@@ -191,10 +191,8 @@ $\rightarrow \rho_{\rm R}(k)=\frac{L}{2\pi}$, which is lower than for the case o
## Summary
* By using plane waves in real space as an Ansatz we found all normal modes and eigenvectors
* Dispersion relation of a system with period $a$ in real space is periodic with period $2\pi/a$ in $k$-space
* Computing dispersions explains all the problems we listed before (need for cutoff, lack of scattering with every single atom, existence of insulators).
* Electrons and phonons have a complicated nonlinear relation between momentum and velocity (**group velocity**), effective mass, and density of states
* By using plane waves in real space as an Ansatz, we found all normal modes and eigenvectors. (Just like in the case of 1 degree of freedom per unit cell).
* The dispersion relation of a system with period $a$ in real space is periodic with period $2\pi/a$ in $k$-space
* In a system with more than one degree of freedom per unit cell we need to consider independent amplitudes for each degree of freedom, and get multiple bands.
