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# 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:

 (602)

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

 (604)

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

 (614)

and a eigenket

 (615)

Note that

 (616)

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