Fermion

A fermion is a particle that obeys Fermi–Dirac statistics, meaning that exchanging two identical particles reverses the sign of their quantum state. In ordinary three-dimensional space, fermions have half-integer spin, such as $1/2$, $3/2$, or $5/2$. They obey the Pauli exclusion principle: no two identical fermions can occupy the same complete single-particle quantum state. Examples include the electron, quarks, and neutrinos, as well as composite particles such as protons and neutrons. Fermions contrast with bosons, which have integer spin and different exchange statistics. (atlas-public.web.cern.ch)

Exchange symmetry and exclusion

In quantum mechanics, identical particles cannot be distinguished by permanent labels. Their joint wave function must reflect this indistinguishability. For two identical fermions, exchanging their complete coordinates gives

where each $x_i$ includes position and internal variables such as spin. This property is called antisymmetry. The overall sign does not change measurement probabilities, but the relative signs between terms affect interference and determine which many-particle states are possible. (damtp.cam.ac.uk)

If two fermions are assigned the same single-particle state, antisymmetry makes the corresponding wave function vanish. Exclusion therefore concerns an entire quantum state, not merely a location or an energy value. Two electrons can occupy the same spatial orbital if their spin states differ; these are two distinct complete states. For independent fermions, an antisymmetric many-particle wave function can be constructed as a Slater determinant of occupied single-particle states. (tsapps.nist.gov)

The spin–statistics theorem connects half-integer spin with fermionic exchange behavior within relativistic quantum field theory. Fermionic fields are quantized using anticommutation relations. These relations encode both the exchange sign and the restriction to zero or one particle in each complete state. Exclusion is thus not an additional repulsive force between particles. (damtp.cam.ac.uk)

Elementary fermions

The Standard Model groups elementary fermions into quarks and leptons, all with spin $1/2$. There are six quark flavors—up, down, charm, strange, top, and bottom—and six lepton types: the electron, muon, tau, and their three associated neutrinos. Each family is organized into three generations. The familiar constituents of atoms belong to the first generation. (home.web.cern.ch)

Quarks participate in the strong interaction and combine into color-neutral particles. Leptons do not participate in that interaction. Charged leptons interact electromagnetically, whereas neutrinos are electrically neutral. These differences in interactions do not alter their shared fermionic statistics. (home.web.cern.ch)

Antiparticles are also fermions when their corresponding particles are fermions. For example, the positron, the electron’s antiparticle, has spin $1/2$. Particle and antiparticle have the same spin and mass, although their electric charges are opposite when nonzero. Consequently, “fermion” is not synonymous with electrically charged particle or exclusively with ordinary matter rather than antimatter. (atlas-public.web.cern.ch)

Composite fermions

Fermionic behavior is not limited to elementary particles. The proton and neutron are composite spin-$1/2$ particles whose constituent structure includes three valence quarks. They nevertheless obey fermionic exchange statistics when treated as whole particles. (damtp.cam.ac.uk)

A composite object containing an odd number of fermionic constituents behaves as a fermion; one containing an even number behaves as a boson, provided the description treats the object as a whole in a specified internal state. Neutral atoms illustrate this rule: equal numbers of protons and electrons contribute an even total, leaving neutron-number parity to determine their statistics. Helium-3 atoms are fermions, whereas helium-4 atoms are bosons. Their ground-state spins are $1/2$ and zero, respectively. (damtp.cam.ac.uk)

Statistical distribution

In statistical mechanics, the average occupation of a single-particle state of energy $E$ in an ideal fermion gas at equilibrium is

Here $T$ is temperature, $k_{\mathrm B}$ is the Boltzmann constant, and $\mu$ is the chemical potential. The denominator’s plus sign distinguishes this distribution from its bosonic counterpart and ensures that occupation remains between zero and one. (ocw.mit.edu)

At zero temperature, states below the Fermi energy are occupied and those above it are empty. At low nonzero temperatures, occupation changes mainly within an energy interval of order $k_{\mathrm B}T$ around the chemical potential. This restricted availability of states is central to understanding electrons in metals and other degenerate fermion systems. (damtp.cam.ac.uk)

Physical consequences and pairing

Electron exclusion shapes atomic shell structure and the organization of the periodic table. Because additional electrons cannot all enter the lowest complete state, they occupy different orbitals and spin states. Fermionic state filling therefore underlies important aspects of atomic structure and chemical behavior. (ocw.mit.edu)

Fermion gases also develop degeneracy pressure: compression forces particles to occupy higher-momentum states even when thermal motion is small. Electron degeneracy pressure supports white dwarfs against gravitational contraction. Fermionic degeneracy is also important in neutron stars, although their dense matter requires interactions beyond an ideal-gas description. (damtp.cam.ac.uk)

Fermions can form correlated pairs without violating exclusion. In conventional superconductivity, electrons form Cooper pairs, whose collective behavior is described by BCS theory. The paired system can exhibit bosonic collective behavior while its constituent electrons remain fermions. Pairing likewise enables helium-3 to become superfluid and allows ultracold fermionic atoms to form condensates of pairs. (nobelprize.org)

Historical development

The name honors Enrico Fermi, who developed statistical laws for particles subject to exclusion in 1926. Wolfgang Pauli had formulated the exclusion principle in 1925 while investigating atomic structure. The subsequent connection between exclusion, antisymmetric wave functions, and half-integer spin established fermions as a general class of particles rather than a category restricted to electrons in atoms. (nobelprize.org)