Non-degenerate perturbation theory

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

$$H_0 \vert n\rangle = E_n \vert n\rangle,$$ (605)

where $n$ runs from 1 to $N$. The eigenkets $\vert n\rangle$ are orthonormal, 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:

$$(H_0 + H_1) \vert E\rangle = E \vert E\rangle.$$ (606)

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

$$\vert E\rangle = \sum_k \langle k \vert E\rangle \vert k\rangle,$$ (607)

where the summation is from $k=1$ to $N$. Substituting the above equation into Eq. (606), and right-multiplying by $\langle m\vert$, we obtain

$$(E_m + e_{mm} - E) \langle m\vert E\rangle + \sum_{k\neq m} e_{mk} \langle k\vert E\rangle = 0,$$ (608)

where

$$e_{mk} = \langle m \vert H_1\vert k\rangle.$$ (609)

Let us now develop our perturbation expansion. We assume that

$$\frac{\vert e_{mk}\vert}{E_m - E_k} \sim O(\epsilon),$$ (610)

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

$$\frac{\vert e_{mm}\vert}{E_m} \sim O(\epsilon),$$ (611)

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

$$E= E_n + O(\epsilon),$$ (612)

and

$$\langle n\vert E\rangle = 1,$$ (613)

$$\langle m\vert E\rangle \sim O(\epsilon),$$ (614)

for $m\neq n$. Suppose that we write out Eq. (608) for $m\neq n$, neglecting terms which are $O(\epsilon^2)$ according to our expansion scheme. We find that

$$(E_m - E_n) \langle m \vert E \rangle + e_{mn} \simeq 0,$$ (615)

giving

$$\langle m\vert E\rangle \simeq -\frac{e_{mn}}{E_m - E_n}.$$ (616)

Substituting the above expression into Eq. (608), evaluated for $m=n$, and neglecting $O(\epsilon^3)$ terms, we obtain

$$(E_n + e_{nn} - E) - \sum_{k\neq n} \frac{\vert e_{nk}\vert^2} {E_k-E_n} = 0.$$ (617)

Thus, the modified $n$th energy eigenstate possesses an eigenvalue

$$E_n' = E_n + e_{nn} + \sum_{k\neq n} \frac{\vert e_{nk}\vert^2} {E_n-E_k} + O(\epsilon^3),$$ (618)

and a eigenket

$$\vert n\rangle' = \vert n\rangle +\sum_{k\neq n}\frac{e_{kn}}{E_n - E_k} \vert k\rangle + O(\epsilon^2).$$ (619)

Note that

$$\langle m\vert n\rangle' = \delta_{mn} + \frac{e_{nm}^\ast}{... ...{mn}} {E_n-E_m} + O(\epsilon^2) = \delta_{mn} + O(\epsilon^2).$$ (620)

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