Wave packet description of electrons in solid state physics

[taishizhiqiu]

In solid state physics, I learned that the velocity of a bloch electron is $\frac{\partial E(k)}{\partial k}$, where $E(k)$ is the energy dispersion. This formula is derived on the basis of the assumption that electrons is a wave packet of bloch state in solids.

However, I have a question concerning this statement:

I learned solid state physics three years ago and I am now a Ph.D. student. However, I still cannot convince me of the original statement because I know a wave packet BREAKS DOWN with time evolution. I can't imagine this formula can describe long time evolution of electron states.

Can anyone give me some confidence of this formula?

 

[DrDu]

This is the velocity for a very broad wavepacket (in real space), i.e. a very localized state in k-space. (If not, for which k-state would you have to take the derivative?) But an infinitely broad wavepacket can't disperse any further.