2.1.2.2 Force Expression

The external force in (2.14) includes both electric and magnetic forces. It represents their vector sum:

$$\vec{F}(\vec{r},\vec{k},t)=q\biggl(\vec{E}(\vec{r},t)+\vec{v}_{n}(\vec{k})\times\vec{B}(\vec{r},t)\biggr),$$ (2.15)

where $q$ is the particle charge. In the case of a time-independent electric field such an expression for the external force can be justified by the energy conservation law. If the electric field is fixed and $\phi(\vec{r})$ is the electrostatic potential then the Bloch wave packets travel in such a way that $\epsilon_{n}(\vec{k}(t))+q\phi(\vec{r}(t)) = \mathrm{const}$. The time derivative of this expression vanishes and taking (2.13 into account results in

$$\vec{v}_{n}(\vec{k})\cdot(\hbar\Dot{\vec{k}}+q\nabla\phi(\vec{r}))=0.$$ (2.16)

This is the equation of motion in quasi-momentum space with the electric force according to (2.15). However, (2.16) is not a unique expression for the energy conservation as the expression $\hbar\Dot{\vec{k}}+q\nabla\phi(\vec{r})+\vec{f}$, where $\vec{f}$ is a vector perpendicular to the average electron velocity $\vec{v}_{n}$, can also fulfill this requirement. It is possible to show that the only possible additional term is $\vec{f}=\vec{v}_{n}(\vec{k})\times\vec{B}$, the Lorenz force, and that (2.15) is valid for time dependent external fields [15]. S. Smirnov: