How to Find the Energy of a Perturbed System Based on the Wavelength Parameter

In quantum physics, in order to find the energy of a perturbed system, E_n, you need to start by calculating the energy and wave function of an unperturbed system. You start with the energy:

Then add the first-order correction to the energy,

and add the second-order correction to the energy,

Now what about the wave function of the perturbed system,

Start with the wave function of the unperturbed system,

Add to it the first-order correction,

And then add to that the second-order correction to the wave function,

Note that when

becomes the unperturbed energy:

and

becomes the unperturbed wave function:

So your task is to calculate E⁽¹⁾_n and E⁽²⁾_n, as well as

So how do you do that in general? Time to start slinging some math. You start with three perturbed equations:

You then combine these three equations to get this jumbo equation: