When working in quantum mechanics, an operator can be written as a "ket". $\left| n \right> $
However, what does it mean when we have two items inside the ket? Such as $\left| l ,m \right>$. My notes say that "Here we adopt the Dirac notation in a way such that $\left|l,m\right> \equiv \psi^l_m$". I am confused as to what $\left| l ,m \right>$ is. Does it have something to do with there now being two dimensions? How would lowering and raising operators work on it?
Hamiltonians with symmetrical 2D potentials that we can solve explicitly often have eigenfunctions that can be naturally labeled by two parameters. If we have an irregular potential ("random" bumps and valleys), there will not necessarily be a natural two parameter labeling. In either case, they are just labels - we could label them $\left| a \right>, \left| b \right>, \left| c \right>, $ ... .