# 7.3: Non-Degenerate Perturbation Theory

Let us now generalize our perturbation analysis to deal with systems possessing more than two energy eigenstates. The energy eigenstates of the unperturbed Hamiltonian, , are denoted

(601) |

where runs from 1 to . The eigenkets are orthogonal, form a complete set, and have their lengths normalized to unity. 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,

(603) |

where the summation is from to . Substituting the above equation into Equation (602), and right-multiplying by , we obtain

where

(605) |

Let us now develop our perturbation expansion. We assume that

(606) |

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

(607) |

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

(608) |

and

(609) | |||

(610) |

for . Suppose that we write out Equation (604) for , neglecting terms that are according to our expansion scheme. We find that

(611) |

giving

(612) |

Substituting the above expression into Equation (604), evaluated for , and neglecting terms, we obtain

(613) |

Thus, the modified th energy eigenstate possesses an eigenvalue

and a eigenket

Note that

(616) |

Thus, the modified eigenkets remain orthogonal and properly normalized to .

### Contributors

- Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)