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:
