Bloch's theorem

Bloch's theorem answers a puzzle that should worry you: a metal is packed with ions, a dense forest of scattering centres, so why do electrons sail through and conduct at all rather than getting bounced to a standstill? The answer is that a perfectly periodic potential does not scatter an electron randomly. Instead the electron settles into a special travelling wave, a plane wave gently modulated with the rhythm of the lattice, and glides through undisturbed.

Precisely, for any potential with the lattice periodicity, V(r + R) = V(r), the energy eigenstates of the Schrodinger equation can be chosen in the Bloch form psi_k(r) = exp(i k . r) u_k(r), where u_k(r + R) = u_k(r) has the full periodicity of the lattice. Equivalently, translating by a lattice vector only multiplies the wavefunction by a phase: psi_k(r + R) = exp(i k . R) psi_k(r). The label k is the crystal momentum (or Bloch wavevector), and the theorem follows because the Hamiltonian commutes with the lattice translation operators, so they can be simultaneously diagonalized. The allowed energies organize into bands E_n(k), indexed by a band number n and a wavevector k in the Brillouin zone.

This single result is the doorway to all of band theory. One honest subtlety: the crystal momentum hbar k is not the true mechanical momentum, and it is conserved only modulo a reciprocal lattice vector G (this is why phonon-assisted or umklapp processes exist). The theorem is also an idealization; it assumes a perfect, infinite, static periodic lattice. Real resistance comes precisely from what breaks that periodicity: impurities, defects, boundaries, and thermal phonons.