Crystal electrons in an electric field

Taking a derivative of the velocity in state with respect to time, and using the 

We should note two important things. First, the identification of the time derivative of k with the external force at this point cannot be considered a proper proof it is only an inference from the dimensions of the different quantities that enter into Eq. (3.52). In particular, so far we have discussed the dynamics of electrons in the crystal without the presence of any external forces, which could potentially change the wavefunctions and the eigenvalues of the single-particle hamiltonian we examine this issue in detail in the next section. Second, to the extend that the above relation actually holds, it is a state-independent equation, so the wave-vectors evolve in the same manner for all states  

We consider next what happens to crystal electrons when they are subjected to a constant external electric field E, which gives rise to an electrostatic potential 4 (r) = —E r. External electric fields are typically used to induce electron transport, for instance in electronic devices. The periodicity of the crystal has profound effects on the transport properties of electrons, and influences strongly their response to external electric fields. 

For the present discussion we will include the effects of the external held from the beginning, starting with the new hamiltonian in the presence of the external electric held, which is 

The hrst new term in this hamiltonian has non-vanishing matrix elements between states of the same k only. To show this, consider the matrix element of this term 

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