Wave localization by disorder — quantum diffusion suppressed by interference
In a 1D disordered lattice, all eigenstates are exponentially localized for any nonzero disorder W:
|ψ(x)| ~ exp(−|x−x₀|/ξ)
Localization length: ξ ≈ (t/W)² · 24 (Born approx). In 3D there is a mobility edge: extended states for low energy, localized for high disorder.
IPR = Σ|ψ|⁴ measures localization: IPR→1/N for extended, IPR→1 for fully localized.