Appendix B

Perturbation

Simple physical systems such as simple harmonic oscillator and hydrogen atoms can be solved exactly where the Hamiltonian is simple enough to generate exact eigenvalues. In general, the Hamiltonian is very complicated for systems, such as many electron atoms, semiconductor nanostructures, multiple quantum wells, and quantum dots. Solving the Schrödinger equation for such complicated systems is difficult to handle, and, therefore, one needs to make several approximations to reach a reasonable answer. One of these approximations is the perturbation theory. In this appendix we treat the time-independent perturbation (stationary) approximation, which is widely used in many systems such as solid state physics. To understand this approximation, one needs to define a physical system and isolate the main effects that are responsible for the main features of the system. Once these features are understood, the finer details can be discussed by considering the less-important effects that were neglected in the first approximation. Treating these secondary effects can be performed using the perturbation theory. Thus, the Hamiltonian of the system can be presented in the following form:

where H0 is the unperturbed Hamiltonian with known eigenvectors and eigenvalues, and H1 is the perturbation that describes the secondary effects in the system. The problem now is to ...