= Solution
The <Bloch theorem> says that for a translation-invariant lattice <Hamiltonian>, eigenstates can be chosen as $\psi_k(x+a)=e^{ika}\psi_k(x)$, or $\psi_k(x)=e^{ikx}u_k(x)$ with $u_k$ periodic of period $a$. The localized site orbitals are translates, $\phi_n(x)=\phi_0(x-na)$ and $\chi_n(x)=\chi_0(x-na)$. In the <tight-binding model>, neglect intersite overlap and use
$$
\psi(x,t)=\sum_n\bigl(b_n(t)\phi_n(x)+c_n(t)\chi_n(x)\bigr).
$$
Projecting the <Schrödinger equation> onto each localized orbital gives its onsite energy times its own coefficient, plus the four stated neighbor matrix elements. Thus
$$
\boxed{\begin{aligned}
i\hbar\dot b_n&=E_0b_n-A(b_{n-1}+b_{n+1}+c_{n-1}+c_{n+1}),\\
i\hbar\dot c_n&=E_1c_n-A(c_{n-1}+c_{n+1}+b_{n-1}+b_{n+1}).
\end{aligned}}
$$
For the Bloch amplitudes, set $z=\cos ka$. The coefficient eigenproblem is
$$
\begin{pmatrix}E_0-2Az&-2Az\\-2Az&E_1-2Az\end{pmatrix}\binom BC=E\binom BC,
$$
so the two <energy bands> are
$$
\boxed{E_\pm(k)=\frac{E_0+E_1}{2}-2A\cos ka
\pm\frac12\sqrt{(E_1-E_0)^2+16A^2\cos^2ka}.}
$$
A first <Brillouin zone> is $-\pi/a\le k<\pi/a$; wavevectors differing by $2\pi/a$ are equivalent. Put $\Sigma=E_0+E_1$, $D=\sqrt{(E_1-E_0)^2+16A^2}$. Each band is nonincreasing as a function of $z\in[-1,1]$, because $dE_\pm/dz=-2A\pm8A^2z/\sqrt{(E_1-E_0)^2+16A^2z^2}\le0$. Hence
$$
\boxed{E_{\pm,\min}=\Sigma/2-2A\pm D/2,\qquad
E_{\pm,\max}=\Sigma/2+2A\pm D/2,\qquad
\Delta E=E_{+,\min}-E_{-,\max}=D-4A.}
$$
The extrema occur at $k=0$ and at the zone edge; they may be nonunique when $E_0=E_1$, in which case the gap vanishes. In the trial wavefunction, replacing $x$ by $x+a$ and reindexing $n$ produces exactly the factor $e^{ika}$, verifying <Bloch theorem> directly.
An <electrical conductor> has available nearby unoccupied states in a partially filled band or overlapping bands. A <band insulator> has filled lower bands separated from empty higher bands by a large gap. A <semiconductor> has the same filled-band structure with a smaller gap, so thermal excitation or doping supplies mobile electrons and holes. Band filling and carrier availability, not just the existence of two bands, determine which description applies.
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