Translation-character proof of Bloch theorem
= Translation-character proof of Bloch theorem
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The lattice translations commute with one another and with a lattice-periodic Hamiltonian. They can therefore be diagonalized simultaneously within each energy eigenspace. Every unitary character of a <Bravais lattice> has the form $r\mapsto e^{-ik\cdot r}$, with $k$ defined modulo the <reciprocal lattice>. Hence $T_r\psi_k=e^{-ik\cdot r}\psi_k$, so $u_k(x)=e^{-ik\cdot x}\psi_k(x)$ is lattice-periodic.