In 1D, any amount of disorder localizes all eigenstates — no metallic conduction is possible (Anderson 1958, Nobel 1977). Explore how random on-site potentials trap quantum probability amplitudes.
Tight-binding Hamiltonian: H = Σᵢ εᵢ|i⟩⟨i| − t Σᵢ(|i⟩⟨i+1|+h.c.) where εᵢ ∈ [−W/2, W/2] uniform random. In 1D, the localization length ξ is finite for any W>0. Localized states have exponentially decaying envelopes |ψ(x)|² ~ e^{-2|x−x₀|/ξ}. The Inverse Participation Ratio (IPR = Σ|ψᵢ|⁴) distinguishes localized (IPR ~ 1/ξ) from extended (IPR ~ 1/N) states.