The Pauli exclusion principle

The antisymmetry of the fermionic wave function under exchange has an immediate consequence.

If two identical fermions occupy the same quantum state, the wave function must be simultaneously:

ψ(1, 2) = the state with both fermions in state φ
ψ(2, 1) = the same state (particles are identical, in the same state)

But antisymmetry requires ψ(1, 2) = −ψ(2, 1).

Since ψ(1, 2) = ψ(2, 1), we need ψ = −ψ.

Which gives ψ = 0.

The state does not exist.

Two identical fermions cannot occupy the same quantum state.

You have just watched the Pauli exclusion principle emerge from ψ = −ψ → ψ = 0.

The most consequential equation in chemistry, derived from two axioms.

The Pauli exclusion principle is a theorem of the record algebra.

It follows from S (two sectors → ℤ₂), B (one unpaired element → non-trivial ℤ₂ action → antisymmetric wave function), and the structure of the Hilbert space (Chapter 1 — linearity, which makes ψ = −ψ imply ψ = 0).

Not a minor result.

The Pauli exclusion principle is what prevents all electrons from falling into the lowest energy state.

It is what gives atoms their structure.

What makes chemistry possible. What makes the periodic table work. What prevents matter from collapsing.

And it follows from the axiom: two sectors (S), one break (B), one minimum element with no partner.

You exist because of this result.

Your atoms have structure because two identical fermions cannot share a state.

Chemistry, biology, you — all downstream of the Pauli exclusion principle, which is downstream of the axiom.