Dynamics of electron in crystal lattice

[paweld]

I'm not sure what is the correct equation for motion of electron in a crystal
lattice under the influence of magnetic force. On may easily proof that for
electric force the following equation holds (the proof might be found in
http://ajp.aapt.org/resource/1/ajpias/v54/i2/p177_s1" ):
$$\hbar \frac{d}{d t} \langle T \rangle = -e E$$
(T - lattice translation opertor; its eigenvalues are called usually quasimomentum;
here we have average value). Straightforward modification of above equation
which incorporate magnetic force would be (E(k) means energy):
$$\hbar \frac{d}{d t} \langle T \rangle = -e (\vec{E} + \frac{1}{\hbar}\nabla_{\vec{k}}E(k)\times \vec{B})$$
And this equation is stated by most textbooks concering solid state physics (however
without proof or with proof which is not rigorous). Does anyone know a good proof
of these equation?

 

[kanato]

If you look in appendix E of Ashcroft and Mermin, there is a rigorous proof that the expectation value of the velocity operator is $$\nabla_k E(k)$$. Since the equation you wrote is basically the Lorentz force equation, I suspect you could derive it from that and Ehrenfest's theorem.

 

[paweld]

Using Ehrenfest theorem one would obtain the following formula
$$\hbar \frac{d}{d t} \langle p\rangle = -e (\vec{E} + \frac{1}{\hbar}\nabla_{\vec{k}}E(k)\times \vec{B})$$
where p is momentum of electron and it is different from quasimomentum
which is usually denoted by k. Most textbooks gives this formula with k
not p and this is why I'm wondering why it's true.

 

[DrDu]

Ashcroft and Mermin discuss this topic quite at length and, more importantly, give many links to relevant literature.