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Physics LibreTexts

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=kk|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+emmE)m|E+kmemkk|E=0, ???

where

emk=m|H1|k. ???

Let us now develop our perturbation expansion. We assume that

|emk|EmEkO(ϵ), ???

for all mk , where ϵ1 is our expansion parameter. We also assume that

|emm|EmO(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 mn . Suppose that we write out Equation ??? 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 above expression into Equation ???, evaluated for m=n , and neglecting O(ϵ3) terms, we obtain

(En+ennE)kn|enk|2EkEn0. ???

Thus, the modified n th energy eigenstate possesses an eigenvalue

En=En+enn+kn|enk|2EnEk+O(ϵ3), ???

and a eigenket

|n=|n+kneknEnEk|k+O(ϵ2). ???

Note that

m|n=δmn+enmEmEn+emnEnEm+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)


This page titled 7.3: 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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