Semiclassical dynamics of Bloch electrons

Electron wave packets and group velocities

In quantum mechanics we cant define the momentum and position of a particle simultaneously which makes the math quite complicated. To bypass this we use wave packets, which are superpositions of Bloch waves with a small range $\Delta k$ of wave vectors.

🧽 Assume: treat wave packet as point particle because electric fields and radiation have $\lambda\gg \Delta x$

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Motion in an applied electric field

💼 Case: electron in an applied electric field $\mathcal E_x$

🗒️ Note: $\hbar k$ is not the momentum of the electron, the forces due to the ions are inside $v(\vec k)$

Bloch oscillations


💼 Case: consider a ring of identical atoms with one electron in the band which starts at $k_0=0$ at $t=0$

Lets start by considering reciprocal space since our equation uses $\vec k$ and not real space