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](https://www.dummies.com/wp-content/uploads/398098.image0.png)
So how do you do that? You start with three perturbed equations:
![image1.png](https://www.dummies.com/wp-content/uploads/398099.image1.png)
You then combine these three equations to get this jumbo equation:
![image2.png](https://www.dummies.com/wp-content/uploads/398100.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](https://www.dummies.com/wp-content/uploads/398101.image3.png)
This looks like a tough equation until you realize that
![image4.png](https://www.dummies.com/wp-content/uploads/398102.image4.png)
is equal to zero, so you get
![image5.png](https://www.dummies.com/wp-content/uploads/398103.image5.png)
Because
![image6.png](https://www.dummies.com/wp-content/uploads/398104.image6.png)
is also equal to zero, and again neglecting the first term, you get
![image7.png](https://www.dummies.com/wp-content/uploads/398105.image7.png)
E(2)n is just a number, so you have
![image8.png](https://www.dummies.com/wp-content/uploads/398106.image8.png)
And of course, because
![image9.png](https://www.dummies.com/wp-content/uploads/398107.image9.png)
you have
![image10.png](https://www.dummies.com/wp-content/uploads/398108.image10.png)
Note that if
![image11.png](https://www.dummies.com/wp-content/uploads/398109.image11.png)
is an eigenstate of W, the second-order correction equals zero.
Okay, so
![image12.png](https://www.dummies.com/wp-content/uploads/398110.image12.png)
How can you make that simpler? Well, from using
![image13.png](https://www.dummies.com/wp-content/uploads/398111.image13.png)
Substituting that equation into
![image14.png](https://www.dummies.com/wp-content/uploads/398112.image14.png)
gives you
![image15.png](https://www.dummies.com/wp-content/uploads/398113.image15.png)
Now you have
![image16.png](https://www.dummies.com/wp-content/uploads/398114.image16.png)
Here's the total energy with the first- and second-order corrections:
![image17.png](https://www.dummies.com/wp-content/uploads/398115.image17.png)
So from this equation, you can say
![image18.png](https://www.dummies.com/wp-content/uploads/398116.image18.png)
That gives you the first- and second-order corrections to the energy, according to perturbation theory.