Introduction: Crystals and Periodicity
In crystals, atoms are arranged periodically: \(V(x + a) = V(x)\). Bloch's theorem tells us the special form that wavefunctions must take in periodic potentials.
Bloch's Theorem
If \(V(x + a) = V(x)\), then eigenstates can be written as:
\[\psi_k(x) = e^{ikx}u_k(x)\]where \(u_k(x + a) = u_k(x)\) is periodic with the lattice.
Alternative Form
Bloch's theorem can also be stated as:
\[\psi(x + a) = e^{ika}\psi(x)\]The wavefunction picks up a phase factor when translated by one lattice spacing.
The Crystal Momentum
\(\hbar k\) is called the crystal momentum (or quasi-momentum). It's defined modulo \(2\pi/a\)—values differing by a reciprocal lattice vector are equivalent. This leads to the Brillouin zone picture.
The Quantum Connection
Bloch's theorem is the foundation of solid-state physics. It explains why electrons in metals can flow despite the "obstacle course" of atomic nuclei—they form extended Bloch waves. The energy spectrum forms bands, not discrete levels, leading to the distinction between conductors, insulators, and semiconductors.