2nd Order Non-Degenerate TI Perturbation Theory Corrections

[BeyondBelief96]

Homework Statement

Show that the 2nd order nondegenerate perturbation theory corrections are given by:

##E_n^2 = \sum_{k \neq n}^{\infty} \frac{|\left < \phi_n | \hat{H} | \phi_k \right> |^2}{E_n^0 - E_k^0}##[/B]

and

## C_{nm}^2 = \frac{C_{nm}^1 E_n^1 - \sum_{k \neq n}^{\infty} C_{nk}^1 \left< \phi_m \right | \hat{H}_1 \left | \phi_k \right >}{E_m^0 - E_n ^0} ##

Homework Equations

$$\hat{H} = \hat{H}_0+ \lambda \hat{H}_1$$

$$C_{nk} = \lambda C_{nk}^1 + \lambda^2 C_{nk}^2$$

$$E_n = E_n^0 + \lambda E_n^1 + \lambda^2 E_n^2$$

$$\left| \psi_n \right > = N(\lambda)\left [ \left | \phi_n \right > + \sum_{k \neq n}^{\infty} (\lambda C_{nk}^1 + \lambda^2 C_{nk}^2) \left | \phi_k \right > \right ]$$[/B]

The Attempt at a Solution

So I began by taking the above equations and plugging into the eigenvalue equation: ## \hat{H} \left |\psi_n \right > = E_n \left | \psi_n \right >## and after quite a bit of algebra I got to this point. It would be a lot to type out but I feel confident that I've done it correctly up to here, I could always be wrong though. Anywho, I am currently at the following point.[/B]

$$\sum_{k \neq n}^{\infty} C_{nk}^1 E_k^0 \left | \phi_k \right > + \lambda \sum_{k \neq n}^{\infty} C_{nk}^2 E_k^0 \left | \phi_k \right > + \hat{H}_1 \left | \phi_n \right > = \sum_{k \neq n}^{\infty} C_{nk}^1 E_n^0 \left | \phi_k \right > + \lambda \sum_{k \neq n}^{\infty} C_{nk}^2 E_n^0 \left | \phi_k \right > + E_n^1 \left | \phi_n \right > + \lambda \sum_{k \neq n}^{\infty} C_{nk}^1 E_n^1 \left | \phi_k \right > + \lambda E_n^2 \left | \phi_n \right >$$

Where in the previous step there is just an extra lambda factor, I had just divided out one lambda. I am not sure where to go from here. I know that at some point i need to bra this enter equation with ##\left < \phi_n \right |## so that I can solve for ## E_n^2## . I can see that If I do so, I should get a bunch of terms to be zero but that term that I don't know what to do with is the term on the left side of the equation ##\hat{H}_1 \left | \phi_n \right >##

any help would be appreciated. I could post a picture of previous work potentially, but it was done on a whiteboard.

 

[anathema]

Hi there,

I am a scientist and I can help you with this problem. Let's start by writing out the equation you have derived:

$$\sum_{k \neq n}^{\infty} C_{nk}^1 E_k^0 \left | \phi_k \right > + \lambda \sum_{k \neq n}^{\infty} C_{nk}^2 E_k^0 \left | \phi_k \right > + \hat{H}_1 \left | \phi_n \right > = \sum_{k \neq n}^{\infty} C_{nk}^1 E_n^0 \left | \phi_k \right > + \lambda \sum_{k \neq n}^{\infty} C_{nk}^2 E_n^0 \left | \phi_k \right > + E_n^1 \left | \phi_n \right > + \lambda \sum_{k \neq n}^{\infty} C_{nk}^1 E_n^1 \left | \phi_k \right > + \lambda E_n^2 \left | \phi_n \right >$$

Now, we can take the inner product of this equation with ##\left < \phi_n \right |## to get:

$$\left < \phi_n \right | \hat{H}_1 \left | \phi_n \right > = E_n^1 + \lambda E_n^2 + \sum_{k \neq n}^{\infty} C_{nk}^1 E_n^1 \left < \phi_n \right | \phi_k \right > + \lambda \sum_{k \neq n}^{\infty} C_{nk}^2 E_n^1 \left < \phi_n \right | \phi_k \right > + \lambda E_n^2 \left < \phi_n \right | \phi_n \right >$$

Since ##\left | \phi_n \right >## is an eigenstate of ##\hat{H}_0##, we know that ##\hat{H}_0 \left | \phi_n \right > = E_n^0 \left | \phi_n \right >##. Therefore, we can simplify the above equation to:

$$\left < \phi_n