2.1.3.1 Phase Space Domain Evolution.

The equations of motion can be rewritten in a canonical form using the Hamiltonian for electrons in band $n$:

$$\mathcal{H}(\vec{r},\vec{p}_{c})=\epsilon_{n}\biggl(\frac{1}{\hbar}\bigl\{\vec{p}_{c}-q\vec{A}(\vec{r},t)\bigr\}\biggr)+q\phi(\vec{r},t),$$ (2.17)

where $\vec{p}_{c}$ stands for the quasi-momentum canonical conjugate to the electron coordinate $\vec{r}$ and $\vec{A}(\vec{r},t)$ is the vector potential. This canonical conjugate quasi-momentum differs from the quasi-momentum:

$$\vec{p}=\vec{p}_{c}-q\vec{A}(\vec{r},t).$$ (2.18)

Therefore the semiclassical evolution process conserves volume of a domain moving in the $(\vec{r},\vec{p}_{c})$ space. Since the difference between quasi-momentum and conjugate quasi-momentum is a vector independent of $\vec{p}_{c}$, the domain volumes are also conserved in $(\vec{r},\vec{k})$ space if the evolution process is described by the semiclassical equations of motion. This statement is schematically depicted in the Fig. 2.3.

Figure: Evolution of a domain in the $(\vec{r},\vec{k})$ space. The domain changes its form but its volume is the same during the semiclassical evolution process.

From this volume conservation law it follows that bands which are fully occupied do not contribute to the electrical conductivity and thus conduction is only possible for materials with partially filled bands. S. Smirnov: