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Second Order Perturbation Theory

If \bgroup\color{col1}$ E_i$\egroup is a (non-degenerate) eigenvalue of \bgroup\color{col1}$ \hat{H}_0$\egroup with (normalised) eigenvector \bgroup\color{col1}$ \vert i\rangle$\egroup, the second order approximation \bgroup\color{col1}$ E_i^{(2)}$\egroup of the corresponding new eigenvalue of \bgroup\color{col1}$ \hat{H}_0+\hat{H}_1$\egroup is given by

$\displaystyle {\color{col1}E_i^{(2)} = E_i + \langle i \vert \hat{H}_1 \vert i ...
...i\ne j} \frac{\vert\langle i \vert \hat{H}_1 \vert j \rangle \vert^2}{E_i-E_j}}$     (2.6)

Note that in our case here the unperturbed states are \bgroup\color{col1}$ \vert i=1\rangle = \vert L\rangle$\egroup and \bgroup\color{col1}$ \vert i=2\rangle = \vert R\rangle$\egroup, and the energies are \bgroup\color{col1}$ E_1=\varepsilon/2$\egroup and \bgroup\color{col1}$ E_2=-\varepsilon/2$\egroup. We have \bgroup\color{col1}$ \langle i \vert \hat{H}_1 \vert i \rangle =0$\egroup whence the first order correction vanishes. We furthermore have
$\displaystyle \langle L \vert \hat{H}_1\vert R \rangle = \langle R \vert \hat{H}_1 \vert L \rangle =T_c,$     (2.7)

which leads to
$\displaystyle E_1^{(2)}$ $\displaystyle =$ $\displaystyle \frac{\varepsilon}{2} + \frac{T_c^2}{E_1-E_2}= \frac{\varepsilon}{2} + \frac{T_c^2}{\varepsilon}$ (2.8)
$\displaystyle E_2^{(2)}$ $\displaystyle =$ $\displaystyle -\frac{\varepsilon}{2} + \frac{T_c^2}{E_2-E_1}= -\frac{\varepsilon}{2} - \frac{T_c^2}{\varepsilon}.$ (2.9)

We compare this to a Taylor expansion of the exact result, Eq. (II.2.4), for the eigenvalues \bgroup\color{col1}$ \varepsilon_\pm$\egroup:
$\displaystyle \varepsilon_{\pm}$ $\displaystyle =$ $\displaystyle \pm \frac{1}{2}\sqrt{\varepsilon^2+4T_c^2}
= \pm \frac{1}{2}\vare...
...2 \frac{T_c^2}{\varepsilon^2} + O\left(\frac{T_c}{\varepsilon}\right)^4\right],$ (2.10)

which means that
$\displaystyle \varepsilon_+$ $\displaystyle =$ $\displaystyle \frac{1}{2}\varepsilon + \frac{T_c^2}{\varepsilon}+ O\left(\frac{...
...arepsilon - \frac{T_c^2}{\varepsilon}+ O\left(\frac{T_c}{\varepsilon}\right)^4,$ (2.11)

which co-incides with our perturbation theory, i.e. the expressions Eq. (II.2.8) for \bgroup\color{col1}$ E_1^{(2)}$\egroup and \bgroup\color{col1}$ E_2^{(2)}$\egroup! At the same time, we make the following observations:

Backup literature for this section: textbook Gasiorowisz [3] cp. 11 for time-independent perturbation theory (revise if necessary). Lecture notes QM 1 chapter 3

http://brandes.phy.umist.ac.uk/QM/http://brandes.phy.umist.ac.uk/QM/ for two-level system.


next up previous contents index
Next: Hydrogen Atom: Fine Structure Up: Example: Two-Level System Previous: Exact solution   Contents   Index
Tobias Brandes 2005-04-26