Introduction

Consider the following very commonly occurring problem. The Hamiltonian of a quantum mechanical system is written

$$H = H_0 + H_1.$$ (833)

Here, $H_0$ is a simple Hamiltonian whose eigenvalues and eigenstates are known exactly. $H_1$ introduces some interesting additional physics into the problem, but is sufficiently complicated that when we add it to $H_0$ we can no longer find the exact energy eigenvalues and eigenstates. However, $H_1$ can, in some sense (which we shall specify more precisely later on), be regarded as being small compared to $H_0$. Can we find approximate eigenvalues and eigenstates of the modified Hamiltonian, $H_0+H_1$, by performing some sort of perturbation expansion about the eigenvalues and eigenstates of the original Hamiltonian, $H_0$? Let us investigate.

Incidentally, in this section, we shall only discuss so-called time-independent perturbation theory, in which the modification to the Hamiltonian, $H_1$, has no explicit dependence on time. It is also assumed that the unperturbed Hamiltonian, $H_0$, is time-independent.