If we have a lagrangian with a symmetry then we get a conserved quantity: $$Q=Q(p,q)$$ which is a function of the conjugate momentum and the coordinates.
If we move over to quantum mechanics then we promote $Q$ to an operator by promoting $p,q$ to operators but otherwise keeping $Q$ as the same functional form of $p,q$ (assuming no ordering ambiguities).Then if we want to apply the symmetry to a state $|\psi\rangle$ then we use the conserved quantity, $$|\psi\rangle\to|\psi'\rangle=e^{iQ}|\psi\rangle$$.
Why is the conserved quantity the generator of the transformation ?
