In quantum physics, you can find the eigenvalues of the raising and lowering angular momentum operators, which raise and lower a state’s z component of angular momentum.
Start by taking a look at L+, and plan to solve for c:
L+| l, m > = c | l, m + 1 >
So L+ | l, m > gives you a new state, and multiplying that new state by its transpose should give you c2:
![image0.png](https://www.dummies.com/wp-content/uploads/395035.image0.png)
To see this equation, note that
![image1.png](https://www.dummies.com/wp-content/uploads/395036.image1.png)
On the other hand, also note that
![image2.png](https://www.dummies.com/wp-content/uploads/395037.image2.png)
so you have
![image3.png](https://www.dummies.com/wp-content/uploads/395038.image3.png)
What do you do about L+ L–? Well, you assume that the following is true:
![image4.png](https://www.dummies.com/wp-content/uploads/395039.image4.png)
So your equation becomes the following:
![image5.png](https://www.dummies.com/wp-content/uploads/395040.image5.png)
Great! That means that c is equal to
![image6.png](https://www.dummies.com/wp-content/uploads/395041.image6.png)
So what is
![image7.png](https://www.dummies.com/wp-content/uploads/395042.image7.png)
Applying the L2 and Lz operators gives you this value for c:
![image8.png](https://www.dummies.com/wp-content/uploads/395043.image8.png)
And that’s the eigenvalue of L+, which means you have this relation:
![image9.png](https://www.dummies.com/wp-content/uploads/395044.image9.png)
Similarly, you can show that L– gives you the following:
![image10.png](https://www.dummies.com/wp-content/uploads/395045.image10.png)