= Solution
In the <nearly-free electron model>, the free parabola $\varepsilon_k^0=\hbar^2k^2/(2m)$ is coupled to copies shifted by reciprocal wavevectors $G=2\pi/a$. The first degeneracy is at $k=\pm\pi/a$, where the counterpropagating waves have equal <kinetic energy>. Retaining their coupling gives two branches
$$
E_\pm(k)=V_0+\frac{\varepsilon_k^0+\varepsilon_{k-G}^0}{2}\pm\sqrt{\left(\frac{\varepsilon_k^0-\varepsilon_{k-G}^0}{2}\right)^2+|V_G|^2}.
$$
Consequently a nonzero Fourier component $V_G$ opens a gap $\boxed{\Delta E=2|V_G|}$ at the zone edge. The dispersion remains approximately free-electron-like away from the crossing, but the branches avoid crossing and their slopes flatten at the band edges. The horizontal axis can be labeled either <wavevector> $k$, with edges at $\pm\pi/a$, or physical <momentum> $p=\hbar k$, with edges at $\pm\pi\hbar/a$.
At the edge, an appropriate choice of spatial origin makes $V_G$ real. The eigenstates are equal-weight standing waves. In a normalization length $L$ containing an integer number of periods, representative real forms are
$$
\boxed{\psi_c(x)=\sqrt{\frac2L}\cos\frac{\pi x}{a},\qquad \psi_s(x)=\sqrt{\frac2L}\sin\frac{\pi x}{a}.}
$$
They come from the sum and difference of $e^{i\pi x/a}$ and $e^{-i\pi x/a}$, up to a global phase. Their densities have maxima and nodes displaced by $a/2$ relative to one another. Both <wavefunctions> change sign under translation by $a$, as required by their zone-edge Bloch phase, while their densities are lattice-periodic.
\Image[/past-exam-of-the-quantum-condensed-matter-course-of-the-university-of-cambridge/2013-zone-edge-states.png]
{title=An illustrative nearly-free-electron gap and its two zone-edge standing waves}
For the figure's choice $V(x)=2V_G\cos(2\pi x/a)$ with $V_G<0$, the potential minima occur at $x=na$. The cosine state puts density there and is lower, while the sine state puts a node there and is higher. Reversing the potential's phase relative to the coordinate origin exchanges that assignment. Since the question supplies only the spatial period, it fixes the zone-edge locations and qualitative standing-wave structure, but not a numerical gap or an origin-independent choice of which drawn sine/cosine is lower.
Back to article page
