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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" [Broken] ):

[tex]

\hbar \frac{d}{d t} \langle T \rangle = -e E

[/tex]

(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):

[tex]

\hbar \frac{d}{d t} \langle T \rangle = -e (\vec{E} + \frac{1}{\hbar}\nabla_{\vec{k}}E(k)\times \vec{B})

[/tex]

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?

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" [Broken] ):

[tex]

\hbar \frac{d}{d t} \langle T \rangle = -e E

[/tex]

(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):

[tex]

\hbar \frac{d}{d t} \langle T \rangle = -e (\vec{E} + \frac{1}{\hbar}\nabla_{\vec{k}}E(k)\times \vec{B})

[/tex]

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?

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