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.
Using the uncertainty principle $\Delta x\sim 1/\Delta k$ + we our wave packet to be sharp
$$ \Delta k\ll 1/a \quad \text{where $1/a$ is a $\sim$ size of a Brillouin zone} $$
We can flip this relation to $\Delta x \gg a$ so the wave packet is spread over many unit cells
🧽 Assume: treat wave packet as point particle because electric fields and radiation have $\lambda\gg \Delta x$
We can write its group velocity in component form as#
$$ v_x=\frac{1}{\hbar} \frac{\partial E}{\partial k_x} \quad v_y=\frac{1}{\hbar} \frac{\partial E}{\partial k_y} \quad v_z=\frac{1}{\hbar} \frac{\partial E}{\partial k_z} $$
where we used the quantum-mechanical expression $\omega=E/\hbar$ for the angular frequency
We can rewrite these expressions as
$$ \boxed{v(\vec k)=\frac{1}{\hbar} \frac{\partial E(\vec k)}{\partial \vec k} \qquad \text{or} \qquad v(\vec k) = \hbar ^{-1} \vec \nabla _k E(\vec k)} $$
🗒️ Note: electron velocity around minima/maxima points radially outwards/inwards
The group velocity is an oscillatory function of the wave vector
💼 Case: electron in an applied electric field $\mathcal E_x$
The electron gains energy at rate
$$ \dot E=\text{[force]$\cdot$[velocity]}=-e\mathcal E_x v_x $$
We can re-write this as
$$ \dot E=\frac{\partial E}{\partial k_x}\dot k _x = \hbar \dot k_x v_x= -e\mathcal E_x v_x $$
This gives the relation $\hbar \dot k_x = -e\mathcal E_x$ which we can generalise as
$$ \boxed{\hbar \dot {\vec k} =-e\vec {\mathcal E}} $$
🗒️ Note: $\hbar k$ is not the momentum of the electron, the forces due to the ions are inside $v(\vec k)$
Integrating our previous equation for a constant electric field gives us:
$$ \vec k = \vec k_0 - \frac{e \vec {\mathcal E} t}{\hbar} $$
where $\vec k_0$ is the initial wave vector of the electron, hence $\vec k-\vec k_0 \propto -t$
💼 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