It is assumed that these states, and their associated eigenvalues, are known. We also expect the states to be orthonormal, and to form a complete set.

Let us now try to solve the modified energy eigenvalue problem

It follows from (877) that

where or . Equations (875), (876), (878), (879), and the orthonormality condition

(880) |

where

(882) | |||

(883) | |||

(884) |

Here, use has been made of the fact that is an Hermitian operator.

Consider the special (but not uncommon) case of a perturbing Hamiltonian
whose diagonal matrix elements are zero, so that

(885) |

(886) |

(887) |

(888) |

(889) | |||

(890) |

Note that causes the upper eigenvalue to rise, and the lower to fall. It is easily demonstrated that the modified eigenstates take the form

(891) | |||

(892) |

Thus, the modified energy eigenstates consist of one of the unperturbed eigenstates, plus a slight admixture of the other. Now our expansion procedure is only valid when . This suggests that the condition for the validity of the perturbation method as a whole is

(893) |