7.3: Non-Degenerate Perturbation Theory
( \newcommand{\kernel}{\mathrm{null}\,}\)
Let us now generalize our perturbation analysis to deal with systems possessing more than two energy eigenstates. The energy eigenstates of the unperturbed Hamiltonian, H0 , are denoted
H0|n⟩=En|n⟩, ???where n runs from 1 to N . The eigenkets |n⟩ 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:
(H0+H1)|E⟩=E|E⟩. ???We can express |E⟩ as a linear superposition of the unperturbed energy eigenkets,
|E⟩=∑k⟨k|E⟩|k⟩, ???where the summation is from k=1 to N . Substituting the above equation into Equation ???, and right-multiplying by ⟨m| , we obtain
(Em+emm−E)⟨m|E⟩+∑k≠memk⟨k|E⟩=0, ???where
emk=⟨m|H1|k⟩. ???Let us now develop our perturbation expansion. We assume that
|emk|Em−Ek∼O(ϵ), ???for all m≠k , where ϵ≪1 is our expansion parameter. We also assume that
|emm|Em∼O(ϵ), ???for all m . Let us search for a modified version of the n th unperturbed energy eigenstate, for which
E=En+O(ϵ), ???and
⟨n|E⟩ = 1, ??? ⟨m|E⟩ ∼ O(ϵ), ???
for m≠n . Suppose that we write out Equation ??? for m≠n , neglecting terms that are O(ϵ2) according to our expansion scheme. We find that
giving
⟨m|E⟩≃−emnEm−En. ???Substituting the above expression into Equation ???, evaluated for m=n , and neglecting O(ϵ3) terms, we obtain
(En+enn−E)−∑k≠n|enk|2Ek−En≃0. ???Thus, the modified n th energy eigenstate possesses an eigenvalue
E′n=En+enn+∑k≠n|enk|2En−Ek+O(ϵ3), ???and a eigenket
|n⟩′=|n⟩+∑k≠neknEn−Ek|k⟩+O(ϵ2). ???Note that
⟨m|n⟩′=δmn+e∗nmEm−En+emnEn−Em+O(ϵ2)=δmn+O(ϵ2). ???Thus, the modified eigenkets remain orthogonal and properly normalized to O(ϵ2) .
Contributors
Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)