TABLE OF CONTENTS In the chapters on transport and optical properties we have seen that scattering of electrons from one state to another is critical in the understanding of almost all physical properties of materials. Optical and transport phenomena are linked to scattering processes. We have used the Fermi golden rule to evaluate several important scattering processes in materials. In this appendix we will give a derivation of this important equation in quantum physics. The general Hamiltonian of interest is of the form
| (C.1) |  |
where H 0 is a simple Hamiltonian with known solutions
| (C.2) |  |
and E k , u k are known. For example the solutions to H would be the bandstructure of electrons in a crystal. In the absence of H ? (t), if a particle is placed in a state u k , it remains there forever. The effect of H ? is to cause time-dependent transitions between the states u k . The time-dependent Schr dinger equation is
| (C.3) |  |
The approximation will involve expressing ? as an expansion of the eigenfunctions u n exp( ?i E n t/?) of the unperturbed time-dependent functions
| (C.4) |  |
The time-dependent problem is solved when the coefficients a n (t) are known. In the spirit of the perturbation approach, these coefficients are determined to different orders. Hopefully, the first- or second-order terms would suffice and higher-order terms would be negligible.
Substituting for ? (given by Eq. C.4) in Eq. C.3, using...
