Find the Eigenvalues of the Raising and Lowering Angular Momentum Operators

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 L_z 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: