I learnt from my quantum mechanics courses that angular momentum operator is the generator of rotation, and applying angular momentum operator is like performing an infinitesimal rotation in 3D space.
Here is my question: why would a wavefunction "lengthen" under infinitesimal rotation? For instance, let's take a wavefunction $\psi$ with $l=2$ as an example. Obviously, it is invariant under rotation in z-axis and $l_{z}\psi=2\hbar\psi$, but I don't understand why a function will be scaled by a factor of $2\hbar$ when it is rotated. How can rotation reduce (or enhance) the amplitude of a function?
I understand that in this case it doesn't matter because wavefunction must be normalized and a constant factor in front won't change any observables. However, if I understand correctly, the use of rotational generator should be something universal, i.e. I could just as well apply the generators of rotation to any scalar field that maybe changing the amplitude do carry physical significance. This brings back to may question, how can an infinitesimal rotation change the amplitude of a function?
I am guessing that my interpretation of angular momentum operator is wrong. I can answer my homework's questions just fine as they mostly ask for mathematical calculations, which I have less trouble with, but I am having a hard time understanding intuitively how different operators operate. Thank you very much for your help.