Pauli exclusion principle

The Pauli exclusion principle is a fundamental concept in quantum mechanics that governs the behaviour of fermions—particles with half-integer spin. First proposed in 1925 by Austrian physicist Wolfgang Pauli and later extended within the framework of the spin–statistics theorem (1940), the principle states that no two identical fermions can simultaneously occupy the same quantum state in a quantum system. Initially formulated for electrons in atoms, it is now understood to apply universally to all fermionic particles.
The principle has far-reaching implications across physics and chemistry. It explains the structure of the periodic table, the stability of matter, the formation of electron shells in atoms, and properties of systems ranging from white dwarf stars to ordinary solids.

Fundamental Statement of the Principle

In atomic systems, electrons are described by four quantum numbers: the principal quantum number (n), the azimuthal quantum number (l), the magnetic quantum number (mₗ), and the spin projection quantum number (mₛ). The Pauli exclusion principle asserts that:

• In a multi-electron atom, no two electrons may have an identical set of all four quantum numbers.

For example, electrons occupying the same atomic orbital necessarily share the same values of n, l, and mₗ. Therefore, they must differ in their spin projection, with one electron having mₛ = +½ and the other mₛ = –½. This requirement imposes strict limits on how electrons can be arranged within atoms and underpins the familiar orbital-filling patterns of chemical periodicity.
The principle applies only to fermions. Bosons, which possess integer spin, are exempt from this restriction. Any number of identical bosons may occupy the same quantum state, as observed in coherent photons generated by lasers and in Bose–Einstein condensates.

Quantum Mechanical Interpretation

The exclusion principle arises from the symmetry properties of the many-particle wave function describing identical particles. Quantum mechanics requires that:

• Fermions possess antisymmetric total wave functions under particle exchange.
• Bosons possess symmetric total wave functions under particle exchange.

If two identical fermions were placed in the same quantum state, exchanging them would leave their configuration unchanged. However, antisymmetry demands that the wave function change sign upon exchange. The only function that is simultaneously unchanged and changes sign is a function that is identically zero. Thus, such a configuration cannot occur.
Bosons do not experience this constraint because their symmetric wave functions are unaffected by exchange, allowing multiple bosons to occupy the same state.

Fermions and Bosons

Particles are classified according to their intrinsic spin:

• Fermions: half-integer spin (½, 3⁄2, 5⁄2, …)Includes quarks, electrons, neutrinos, protons, neutrons, and certain atoms such as helium-3.
• Bosons: integer spin (0, 1, 2, …)Includes photons, W and Z bosons, and composite particles such as helium-4 atoms.

Whether an atom behaves as a fermion or a boson depends on its total spin, which is determined by the combined spins of its constituent particles. Helium-3, with spin ½, behaves as a fermion; helium-4, with spin 0, behaves as a boson.

Historical Development

Early twentieth-century atomic theory revealed that atoms with even numbers of electrons exhibited greater chemical stability than those with odd numbers. Gilbert N. Lewis, in 1916, proposed that atoms tended to hold electrons in pairs within a shell. Irving Langmuir expanded on this concept in 1919, suggesting that the periodic table arose from the arrangement of electrons in structured shells around the nucleus.
Niels Bohr refined his atomic model in 1922, noting that specific electron counts such as 2, 8, and 18 corresponded to closed shells. Yet the origin of these numbers remained unexplained.
A key insight emerged from Edmund C. Stoner’s 1924 analysis of the splitting of atomic energy levels in magnetic fields. He observed that the number of possible electron energy levels for a given principal quantum number matched the number of electrons in a closed shell. Pauli interpreted this observation as evidence that electrons must occupy distinct states defined by four quantum numbers.
To account for the required multiplicity of states, Pauli introduced a new two-valued quantum number. Shortly afterwards, Samuel Goudsmit and George Uhlenbeck identified this property as electron spin, providing crucial support for Pauli’s proposal.

Symmetry and the Structure of the Quantum State

In his Nobel lecture, Pauli emphasised that the exclusion principle is rooted in the symmetry properties of quantum states. The total wave function of two identical particles is constructed from tensor products of one-particle states. Antisymmetry under exchange means that the coefficient of a state in which both particles occupy identical quantum numbers must be zero. This mathematical requirement generalises to systems of N identical fermions: any configuration in which two positions in the many-particle state are identical yields a vanishing wave function, enforcing the exclusion principle.
These symmetry properties are basis-independent; they apply in any representation of the Hilbert space. Thus, the exclusion principle is a general and unavoidable consequence of quantum state antisymmetry.

Advanced Quantum Theory and the Spin–Statistics Connection

The spin–statistics theorem of relativistic quantum field theory provides the deeper basis for the exclusion principle:

• Particles with half-integer spin must have antisymmetric wave functions.
• Particles with integer spin must have symmetric wave functions.

This theorem arises from the behaviour of quantum fields under rotations in imaginary time, connecting spacetime structure with particle statistics.
In one-dimensional systems, even bosons may exhibit exclusion-like behaviour. A Bose gas with infinitely strong delta-function repulsive interactions becomes equivalent to a gas of free fermions; the strong repulsion prevents particle exchange, effectively imposing a fermion-like constraint. This model is described by the quantum nonlinear Schrödinger equation.

Physical and Chemical Implications

The Pauli exclusion principle is fundamental to the structure of matter. Its consequences include:

• Stability of atoms: Electron shells form because electrons cannot share identical quantum states.
• Periodicity in chemistry: Chemical behaviour arises from how electron states fill under the exclusion constraint.
• Electronic properties of solids: The structure of bands and Fermi surfaces relies on fermionic statistics.
• Degeneracy pressure: In white dwarfs and neutron stars, the exclusion principle creates pressure that resists gravitational collapse.
• Superconductivity and superfluidity: These phenomena arise when fermions pair to form composite bosons (Cooper pairs), which are not subject to the exclusion principle.