Quantum wavefunction localization from disorder (Anderson 1958, Nobel 1977)
Anderson localization (P.W. Anderson 1958):
In a 1D tight-binding chain H = −t Σ|i⟩⟨i±1| + Σ εᵢ|i⟩⟨i| with random on-site energies
εᵢ ∈ [−W/2, W/2], all eigenstates are exponentially localized for any W > 0 in 1D and 2D.
The localization length ξ ∝ (t/W)² diverges as W→0. Above W=0 in 3D, a mobility edge
separates localized from extended states. The IPR (inverse participation ratio) = Σ|ψ|⁴
measures localization: IPR → 1 (fully localized), IPR → 1/N (extended). Each eigenstate
is shown below — color by energy, brightness by |ψ|². Increase disorder to watch states shrink.