I understand, mathematically, that the translation operator (both for infinitesimal and finite translations) can be written as a function of the momentum operator. It is said then that momentum "generates" translation. Similiary, the rotation operator can be written as a function of the angular momentum operator.
I can't help but thinking there's some point here that i'm missing. What's the point here, beyond "you can write this in function of that?" Why would anyone even want to construct a translation operator? Why not just changing the coordinates x -> x+a ?