We have used time independent perturbation theory to find the energy shifts of states and to find the change in energy eigenstates in the presence of a small perturbation. We will now consider the case of a perturbation that is time dependent. Such a perturbation can cause transitions between energy eigenstates. We will calculate the rate of those transitions.
This material is covered in Gasiorowicz Chapter 21, in Cohen-Tannoudji et al. Chapter XIII,and briefly in Griffiths Chapter 9.