= Bloch's theorem
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= Bloch theorem
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Bloch's theorem describes <energy eigenstates> of a <Quantum Hamiltonian> invariant under <Bravais lattice> translations. They can be chosen in the form
$$
\psi_{n\mathbf k}(\mathbf r)=e^{i\mathbf k\cdot\mathbf r}u_{n\mathbf k}(\mathbf r),\qquad u_{n\mathbf k}(\mathbf r+\mathbf R)=u_{n\mathbf k}(\mathbf r).
$$
Translation by a lattice vector $\mathbf R$ therefore multiplies the state by $e^{i\mathbf k\cdot\mathbf R}$. The <wavevector> $\mathbf k$ labels <crystal momentum> and is equivalent modulo a <reciprocal lattice> vector.
This converts a problem over an entire ideal crystal into a family of cell-periodic problems, organized by the <Bloch Hamiltonian>. Their <energy eigenvalues> form <energy bands> over a <Brillouin zone>. The periodic function carries the microscopic structure missing from a bare <plane wave>. Boundaries, defects and disorder break exact translation invariance, so the theorem's states serve as a basis or approximation there rather than automatic exact eigenstates.
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