In my condensed matter course my professor writes: "The lattice potential causes scattering that connects k states, and when these states are degenerate a gap opens. For example, the cos(2π/a) term connects the degenerate states at k = −π/a and +π/a (i.e. ⟨ψk=−π/a|Vˆ|ψk=+π/a⟩ ≠ 0), and so a gap opens at these k values." Where by lattice potential he's referring to the 1-D nearly-free electron lattice potential described by a Fourier series:

$V(x) = V_0 + 2V_1cos(\frac{2\pi}{a}x) + 2V_2cos(\frac{4\pi}{a}x) + ...$

which perturbs the eigenenergies in accordance with non-degenerate perturbation theory.

However I don't understand why the scattering "connects" k states somehow and why this leads to a band gap. It was mentioned that the sign of the second order perturbation to the eigenenergies is the culprit for this effect but again, I don't understand why. Any help in understanding this would be greatly appreciated.

In my condensed matter course my professor writes: "The lattice potential causes scattering that connects k states, and when these states are degenerate a gap opens. For example, the $\cos(2\pi/a)$ term connects the degenerate states at $k = −\pi/a$ and $+\pi/a$ (i.e. $⟨\psi_{k=-\pi/a}|\hat{V}|\psi_{k=+\pi/a}\rangle \neq 0$), and so a gap opens at these k values." Where by lattice potential he's referring to the 1-D nearly-free electron lattice potential described by a Fourier series:

$V(x) = V_0 + 2V_1cos(\frac{2\pi}{a}x) + 2V_2cos(\frac{4\pi}{a}x) + ...$

which perturbs the eigenenergies in accordance with non-degenerate perturbation theory.

However I don't understand why the scattering "connects" k states somehow and why this leads to a band gap. It was mentioned that the sign of the second order perturbation to the eigenenergies is the culprit for this effect but again, I don't understand why. Any help in understanding this would be greatly appreciated.

In my condensed matter course my professor writes: "The lattice potential causes scattering that connects k states, and when these states are degenerate a gap opens. For example, the cos(2π/a) term connects the degenerate states at k = −π/a and +π/a (i.e. ⟨ψk=−π/a|Vˆ|ψk=+π/a⟩ ≠ 0), and so a gap opens at these k values." Where by lattice potential he's referring to the 1-D nearly-free electron lattice potential described by a Fourier series:  $V(x) = V_0 + 2V_1cos(\frac{2\pi}{a}x) + 2V_2cos(\frac{4\pi}{a}x) + ...$ which perturbs the eigenenergies in accordance with non-degenerate perturbation theory.

However I don't understand why the scattering "connects" k states somehow and why this leads to a band gap. It was mentioned that the sign of the second order perturbation to the eigenenergies is the culprit for this effect but again, I don't understand why. Any help in understanding this would be greatly appreciated.

In my condensed matter course my professor writes: "The lattice potential causes scattering that connects k states, and when these states are degenerate a gap opens. For example, the cos(2π/a) term connects the degenerate states at k = −π/a and +π/a (i.e. ⟨ψk=−π/a|Vˆ|ψk=+π/a⟩ ≠ 0), and so a gap opens at these k values." Where by lattice potential he's referring to the 1-D nearly-free electron lattice potential described by a Fourier series: 

$V(x) = V_0 + 2V_1cos(\frac{2\pi}{a}x) + 2V_2cos(\frac{4\pi}{a}x) + ...$ 

which perturbs the eigenenergies in accordance with non-degenerate perturbation theory.

However I don't understand why the scattering "connects" k states somehow and why this leads to a band gap. It was mentioned that the sign of the second order perturbation to the eigenenergies is the culprit for this effect but again, I don't understand why. Any help in understanding this would be greatly appreciated.