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Physics LibreTexts

11.4: 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. Consider a system in which the energy eigenstates of the unperturbed Hamiltonian, H0, are denoted H0ψn=Enψn, where n runs from 1 to N. The eigenstates are assumed to be orthonormal, so that m|n=δnm, and to form a complete set. Let us now try to solve the energy eigenvalue problem for the perturbed Hamiltonian: (H0+H1)ψE=EψE. It follows that m|H0+H1|E=Em|E, where m can take any value from 1 to N. Now, we can express ψE as a linear superposition of the unperturbed energy eigenstates: ψE=kk|Eψk, where k runs from 1 to N. We can combine the previous equations to give

(EmE+emm)m|E+kmemkk|E=0, where emk=m|H1|k.

Let us now develop our perturbation expansion. We assume that emkEmEkO(ϵ) for all mk, where ϵ1 is our expansion parameter. We also assume that emmEmO(ϵ) for all m. Let us search for a modified version of the nth unperturbed energy eigenstate for which E=En+O(ϵ), and n|E=1,m|E=O(ϵ) for mn. Suppose that we write out Equation ([e12.45]) for mn, neglecting terms that are O(ϵ2) according to our expansion scheme. We find that (EmEn)m|E+emn0, giving m|EemnEmEn. Substituting the previous expression into Equation ([e12.45]), evaluated for m=n, and neglecting O(ϵ3) terms, we obtain (EnE+enn)kn|enk|2EkEn0. Thus, the modified nth energy eigenstate possesses an eigenvalue

En=En+enn+kn|enk|2EnEk+O(ϵ3) and a wavefunction

ψn=ψn+kneknEnEkψk+O(ϵ2). Incidentally, it is easily demonstrated that the modified eigenstates remain orthonormal to O(ϵ2).

Contributors and Attributions

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


This page titled 11.4: Non-Degenerate Perturbation Theory is shared under a not declared license and was authored, remixed, and/or curated by Richard Fitzpatrick.

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