The Pauli exclusion principle states that no two identical fermions can occupy the same single-particle quantum state. It is a direct consequence of the antisymmetry of the fermionic wavefunction under particle exchange.

For two identical fermions the total state is antisymmetric,

\[ \psi(1,2) = -\,\psi(2,1). \]

If both particles carried identical quantum numbers, i.e. state \(1 = \) state \(2\), then \(\psi=-\psi=0\): the configuration has zero amplitude and is forbidden. Equivalently, occupation numbers of any mode are restricted to \(n=0\) or \(1\).

The principle is the historical reason the color charge quantum number had to be introduced. The baryons

\[ \Omega^-=sss,\qquad \Delta^{++}=uuu \]

are spin-\(\tfrac32\) ground states: three identical quarks with aligned spins in a symmetric spatial state. Without an extra label the total wavefunction would be symmetric, violating the exclusion principle. Endowing each quark with one of three colors and demanding the physical baryon be antisymmetric in color,

\[ |B\rangle \propto \epsilon^{abc}\,q_a q_b q_c , \]

makes the full wavefunction antisymmetric and consistent with Fermi statistics. This is elaborated in Deepseek_color_charge_requirement.

The same principle underlies the shell structure of atoms (and hence the periodic table), the stability of matter, degeneracy pressure in white dwarfs and neutron stars, and the very existence of a Fermi surface in metals.