Fermi–Dirac Statistics

Fermi–Dirac statistics is the quantum statistics of indistinguishable fermions. Each complete one-particle mode can have occupation number $0$ or $1$ , and the thermal mean occupation is

\bar n_i = \frac{1}{ \exp[\beta(\epsilon_i-\mu)]+1 }.

Historically, this statistics grew out of Pauli’s exclusion principle, Fermi’s quantum treatment of the ideal gas, and Dirac’s formulation of antisymmetric many-particle states. It became the state-counting foundation for atoms, metals, white dwarfs, neutron matter estimates, and much of many-body quantum physics.

Pauli Exclusion as Precursor

Pauli introduced the exclusion principle in 1925 to organize atomic spectra, shell closure, and the periodic table. In modern language, no two identical fermions can occupy the same complete one-particle state.

For atomic electrons in a central-field model, a complete one-electron label is schematically

n,\quad \ell,\quad m_\ell,\quad m_s.

Two electrons may share the same spatial orbital only if their spin labels differ. They do not share the same complete spin-orbital.

This rule was historically earlier than the full modern explanation. Pauli’s 1925 principle was not yet a derivation from antisymmetric wavefunctions or the spin-statistics theorem. It was a powerful empirical and theoretical rule that later found its natural place in fermionic quantum statistics.

Fermi and Dirac Statistics

Fermi’s 1926 work applied exclusion-based counting to an ideal gas. Dirac’s 1926 work clarified the role of antisymmetric wavefunctions and exchange signs in quantum mechanics. Together their work gave the statistics now called Fermi–Dirac statistics.

For a single fermionic mode $i$ with energy $\epsilon_i$ , the allowed occupations are only

n_i=0 \quad\text{or}\quad n_i=1.

In the grand-canonical ensemble, the one-mode partition factor is therefore

Z_i = 1+\exp[-\beta(\epsilon_i-\mu)].

The mean occupation is

\bar n_i = \frac{ \exp[-\beta(\epsilon_i-\mu)] }{ 1+\exp[-\beta(\epsilon_i-\mu)] } = \frac{1}{ \exp[\beta(\epsilon_i-\mu)]+1 }.

The plus sign in the denominator is the thermal trace of the occupancy restriction. It prevents the mean occupation from exceeding one.

Occupancy Restrictions

The compact rule is

n_i\in\{0,1\}.

Here $i$ labels a complete one-particle mode. In atomic physics, the mode is a spin-orbital. In a uniform electron gas, a mode includes momentum and spin. In a crystal, it can include band, crystal momentum, and spin or pseudospin labels.

The restriction is not a statement that fermions repel each other by a new force. It is a restriction on the allowed many-particle state space. For two identical fermions in orthonormal one-particle states $a$ and $b$ , the antisymmetric state is

\lvert a,b\rangle_A = \frac{1}{\sqrt2} \bigl( \lvert a\rangle_1\lvert b\rangle_2 - \lvert b\rangle_1\lvert a\rangle_2 \bigr).

If $a=b$ , the state vanishes. That cancellation is the algebraic core of Pauli exclusion.

Electron Gas and Atoms

Fermi–Dirac statistics immediately changed the theory of matter. In atoms, it explained why electrons fill shells instead of all falling into the lowest orbital. In metals, Sommerfeld’s electron theory used Fermi–Dirac filling to replace classical equipartition with a filled Fermi sea plus low-energy excitations near the Fermi surface.

At zero temperature, the ideal Fermi occupation approaches a step function:

\bar n_i \to \begin{cases} 1, & \epsilon_i<\mu,\\ 0, & \epsilon_i>\mu. \end{cases}

For a uniform three-dimensional ideal Fermi gas with internal degeneracy $g$ , the Fermi wave number is

k_F = \left(\frac{6\pi^2 n}{g}\right)^{1/3},

where $n=N/V$ is the number density. The corresponding Fermi energy for nonrelativistic particles of mass $m$ is

E_F = \frac{\hbar^2k_F^2}{2m}.

The reference model Ideal Fermi Gas collects these formulas. The present page emphasizes why the formulas were historically important: exclusion changed the counting of low-temperature matter.

