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11.4: Non-Degenerate Perturbation Theory

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  • 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, \(H_0\), are denoted \[H_0\,\psi_n = E_n\,\psi_n,\] where \(n\) runs from 1 to \(N\). The eigenstates are assumed to be orthonormal, so that \[\langle m|n\rangle = \delta_{nm},\] and to form a complete set. Let us now try to solve the energy eigenvalue problem for the perturbed Hamiltonian: \[(H_0+H_1)\,\psi_E = E\,\psi_E.\] It follows that \[\langle m|H_0+H_1|E\rangle = E\,\langle m |E\rangle,\] where \(m\) can take any value from 1 to \(N\). Now, we can express \(\psi_E\) as a linear superposition of the unperturbed energy eigenstates: \[\psi_E = \sum_k \langle k|E\rangle\,\psi_k,\] where \(k\) runs from 1 to \(N\). We can combine the previous equations to give

    \[\label{e12.45} (E_m-E+e_{mm})\,\langle m|E\rangle + \sum_{k\neq m} e_{mk}\,\langle k|E\rangle = 0,\] where \[e_{mk} =\langle m|H_1| k\rangle.\]

    Let us now develop our perturbation expansion. We assume that \[\frac{e_{mk}}{E_m-E_k} \sim {\cal O}(\epsilon)\] for all \(m\neq k\), where \(\epsilon\ll 1\) is our expansion parameter. We also assume that \[\frac{e_{mm}}{E_m}\sim {\cal O}(\epsilon)\] for all \(m\). Let us search for a modified version of the \(n\)th unperturbed energy eigenstate for which \[E = E_n + {\cal O}(\epsilon),\] and \[\begin{aligned} \langle n|E\rangle &= 1,\\[0.5ex] \langle m|E\rangle&={\cal O}(\epsilon)\end{aligned}\] for \(m\neq n\). Suppose that we write out Equation ([e12.45]) for \(m\neq n\), neglecting terms that are \({\cal O}(\epsilon^{\,2})\) according to our expansion scheme. We find that \[(E_m-E_n)\,\langle m|E\rangle + e_{mn} \simeq 0,\] giving \[\langle m|E\rangle \simeq - \frac{e_{mn}}{E_m-E_n}.\] Substituting the previous expression into Equation ([e12.45]), evaluated for \(m=n\), and neglecting \({\cal O}(\epsilon^{\,3})\) terms, we obtain \[(E_n-E+e_{nn})-\sum_{k\neq n}\frac{|e_{nk}|^{\,2}}{E_k-E_n} \simeq 0.\] Thus, the modified \(n\)th energy eigenstate possesses an eigenvalue

    \[\label{e12.56} E_n' = E_n + e_{nn} + \sum_{k\neq n}\frac{|e_{nk}|^{\,2}}{E_n-E_k} + {\cal O}(\epsilon^{\,3}),\] and a wavefunction

    \[\label{e12.57} \psi_n' = \psi_n + \sum_{k\neq n} \frac{e_{kn}}{E_n-E_k}\,\psi_k + {\cal O}(\epsilon^{\,2}).\] Incidentally, it is easily demonstrated that the modified eigenstates remain orthonormal to \({\cal O}(\epsilon^{\,2})\).

    Contributors and Attributions

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

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