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# 11.4: Non-Degenerate Perturbation Theory

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

$$E_{n}^{\prime}=E_{n}+e_{n n}+\sum_{k \neq n} \frac{\left|e_{n k}\right|^{2}}{E_{n}-E_{k}}+\mathcal{O}\left(\epsilon^{3}\right)$$ 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 $${\cal O}(\epsilon^{\,2})$$.
