= Solution
A <Bravais lattice> is the integer span of three linearly independent primitive vectors. Its <reciprocal lattice> consists of vectors $G$ satisfying $G\mathbin\cdot r\in2π\mathbb Z$ for every lattice vector $r$.
Translations by lattice vectors commute with the <Hamiltonian>. Their simultaneous eigenvalues have the form $e^{ik\cdot r}$, so an energy eigenfunction may be chosen with
$$
ψ_k(x+r)=e^{ik\cdot r}ψ_k(x).
$$
Thus $ψ_k=e^{ik\cdot x}u_k$ with lattice-periodic $u_k$, which is <Bloch theorem>. Wavevectors differing by a reciprocal-lattice vector describe the same translation character, and the <Brillouin zone> is a fundamental Wigner-Seitz cell of the reciprocal lattice.
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