11.4: Non-Degenerate Perturbation Theory
- Page ID
- 15794
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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
\begin{equation}E_{n}^{\prime}=E_{n}+e_{n n}+\sum_{k \neq n} \frac{\left|e_{n k}\right|^{2}}{E_{n}-E_{k}}+\mathcal{O}\left(\epsilon^{3}\right)\end{equation} 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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