## Derivation of the ladder operators for the Orbital Angular Momentum, L

**Vanessa V. Destura**, MSU-IIT

Let be an eigenvector of and with eigenvalues and respectively.

Now, we take the square of the norm of the eigenvalue equations for and

For :

For :

where

Note that

We expand the left-hand side of the equations obtained for (equation ) and (equation ) and substitute equations and respectively, where

So, for :

where Going back to , we have

For :

Going back to , we have

With this, we obtain

Hence, the ladder operators for the Orbital Angular Momentum, are