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7.3: Non-Degenerate Perturbation Theory

  • Page ID
    1218
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    Let us now generalize our perturbation analysis to deal with systems possessing more than two energy eigenstates. The energy eigenstates of the unperturbed Hamiltonian, \( H_0\) , are denoted

    \( H_0\, \vert n\rangle = E_n\, \vert n\rangle,\) \ref{601}

    where \( n\) runs from 1 to \( N\) . The eigenkets \( \vert n\rangle\) 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:

    \( (H_0 + H_1) \,\vert E\rangle = E\,\vert E\rangle.\) \ref{602}

    We can express \( \vert E\rangle\) as a linear superposition of the unperturbed energy eigenkets,

    \( \vert E\rangle = \sum_k \langle k \vert E\rangle \vert k\rangle,\) \ref{603}

    where the summation is from \( k=1\) to \( N\) . Substituting the above equation into Equation \ref{602}, and right-multiplying by \( \langle m\vert\) , we obtain

    \( (E_m + e_{mm} - E)\, \langle m\vert E\rangle + \sum_{k\neq m} e_{mk}\, \langle k\vert E\rangle = 0,\) \ref{604}

    where

    \( e_{mk} = \langle m \vert\,H_1\,\vert k\rangle.\) \ref{605}

    Let us now develop our perturbation expansion. We assume that

    \( \frac{\vert e_{mk}\vert}{E_m - E_k} \sim O(\epsilon),\) \ref{606}

    for all \( m\neq k\) , where \( \epsilon\ll 1\) is our expansion parameter. We also assume that

    \( \frac{\vert e_{mm}\vert}{E_m} \sim O(\epsilon),\) \ref{607}

    for all \( m\) . Let us search for a modified version of the \( n\) th unperturbed energy eigenstate, for which

    \( E= E_n + O(\epsilon),\) \ref{608}

    and

    \( \langle n\vert E\rangle\) \( =\) \( 1,\) \ref{609} \( \langle m\vert E\rangle\) \( \sim\) \( O(\epsilon),\) \ref{610}


    for \( m\neq n\) . Suppose that we write out Equation \ref{604} for \( m\neq n\) , neglecting terms that are \( O(\epsilon^2)\) according to our expansion scheme. We find that

    \( (E_m - E_n) \,\langle m \vert E \rangle + e_{mn} \simeq 0,\) \ref{611}

    giving

    \( \langle m\vert E\rangle \simeq -\frac{e_{mn}}{E_m - E_n}.\) \ref{612}

    Substituting the above expression into Equation \ref{604}, evaluated for \( m=n\) , and neglecting \( O(\epsilon^3)\) terms, we obtain

    \( (E_n + e_{nn} - E) - \sum_{k\neq n} \frac{\vert e_{nk}\vert^{\,2}} {E_k-E_n} \simeq 0.\) \ref{613}

    Thus, the modified \( n\) th energy eigenstate possesses an eigenvalue

    \( E_n' = E_n + e_{nn} + \sum_{k\neq n} \frac{\vert e_{nk}\vert^{\,2}} {E_n-E_k} + O(\epsilon^3),\) \ref{614}

    and a eigenket

    \( \vert n\rangle' = \vert n\rangle +\sum_{k\neq n}\frac{e_{kn}}{E_n - E_k}\,\vert k\rangle + O(\epsilon^2).\) \ref{615}

    Note that

    \( \langle m\vert n\rangle' = \delta_{mn} + \frac{e_{nm}^{\,\ast}}{E_m-E_n} + \frac{e_{mn}} {E_n-E_m} + O(\epsilon^2) = \delta_{mn} + O(\epsilon^2).\) \ref{616}

    Thus, the modified eigenkets remain orthogonal and properly normalized to \( O(\epsilon^2)\) .

    Contributors

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

      \( \newcommand {\ltapp} {\stackrel {_{\normalsize<}}{_{\normalsize \sim}}}\) \(\newcommand {\gtapp} {\stackrel {_{\normalsize>}}{_{\normalsize \sim}}}\) \(\newcommand {\btau}{\mbox{\boldmath$\tau$}}\) \(\newcommand {\bmu}{\mbox{\boldmath$\mu$}}\) \(\newcommand {\bsigma}{\mbox{\boldmath$\sigma$}}\) \(\newcommand {\bOmega}{\mbox{\boldmath$\Omega$}}\) \(\newcommand {\bomega}{\mbox{\boldmath$\omega$}}\) \(\newcommand {\bepsilon}{\mbox{\boldmath$\epsilon$}}\)

    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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