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# 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 (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: (7.20)

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 (7.22)

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 (7.32)

and the eigenket (7.33)

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 ).   Next: Quadratic Stark Effect Up: Time-Independent Perturbation Theory Previous: Two-State System
Richard Fitzpatrick 2016-01-22