Modern Fermion Formalism

Modern quantum mechanics describes identical fermions by antisymmetric states. Fixed- $N$ fermion states live in the exterior-power Hilbert space

\wedge^N\mathcal h.

In occupation-number language, a fermionic basis state is a bitstring

\lvert n_1,n_2,\ldots\rangle_F, \qquad n_i\in\{0,1\}.

Creation and annihilation operators obey anticommutation relations. For example,

\{c_i^\dagger,c_j^\dagger\}=0, \qquad (c_i^\dagger)^2=0.

This operator language is not historical decoration. It is the efficient modern way to encode antisymmetry, exclusion, and signs in many-fermion systems.

The relativistic spin-statistics theorem later explained why half-integer-spin particles are fermions under standard field-theoretic assumptions. Nonrelativistic many-body quantum mechanics uses the fermionic exchange sector as an input for electrons, nucleons, helium-3 atoms, and other fermionic species.

Common Misconceptions

Fermi–Dirac statistics is not just a low-temperature correction to classical statistics; it is the correct exchange statistics for identical fermions.
Pauli exclusion is not a new repulsive force.
“Same state” means same complete one-particle mode, including spin and internal labels.
Opposite-spin electrons in one spatial orbital are not sharing the same complete spin-orbital.
The Fermi surface is a surface in momentum or quantum-number space, not a surface in ordinary position space.
The nonrelativistic formalism uses fermionic statistics; it does not prove the full relativistic spin-statistics theorem.
Fermionic signs are not optional conventions once a mode ordering has been chosen; they encode antisymmetry.

Cross-Links

References

W. Pauli, “Über den Zusammenhang des Abschlusses der Elektronengruppen im Atom mit der Komplexstruktur der Spektren,” Zeitschrift für Physik 31, 765-783, 1925.
E. Fermi, “Sulla quantizzazione del gas perfetto monoatomico,” Rendiconti Lincei 3, 145-149, 1926.
P. A. M. Dirac, “On the Theory of Quantum Mechanics,” Proceedings of the Royal Society A 112, 661-677, 1926, DOI: 10.1098/rspa.1926.0133.
A. Sommerfeld, “Zur Elektronentheorie der Metalle auf Grund der Fermischen Statistik,” Zeitschrift für Physik 47, 1-32, 1928.
M. Jammer, The Conceptual Development of Quantum Mechanics, 2nd ed., American Institute of Physics, 1989.
K. Huang, Statistical Mechanics, 2nd ed., Wiley, 1987.
A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, Dover, 2003.

Exercises

Derive the Fermi–Dirac occupation factor from the one-mode grand partition function $Z_i=1+\exp[-\beta(\epsilon_i-\mu)]$ .

Solution

The allowed occupations are $n_i=0$ and $n_i=1$ . The mean occupation is

\bar n_i = \frac{ 0\cdot 1 + 1\cdot\exp[-\beta(\epsilon_i-\mu)] }{ 1+\exp[-\beta(\epsilon_i-\mu)] }.

Therefore

\bar n_i = \frac{1}{ \exp[\beta(\epsilon_i-\mu)]+1 }.

Explain why two electrons with opposite spin can occupy the same spatial orbital without violating Pauli exclusion.

Solution

The complete one-particle mode includes spin as well as the spatial orbital. Two electrons in the same spatial orbital but with opposite spin occupy different spin-orbitals. The full two-electron state must still be antisymmetric under exchange.

What is the zero-temperature occupation of an ideal fermionic mode with $\epsilon_i<\mu$ ?

Solution

At zero temperature, $\beta\to\infty$ . If $\epsilon_i<\mu$ , then $\beta(\epsilon_i-\mu)\to-\infty$ , so

\exp[\beta(\epsilon_i-\mu)]\to0.

Thus

\bar n_i \to \frac{1}{0+1} = 1.

Why is Pauli exclusion not enough by itself to determine the energy of a many-electron atom?

Solution

Pauli exclusion restricts which many-electron states are allowed. The actual energies also depend on the Hamiltonian: electron-nucleus attraction, electron-electron repulsion, spin-orbit coupling, screening, relativistic corrections, and approximations used to solve the many-body problem. Exclusion supplies the state-counting structure, not the full dynamics.