Week 12: Schrödinger equation as a matrix, the tight-binding chain

Schrödinger's equation as a matrix: a tight-binding chain (Datta Part B, Ch. 17, Figs. 17.5–17.6)

N atoms spaced by a, one orbital each: the Hamiltonian is an N×N matrix with ε + U_n on the diagonal and t between neighbours (Eq. 17.10). Choosing t = −ℏ²/2ma² and ε = E_c − 2t makes the band ε + 2t cos ka match the parabola near k = 0 (Eqs. 17.15–17.16). Diagonalise it live, pick an eigenstate, add a potential.

Number of atoms N Eigenstate shown (1 = lowest) Boundary Potential U(x) on the diagonal Lattice spacing a (nm), m = m₀

t = −ħ²/2ma² (Eq. 17.16a)–

Band: ε + 2t cos ka spans–

Selected level E (matrix) · box/ring formula–

Continuum particle in a box, L = (N+1)a–