@@ -115,7 +115,7 @@ Everything we did can also be applied to the higher-energy crossings seen in the
The key conclusion is that the lattice potential couples plane-wave states that differ by integer multiples of reciprocal lattice vectors. This coupling alters the band structure most strongly where the free-electron eigenenergies cross, opening up gaps of which the magnitudes are determined by the Fourier components of the lattice potential.
??? question "Suppose the lattice potential is $V(x)=A\cos(2\pi/ax)$. At what locations in the dispersion does $V(x)$ lead to the formation of gaps?"
Hint: The Fourier series of $V(x)$ is $V(x)=A(e^{2\pi/ax}+e^{-2\pi/ax})/2$, so the only non-zero Fourier components are $V_1=V_{-1} = A/2$.
Hint: The Fourier series of $V(x)$ is $V(x)=A(e^{i2\pi/ax}+e^{-i2\pi/ax})/2$, so the only non-zero Fourier components are $V_1=V_{-1} = A/2$.
## General description of a band structure in a crystal - Bloch theorem
