# Perturbed Hamiltonian and its affect on the eigenvalues

1. Dec 6, 2015

### pondzo

1. The problem statement, all variables and given/known data

2. Relevant equations

$$E_n^{(2)}=\sum_{k\neq n}\frac{|H_{kn}'|^2}{E_n^{(0)}-e_k^{(0)}}$$

3. The attempt at a solution

Not sure where to start here. The question doesn't give any information about the unperturbed Hamiltonian. Some guidance on the direction would be great! Cheers.

2. Dec 6, 2015

### blue_leaf77

Start by writing out $H_T = H_0 + H'$, where $H_T$ is the total Hamiltonian, the matrix of which is given in the question, $H_0$ the unperturbed Hamiltonian whose elements do not contain $\lambda$, and $H'$ the perturbation term which is proportional to $\lambda$.

3. Dec 6, 2015

### pondzo

Hi blue_leaf! Thanks for that, I just realised that I interpreted the question wrong. I thought the matrix given was only the unperturbed Hamiltonian.

So I have separated it into its un/pertrubed matrices. I just don't think I understand what the question is asking, when it says the change in eigenvalues.

$E_1\approx E_1^{(0)}+E_1^{(1)}+E_1^{(2)}$
$~~~~~=E_0+0-\frac{4}{7}E_0\lambda^2$

$E_2\approx E_2^{(0)}+E_2^{(1)}+E_2^{(2)}$
$~~~~~=8E_0+0+\frac{4}{7}E_0\lambda^2$

$E_3\approx E_3^{(0)}+E_3^{(1)}+E_3^{(2)}$
$~~~~~=3E_0+E_0\lambda+0$

$E_4\approx E_4^{(0)}+E_4^{(1)}+E_4^{(2)}$
$~~~~~=7E_0+0+0$

So are the changes in the eigenvalues $-\frac{4}{7}E_0\lambda^2,~\frac{4}{7}E_0\lambda^2,~E_0\lambda \text{ and } 0$ respectively ?

4. Dec 6, 2015

### blue_leaf77

I think the full expressions of the new energies are what the questions asks.