= Solution
<Bloch theorem> states that energy eigenfunctions in a <Bravais lattice> potential may be chosen as
$$
\boxed{\psi_{n\mathbf k}(\mathbf r)=e^{i\mathbf k\cdot\mathbf r}u_{n\mathbf k}(\mathbf r),\qquad u_{n\mathbf k}(\mathbf r+\mathbf l)=u_{n\mathbf k}(\mathbf r).}
$$
The wavevector is defined modulo a <reciprocal lattice> vector, so one can restrict it to the first <Brillouin zone>. This follows because lattice translations commute with the Hamiltonian and have simultaneous eigenvalues $e^{i\mathbf k\cdot\mathbf l}$.
In one dimension write $V(x)=\sum_GV_Ge^{iGx}$ with $G=2\pi j/a$ and expand a Bloch state as $\psi_k=\sum_Gc_Ge^{i(k+G)x}$. The <Schrödinger equation> becomes
$$
\left[\frac{\hbar^2(k+G)^2}{2m}-E\right]c_G+\sum_{G'}V_{G-G'}c_{G'}=0.
$$
For zero potential, the parabolas $E_G^0(k)=\hbar^2(k+G)^2/(2m)$ fold into the Brillouin zone. A weak potential shifts a nondegenerate level by $V_0$ at first order, with second-order correction $\sum_{G\ne0}|V_G|^2/(E_0^0-E_G^0)$ where the denominators stay away from zero. This expansion fails near crossings, where <degenerate perturbation theory> is required.
At the first zone boundary, put $G=2\pi/a$ and $k=G/2+q$. The nearly degenerate waves with momenta $\hbar k$ and $\hbar(k-G)$ are coupled by $V_G$. Retaining their two-dimensional subspace gives
$$
H_{\rm eff}=\begin{pmatrix}\epsilon_k+V_0&V_G\\V_G^*&\epsilon_{k-G}+V_0\end{pmatrix},\quad\epsilon_p=\frac{\hbar^2p^2}{2m}.
$$
Diagonalizing produces
$$
\boxed{E_\pm(q)=V_0+\frac{\hbar^2}{2m}\left(\frac{G^2}{4}+q^2\right)\pm\sqrt{\left(\frac{\hbar^2Gq}{2m}\right)^2+|V_G|^2}.}
$$
At $q=0$ there is a gap $\boxed{\Delta E=2|V_G|}$: the two standing-wave combinations sample the periodic potential differently. Similar Bragg degeneracies occur at $k=j\pi/a$, with gaps controlled to first order by the corresponding Fourier components. If that component vanishes, a gap need not open at first order. Between the gaps the continuous energy eigenvalues form <energy bands>, periodically represented in the reduced-zone scheme. For the first zone edge, the curvature of the upper and lower branches gives a minimum and a maximum respectively for a sufficiently weak nonzero potential, so the avoided crossing is a genuine local forbidden energy interval.
At zero temperature electrons occupy states up to the <Fermi energy>. The <Fermi surface> is the set $E_n(\mathbf k)=E_F$ separating occupied from unoccupied states within partially filled bands (Fermi points in one dimension). A partially filled band supplies arbitrarily nearby empty states, allowing occupations to respond to a weak electric field and conduct. Completely filled bands give no net current from such a redistribution; if the next empty band is separated by a positive <band gap>, the system is a band insulator. Band overlap can instead yield a metal even at an integer band filling. At finite temperature an insulator can conduct through thermally excited electrons and holes.
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