Anderson localization (1958): adding any amount of disorder to a 1D quantum chain localizes all wavefunctions exponentially. The tight-binding Hamiltonian is:
This recurses as a transfer matrix: [ψₙ₊₁, ψₙ]ᵀ = Mₙ [ψₙ, ψₙ₋₁]ᵀ where Mₙ = [[E−εₙ, −1],[1, 0]]. The product M_N···M_1 has the largest Lyapunov exponent γ = lim_{N→∞} (1/N)log‖∏Mₙ‖ > 0 for all W > 0, giving localization length ξ = 1/γ. The wavefunction |ψ(x)|² decays as exp(−2|x−x₀|/ξ). Near band center (E=0) and weak disorder, ξ ~ (6/W²) for the Anderson model. In d≥3, a mobility edge separates localized from extended states — one of condensed matter's great unsolved problems.