Dynamics of crystal electrons

It can be shown straightforwardly, using second order perturbation theory, that if we know the energy for all n at some point k in the BZ, we can obtain the energy at nearby points. The result is 

Because of the appearance of terms q p (k) in the above expressions, this approach is known as q p perturbation theory. The quantifies defined in Eq. (3.44) are elements of a two-index matrix (n and n y, the diagonal matrix elements are simply the expectation value of the momentum operator in state We can 

The dimensions of this expression are 1/mass. This can then be directly identified as the inverse effective mass of the quasiparticles, which is no longer a simple scalar quantity but a second rank tensor. 

It is important to recogiuze that, as the expression derived above demonstrates, the effective mass of a crystal electron depends on the wave-vector k and band index n of its wavefunction, as well as on the wavefunctions and energies of all other crystal electrons with the same k-vector. This is a demonstration of the quasiparticle nature of electrons in a crystal. Since the effective mass involves complicated dependence on the direction of the k-vector and the momenta and energies of many states, it can have different magnitude and even different signs along different crystallographic directions  

We wish next to derive expressions for the evolution with time of the position and velocity of a crystal electron in the state that is, figure out its dynamics. To this end, we will need to allow the crystal momentum to acquire a time dependence, k = k(r), and include its time derivatives where appropriate. Since we are dealing with a particular band, we will omit the band index n for simplicity. 

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