Slater determinants: antisymmetry for free

Two electrons are not 'electron 1 here, electron 2 there' — they are identical fermions, so the wavefunction must flip sign when you swap them: ψ(r₁,r₂) = −ψ(r₂,r₁).

A plain product φ(r₁)φ(r₂) is symmetric, so it fails. Writing the spin-orbitals as a determinant fixes it automatically: swapping two electrons swaps two rows and flips the sign, and placing two electrons in the same spin-orbital makes two identical columns, so the determinant vanishes. That vanishing is the Pauli exclusion principle.