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 Incoming links: One-particle state

P_{12}

swaps the complete particle slots, including spin. Its antisymmetric projector is

A_{12} = (I - P_{12}) /2

u, v

∣ u \land v ⟩ = \frac{∣ u ⟩ \otimes ∣ v ⟩ - ∣ v ⟩ \otimes ∣ u ⟩}{2} .

But attempting the same construction with

v = u

gives the zero vector:

A_{12} (∣ u ⟩ \otimes ∣ u ⟩) = \frac{∣ u ⟩ \otimes ∣ u ⟩ - ∣ u ⟩ \otimes ∣ u ⟩}{2} = 0.

The zero vector cannot be normalized to a quantum state. This is exclusion as cancellation of probability amplitudes, rather than a repulsive force. More generally, for normalized

u, v

the squared norm of the numerator divided by

2

1 - ∣ ⟨ u ∣ v ⟩ ∣^{2}

; the expression has unit norm only when

u, v

u, v

describes the same one-particle state and also gives zero. Pauli explicitly connects exclusion with a vanishing antisymmetric wavefunction.^[ref]

For

M

complete energy eigenstates of a one-particle Quantum Hamiltonian for identical noninteracting fermions, order their energies

ϵ_{1} \leq \dots \leq ϵ_{M}

0 \leq N \leq M

H = \sum_{α = 1}^{M} ϵ_{α} n_{α}, \sum_{α = 1}^{M} n_{α} = N, n_{α} \in {0, 1} .

The ground state occupies the

N

lowest complete one-particle states. A short minimization argument suffices: moving an occupant from a higher energy to an empty lower energy decreases the total energy. This filling describes the ground state at temperature

T = 0

; exclusion by itself also applies to excited quantum states and at positive temperature. The underlying thermal description is the Fermi-Dirac distribution.^[ref]

If the last occupied energy belongs to a group of

g

q

of them must be occupied, with all lower states filled, the ground state subspace has dimension

(q g)

. Exclusion does not select a particular basis or a particular quantum superposition in that subspace. The factor of two for an Electron spatial orbital assumes both spin projections are included; a model restricted to one spin projection has only one state per spatial orbital. A magnetic field can split the degenerate energy levels, and interactions can change the lowest-energy electron configuration. Thus this finite noninteracting filling rule is not a complete prescription for interacting chemical atoms.