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When reading about electron holes here https://en.wikipedia.org/wiki/Effective_mass_(solid-state_physics), the group velocity is introduced as the reciprocal space gradient of the dispersion relation.

The group velocity is then used in the expression $F=mdv/dt$. This makes no sense to me, shouldn't the group velocity be in position space for this to work? The left hand side of the equation seems to be in position space since it is quoted as being one of Newton's laws. Is the article I am reading hiding away some isomorphism or Fourier transform to save some digital ink? Could someone please write it in position space for me? Perhaps the time derivative commutes with the Fourier transform?

Emil
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The group velocity of a wave packet with dispersion relation $\omega=\omega(k)$ is $v_g = \frac{d\omega}{dk}$. For a quasi-free particle moving through a solid, $\omega = E/\hbar$ and $k=p/\hbar$, so $v_g = \frac{dE}{dp}$.

It's not really clear what you mean about the group velocity being in position space. In the classical physics of a free particle, $E= \frac{p^2}{2m}$, and $\frac{dE}{dp} = \frac{p}{m}=v$. This is just a semi-classical extension of that idea to a quantum mechanical system.

Albatross
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