Non-Degenerate Perturbation Theory

(7.19) |

where runs from 1 to . The eigenkets are orthonormal, and form a complete set. Let us now try to solve the energy eigenvalue problem for the perturbed Hamiltonian:

We can express as a linear superposition of the unperturbed energy eigenkets,

(7.21) |

where the summation is from to . Substituting the previous equation into Equation (7.20), and right-multiplying by , we obtain

where

(7.23) |

Let us now develop our perturbation expansion. We assume that

(7.24) |

for all , where is our expansion parameter. We also assume that

(7.25) |

for all . Let us search for a modified version of the th unperturbed energy eigenstate for which

(7.26) |

and

(7.27) | ||

(7.28) |

for . Suppose that we write out Equation (7.22) for , neglecting terms that are according to our expansion scheme. We find that

(7.29) |

giving

(7.30) |

Substituting the previous expression into Equation (7.22), evaluated for , and neglecting terms, we obtain

(7.31) |

Thus, the modified th energy eigenstate possesses the eigenvalue

and the eigenket

Note that

(7.34) |

Thus, the modified eigenkets remain orthonormal to .

Note, finally, that if the perturbing Hamiltonian, , commutes with the unperturbed Hamiltonian, , then

(7.35) |

and

(7.36) | ||

(7.37) |

The previous two equations are exact (i.e., they hold to all orders in ).