Pauli exclusion principle

The Pauli exclusion principle is the phenomenon where two fermions in the same system cannot simultaneously occupy the same state. It is a consequence of the antisymmetry of a fermion system's wavefunction.

Origin

The principle is directly derived from the antisymmetry of fermion states. Consider a quantum numbers which we use to uniquely identify states. Call aa and bb two possible values for this number1. Then, a fermion can be in two distinct states: ∣a⟩\ket{a} or ∣b⟩\ket{b}. Now say we have two coupled fermions. Their mixed state ∣ψ⟩\ket{\psi} must be a linear combination (due to indistinguishability) and specifically a difference (due to antisymmetry) of pure states:

∣ψ⟩=∣ab⟩−∣ba⟩\ket{\psi} =\ket{ab} -\ket{ba}

If ∣a⟩\ket{a} and ∣b⟩\ket{b} represent different states, this is a valid mixed state. However, if both fermions are in the same state (a=ba=b), we have

∣ψ⟩=∣aa⟩−∣aa⟩=0\ket{\psi} =\ket{aa} -\ket{aa} =0

The mixed state vanishes. Thus, two coupled fermions cannot coexist in the same state.

Footnotes

  1. These need not be the only values for the number, there may be more or even infinite, but they need to be distinct. ↩