Consider the following problem, which is very common. The Hamiltonian of some quantum mechanical system is written

(7.1) |

Here, is a simple Hamiltonian for which we know the exact eigenvalues and eigenstates. introduces some interesting additional physics into the problem, but it is sufficiently complicated that, when we add it to , we can no longer find the exact energy eigenvalues and eigenstates. However, can, in some sense (which we shall specify more exactly later on), be regarded as small compared to . Let us try to find approximate eigenvalues and eigenstates of the modified Hamiltonian, , by performing a perturbation expansion about the eigenvalues and eigenstates of the original Hamiltonian, .

We shall start, in this chapter, by considering *time-independent perturbation theory* [98],
in which the modification to the Hamiltonian,
, has no explicit
dependence on time. It is usually assumed that the unperturbed
Hamiltonian,
, is also time independent.