“Know how to solve every problem that has been solved.” “What I cannot create, I do not understand.” — Richard Feynman

Anticommutators do the antisymmetry

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Lesson 2 of 5 standard ~6 min

The entire fermionic content of the theory lives in two anticommutation relations ({A,B} = AB + BA):

{a_{p}^{†}, a_{q}^{†}} = 0, {a_{p}, a_{q}^{†}} = δ_{pq}

Read the first one. a_p† a_q† = −a_q† a_p† — swapping the order of two creations flips the sign. That is the antisymmetry of the determinant, now an algebraic fact rather than a property you impose. And set p = q: a_p† a_p† = −a_p† a_p† forces (a_p†)² = 0 — you cannot create two electrons in the same spin-orbital. The Pauli principle is a one-line corollary.

standardMultiple choice

The anticommutator {a_p†, a_q†} = 0. Setting p = q gives (a_p†)² = 0. Which physical law is this?

Go deeper ↓Many-electron wavefunctions

The second relation, {a_p, a_q†} = δ_pq, is the bookkeeping that lets you push annihilation operators rightward until they hit the vacuum and vanish — the mechanical engine behind every matrix-element evaluation. No determinants are expanded; you just slide operators past each other, picking up signs and Kronecker deltas.

hardMultiple choice

Why are the Slater-Condon rules said to 'fall out for free' in second quantization?

Go deeper ↓Many-electron wavefunctions