Pauli Exclusion Principle

Statement

No two identical fermions can occupy the same complete one-particle quantum state. Equivalently, the antisymmetric two-fermion state with both slots assigned to the same normalized one-particle state vanishes:

$$\lvert\alpha,\alpha\rangle_A=0.$$

In occupation-number language, each fermionic mode has

$$n_i\in\{0,1\}.$$

In operator language, the same statement is encoded by the fermionic creation operator identity

$$(c_i^\dagger)^2=0,$$

which follows from

$$\{c_i^\dagger,c_i^\dagger\}=0.$$

Status and Canonical Home

This Reference page is a compact lookup card. The canonical explanation, derivation from antisymmetry, atomic shell preview, Fermi-gas preview, and exercises live at Pauli Exclusion Principle.

Use this page when you need the statement, assumptions, and common failure modes quickly. Use the canonical page when you need the actual identical-particle construction.

Assumptions

• The particles are identical fermions of the same species.
• The many-particle state lies in the antisymmetric sector of the tensor-product Hilbert space or in fermionic Fock space.
• “Same state” means the same complete one-particle state, including spin or internal labels.
• The statement is kinematic: it follows from state antisymmetry, not from a repulsive potential in the Hamiltonian.
• For electrons in ordinary nonrelativistic quantum mechanics, the spin-statistics connection is taken as input from relativistic quantum theory.

Quantum-Mechanical Meaning

Pauli exclusion is the structural reason fermions fill distinct modes. It explains why electrons occupy shells, why a free Fermi gas has a filled Fermi sea at zero temperature, and why degeneracy pressure appears in dense fermionic matter.

The statement does not say two fermions can never be found near the same position. It says two identical fermions cannot have the same complete one-particle state. Two electrons can share a spatial orbital when their spin states differ, because the complete spin-orbitals are different.

Consequences

• Fermionic basis states are labeled by occupations $0$ or $1$.
• Slater determinants vanish when two columns represent the same one-particle orbital.
• Atomic shell capacities include spin multiplicity.
• The ideal Fermi gas has a Fermi surface at zero temperature.
• Stability of ordinary matter relies in part on fermionic state filling, though rigorous stability also uses Hamiltonian estimates and Coulomb inequalities.

Canonical Links

• Canonical Pauli Exclusion Page
• Symmetrization Postulate
• Fermions
• Slater Determinants
• Fermionic Fock Space
• Fermionic Anticommutation Relations
• Ideal Fermi Gas

Common Mistakes

• Treating Pauli exclusion as a force.
• Forgetting spin when deciding whether two electrons occupy the same complete state.
• Applying the principle to bosons.
• Confusing the Pauli exclusion principle with Pauli matrices.
• Saying two fermions cannot be at the same point in space without specifying spin and state preparation.
• Treating the nonrelativistic exclusion principle as the full spin-statistics theorem.

Quick Check

Why can a spatial orbital hold two electrons but not three in the simplest shell-model counting?

Solution

An electron spin has two projections. A fixed spatial orbital therefore gives two complete spin-orbitals, one for each spin projection. Pauli exclusion allows one electron per complete spin-orbital, so the spatial orbital can hold two electrons, not three.

References

• W. Pauli, “On the Connection Between the Completion of Electron Groups in the Atom and the Complex Structure of Spectra,” Zeitschrift fur Physik 31, 765-783, 1925.
• P. A. M. Dirac, The Principles of Quantum Mechanics, 4th ed., Oxford University Press, 1958.
• A. Messiah and O. W. Greenberg, “Symmetrization Postulate and Its Experimental Foundation,” Physical Review 136, B248-B267, 1964.
• A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, Dover, 2003.
• E. H. Lieb and R. Seiringer, The Stability of Matter in Quantum Mechanics, Cambridge University Press, 2010.