$$\require{cancel}$$
$H = H_0 + H_1. \tag{583}$
Here, $$H_0$$ is a simple Hamiltonian for which we know the exact eigenvalues and eigenstates. $$H_1$$ introduces some interesting additional physics into the problem, but it 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 exactly later on), be regarded as being small compared to $$H_0$$ . Let us try to find approximate eigenvalues and eigenstates of the modified Hamiltonian, $$H_0+ H_1$$ , by performing a perturbation analysis about the eigenvalues and eigenstates of the original Hamiltonian, $$H_0$$