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How to Find the Second-Order Corrections to Energy Levels and Wave Functions

In quantum physics, in order to find the second-order corrections to energy levels and wave functions of a perturbed system, En, you need to calculate E(2)n, as well as

image0.png

So how do you do that? You start with three perturbed equations:

image1.png

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

image2.png

From the jumbo equation, you can then find the second-order corrections to the energy levels and the wave functions. To find E(2)n, multiply both sides of

image3.png

This looks like a tough equation until you realize that

image4.png

is equal to zero, so you get

image5.png

Because

image6.png

is also equal to zero, and again neglecting the first term, you get

image7.png

E(2)n is just a number, so you have

image8.png

And of course, because

image9.png

you have

image10.png

Note that if

image11.png

is an eigenstate of W, the second-order correction equals zero.

Okay, so

image12.png

How can you make that simpler? Well, from using

image13.png

Substituting that equation into

image14.png

gives you

image15.png

Now you have

image16.png

Here's the total energy with the first- and second-order corrections:

image17.png

So from this equation, you can say

image18.png

That gives you the first- and second-order corrections to the energy, according to perturbation theory.

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