Hi,
I still have trouble thoroughly understanding why the plane wave ansatz was a reasonable solution to the equations of motion. And how can I imagine these plane waves? Are they like discrete waves with constant frequency omega, where atoms are located at the maxima and minima?
Is the group velocity similar to the drift velocity we learned in the Drude model?
Thanks in advance for your help.
Hi!
The plane wave ansatz is a reasonable solution because we are considering a chain of identical atoms and apply periodic boundary conditions. Therefore, all atoms must behave the same way, and can at most differ by a phase factor. As a consequence, the solution is wave-like.
The number of momenta allowed is indeed finite, and it is given by the number of atoms we consider in the periodic chain. For each momenta, there is only one \omega, and so all atoms move with the same frequency back and forth along the chain.
The group velocity is a quantity that depends on momentum, so it’s not the same as drift velocity, but the gif here helps to visualize what it is.
Thank you very much for the explanation. It’s all clear now.