The antisymmetric exchange statistics of identical fermions. Interchanging two one-particle labels reverses the total state, and fermionic creation operators anticommute. Occupation of a one-particle mode is therefore zero or one, giving the Pauli exclusion principle; thermal equilibrium leads to the Fermi-Dirac distribution. Color, flavor, spin and orbital factors must together have the required antisymmetry, even if one factor is symmetric.
The lowest even quantum harmonic oscillator quantum state has density , and the lowest odd quantum state has density . Since , integration by parts gives whenever the boundary terms vanish. For , this proves the displayed identity. Odd moments vanish in both quantum states by parity; even moments quantify the larger spatial spread in the odd quantum state. In particular the relative-coordinate variance is three times as large, and the odd density vanishes at particle coincidence.
The convention is useful for parabolic-band carrier densities. In three dimensions, a spin-degenerate single valley has . Other conventions omit the gamma normalization, so prefactors must be stated.
For equal-mass conduction and valence edges at and fixed , and , where . Other states pinning the chemical potential can compensate the unequal carrier charges; imposing intrinsic charge neutrality in addition would change the problem.
Expanding the Fermi denominator as an alternating exponential series gives . Thus a band with chemical potential pinned exactly at its lower edge has , rather than the classical result .
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Fermi–Dirac statistics is a quantum statistical framework that describes the distribution of particles, specifically fermions, which are particles that obey the Pauli exclusion principle. Fermions include particles like electrons, protons, and neutrons, and they have half-integer spin (e.g., 1/2, 3/2). In systems of indistinguishable fermions, no two particles can occupy the same quantum state simultaneously.
