2.1.1.1 Equation of Motion

The motion in real space is described by the following equation:

$$\frac{d\vec{r}}{dt}=\vec{v}_{n}(\vec{k}),$$ (2.7)

where $\vec{v}_{n}(\vec{k})$ is the quantum mechanical average of the velocity operator over an electron state represented as a Bloch wave packet for band index n:

$$\vec{v}_{n}(\vec{k})=\int \psi_{n\vec{k}}^{*}(\vec{r})\hat{\vec{v}}\psi_{n\vec{k}}(\vec{r})\,d\vec{r},$$

$$\hat{\vec{v}}=-i\frac{\hbar}{m_{0}}\nabla_{\vec{r}}.$$ (2.8)

Using Bloch's wave function with the periodic amplitude $u_{n\vec{k}}(\vec{r})$ and the ortho-normality condition, equation (2.8) gives:

$$\vec{v}_{n}(\vec{k})=\frac{\hbar\vec{k}}{m_{0}}-\frac{i\hbar}{m_{0}}\int u_{n\vec{k}}^{*}(\vec{r})\nabla u_{n\vec{k}}(\vec{r})\,d\vec{r}.$$ (2.9)

The important difference of this expression from that of the free electron theory consists in the second term which is in general not equal to zero. Thus $\vec{v}_{n}(\vec{k})\neq\frac{\hbar\vec{k}}{m_{0}}$ and the directions of the quantum mechanical average of the velocity and quasi-momentum do not coincide in general as it is shown in Fig. 2.1.

Figure: Momentum and quasi-momentum differ by the vector $\vec {D}=\frac{i\hbar}{m_{0}}\int u_{n\vec {k}}^{*}(\vec {r})\nabla u_{n\vec {k}}(\vec {r})\,d\vec {r}$.

In the limiting case of free electrons the amplitude $u_{n\vec{k}}$ does not depend on position and thus the second term in (2.9) vanishes. In this case $\vec{v}_{n}(\vec{k})=\frac{\hbar\vec{k}}{m_{0}}$ as in the theory of free electrons. This is another consequence of the quantum mechanical aspect of the semiclassical model, which uses the concept of quasi-momentum.

Note that the quasi-momentum does not coincide with the momentum of a Bloch electron. The point is that the momentum $\vec{P}_{n}(\vec{k})=m_{0}\vec{v}_{n}(\vec{k})$ of an electron changes under the action of the total force which also includes the periodic potential. The quasi-momentum $\vec{p}=\hbar\vec{k}$ of an electron only changes under the action of external fields and the periodic field of a crystal does not change the quasi-momentum of an electron.

S. Smirnov: