= Solution
{c}
Write $U(x)=-e\phi(x)$ for the <potential energy> of an <electron> in an <electrostatic potential> $\phi$, where $e>0$. With $G=2\pi/a$, the coupling in the <nearly-free electron model> is the <Fourier coefficient>
$$
\boxed{V_G=\frac1a\int_0^a U(x)e^{-iGx}\,dx
=-\frac ea\int_0^a\phi(x)e^{-iGx}\,dx.}
$$
If a symbol $V(x)$ already denotes the <electron> <potential energy>, as in part (c), it replaces $U(x)$ without another factor $-e$.
At the <Brillouin zone> boundary the two resonant <plane waves> have unperturbed <energy> $E_{\rm ZB}$. Their <Hamiltonian operator> in the two-state basis has diagonal entries $E_{\rm ZB}$ and off-diagonal entries $V_G,V_G^*$; its <eigenvalues> are $E_{\rm ZB}\pm|V_G|$. Thus
$$
\boxed{\Delta=2|V_G|=2\epsilon E_{\rm ZB}.}
$$
This is the local <band gap> in the two-wave <nearly-free electron model>.
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