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Bosons have symmetric many-particle states, so any number may occupy the same one-particle state. At low temperature, an ideal Bose gas therefore places a macroscopic population in its lowest one-particle state; below the critical temperature this is Bose-Einstein condensation.

Fermions have antisymmetric many-particle states and obey the Pauli exclusion principle, so each one-particle state can hold at most one fermion of each available internal label. At zero temperature, a noninteracting Fermi gas fills distinct states in increasing order of energy up to the Fermi energy. Its many-particle ground state is thus the filled Fermi sea.

The zero-temperature internal energy isUsing part (i),Consequently the adiabatic equation of state isThe pressure is degeneracy pressure: it remains nonzero at zero temperature because of the Pauli exclusion principle. By comparison, a classical ultrarelativistic ideal gas at fixed temperature obeys $pV=N{k}_{B}T$, so its corresponding volume exponent is one and its pressure vanishes as $T\to 0$.

Assume $eB>0$; reversing the sign only exchanges the spin labels. Since $\mathbf{B}\cdot \mathbf{\sigma }=B{\sigma }_3$, the Pauli term shifts the spinless energies by $\pm \hslash {\omega }_c/2$. ThusAfter collecting equal energies, the spin splitting of Landau levels isThe level $\ell =0$ contains only $(n,sigma)=(0,-1)$ and has degeneracy $D$. Every level $\ell \ge 1$ contains $(\ell ,-1)$ and $(\ell -1,+1)$ and has degeneracy $2D$.

For $N$ noninteracting electrons, the Pauli exclusion principle fills these states from the bottom. If $0\le N\le D$, every electron fits in the zero-energy level andFor $N>D$, writeThen the first $k$ positive levels are full and $r$ states in level $k+1$ are occupied. The ground-state energy isEquivalently, onThe graph is continuous and piecewise linear: it is flat up to $N=D$, then has slopes $\hslash {\omega }_c,2\hslash {\omega }_c,3\hslash {\omega }_c,\dots$, with kinks at

The invariant colour state of three fundamental quarks is the normalized completely antisymmetric tensor ${ϵ}_{abc}/\sqrt{6}$. With this colour factor, the Pauli exclusion principle requires the remaining spatial-spin-flavour wavefunction to be symmetric. Colour and approximate flavour symmetry are distinct SU(3) actions.