Next: Quadratic Stark Effect
Up: Time-Independent Perturbation Theory
Previous: Two-State System
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,
, are denoted
![$\displaystyle H_0\, \vert n\rangle = E_n\, \vert n\rangle,$](img1467.png) |
(601) |
where
runs from 1 to
. The eigenkets
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:
![$\displaystyle (H_0 + H_1) \,\vert E\rangle = E\,\vert E\rangle.$](img1442.png) |
(602) |
We can express
as a linear superposition of the unperturbed energy
eigenkets,
![$\displaystyle \vert E\rangle = \sum_k \langle k \vert E\rangle \vert k\rangle,$](img1468.png) |
(603) |
where the summation is from
to
. Substituting the above
equation into Equation (602), and right-multiplying by
, we obtain
![$\displaystyle (E_m + e_{mm} - E)\, \langle m\vert E\rangle + \sum_{k\neq m} e_{mk}\, \langle k\vert E\rangle = 0,$](img1471.png) |
(604) |
where
![$\displaystyle e_{mk} = \langle m \vert\,H_1\,\vert k\rangle.$](img1472.png) |
(605) |
Let us now develop our perturbation expansion. We assume that
![$\displaystyle \frac{\vert e_{mk}\vert}{E_m - E_k} \sim O(\epsilon),$](img1473.png) |
(606) |
for all
, where
is our expansion parameter. We also
assume that
![$\displaystyle \frac{\vert e_{mm}\vert}{E_m} \sim O(\epsilon),$](img1476.png) |
(607) |
for all
. Let us search for a modified version of the
th unperturbed energy
eigenstate, for which
![$\displaystyle E= E_n + O(\epsilon),$](img1477.png) |
(608) |
and
for
. Suppose that we
write out Equation (604) for
, neglecting terms that
are
according to our expansion scheme. We find that
![$\displaystyle (E_m - E_n) \,\langle m \vert E \rangle + e_{mn} \simeq 0,$](img1485.png) |
(611) |
giving
![$\displaystyle \langle m\vert E\rangle \simeq -\frac{e_{mn}}{E_m - E_n}.$](img1486.png) |
(612) |
Substituting the above expression into Equation (604),
evaluated for
, and neglecting
terms, we obtain
![$\displaystyle (E_n + e_{nn} - E) - \sum_{k\neq n} \frac{\vert e_{nk}\vert^{\,2}} {E_k-E_n} \simeq 0.$](img1489.png) |
(613) |
Thus, the modified
th energy eigenstate possesses an eigenvalue
![$\displaystyle E_n' = E_n + e_{nn} + \sum_{k\neq n} \frac{\vert e_{nk}\vert^{\,2}} {E_n-E_k} + O(\epsilon^3),$](img1490.png) |
(614) |
and a eigenket
![$\displaystyle \vert n\rangle' = \vert n\rangle +\sum_{k\neq n}\frac{e_{kn}}{E_n - E_k}\,\vert k\rangle + O(\epsilon^2).$](img1491.png) |
(615) |
Note that
![$\displaystyle \langle m\vert n\rangle' = \delta_{mn} + \frac{e_{nm}^{\,\ast}}{E_m-E_n} + \frac{e_{mn}} {E_n-E_m} + O(\epsilon^2) = \delta_{mn} + O(\epsilon^2).$](img1492.png) |
(616) |
Thus, the modified eigenkets remain orthogonal and properly normalized
to
.
Next: Quadratic Stark Effect
Up: Time-Independent Perturbation Theory
Previous: Two-State System
Richard Fitzpatrick
2013-04-08