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:

To see this equation, note that

On the other hand, also note that

so you have

What do you do about L+ L? Well, you assume that the following is true:

So your equation becomes the following:

Great! That means that c is equal to

So what is

Applying the L2 and Lz operators gives you this value for c:

And that’s the eigenvalue of L+, which means you have this relation:

Similarly, you can show that L gives you the following: