Physics: Nearly-free electron model: gap opens at Brillouin zone boundary k = ±nπ/a with magnitude 2|V_n|. Tight-binding 1D: E(k) = ε₀ − 2t·cos(ka), bandwidth W = 4t. Bloch theorem: ψ_k(r) = e^(ik·r)·u_k(r) — crystal momentum k is a good quantum number. Metals have the Fermi level inside a band; insulators have it in a gap (band insulator) or correlation-driven gap (Mott insulator). Fermi velocity v_F = (1/ℏ)∂E/∂k at E_F determines transport. Zone folding in superlattices creates mini-bands with gaps at the mini-zone boundary.