the total $$N$$-particle energy $$E$$?
This is easy to calculate using the density of states:

$$\begin{aligned}
    E
    &= \int_0^{\varepsilon_F} \varepsilon \: g(\varepsilon) \dd{\varepsilon}
    \\
    &= \frac{S L^3}{4 \pi^2} \bigg( \frac{2 m}{\hbar^2} \bigg)^{3/2}
    \int_0^{\varepsilon_F} \varepsilon^{3/2} \dd{\varepsilon}
    \\
    &= \frac{3}{2} \frac{S L^3}{6 \pi^2} \bigg( \frac{2 m}{\hbar^2} \bigg)^{3/2} \: \frac{2}{5} \varepsilon_F^{5/2}
\end{aligned}$$

Here, we recognize $$N(\varepsilon_F)$$ from earlier,
leading to the following expression for the total $$E$$:

$$\begin{aligned}
    \boxed{
        E
        = \frac{3}{5} N \varepsilon_F
    }
\end{aligned}$$

This model is a strong foundation for many more advanced calculations.



## References
1.  H. Bruus, K. Flensberg,
    *Many-body quantum theory in condensed matter physics*,
    2016, Oxford.