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NonDegenerate Perturbation Theory
Let us now generalize our perturbation analysis to deal
with systems possessing more than two energy eigenstates.
Consider a system in which the energy
eigenstates of the unperturbed Hamiltonian, , are denoted

(894) 
where runs from 1 to . The eigenstates are assumed to
be orthonormal, so that

(895) 
and to form a complete set. Let us now try to
solve the energy eigenvalue problem for the perturbed Hamiltonian:

(896) 
If follows that

(897) 
where can take any value from 1 to . Now, we can express
as a linear superposition of the unperturbed energy eigenstates:

(898) 
where runs from 1 to . We can combine the above
equations to give

(899) 
where

(900) 
Let us now develop our perturbation expansion. We assume that

(901) 
for all , where is our expansion parameter.
We also assume that

(902) 
for all . Let us search for a modified version of the th unperturbed
energy eigenstate for which

(903) 
and
for . Suppose that we write out Eq. (899) for ,
neglecting terms which are
according to our expansion
scheme. We find that

(906) 
giving

(907) 
Substituting the above expression into Eq. (899),
evaluated for , and neglecting
terms, we obtain

(908) 
Thus, the modified th energy eigenstate possesses an eigenvalue

(909) 
and a wavefunction

(910) 
Incidentally, it is easily demonstrated that the modified eigenstates remain orthonormal
to
.
Next: Quadratic Stark Effect
Up: TimeIndependent Perturbation Theory
Previous: TwoState System
Richard Fitzpatrick
20100720