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+---
+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.