Hello lugita15,
I think the operator " generator of temporal translation " is [itex]g_t = i\hbar \partial/\partial t[/itex], in the same sense as the operator " generator of coordinate translation " is [itex]g_k = i\hbar \partial/\partial q_k [/itex].
Schroedinger's equation says that the actual psifunction [itex]\psi(q,t)[/itex] is such that the operator [itex]i\hbar \partial/\partial t[/itex] has the same effect as the Hamiltonian operator [itex]H(q,p,t)[/itex]. You can write then [itex]e^{\Delta t g_t/i\hbar} \psi = e^{\Delta t H(q,p,t)/i\hbar}\psi [/itex]. But mathematically, these operators are not the same thing.
Group theory and symmetries are important concepts, but personally I am sceptical about the role of symmetry in " deriving " quantum mechanics. It is better to study historical papers and try to understand how people came to it. Besides, the whole concept of translation generators works only for functions that are equal to their Taylor expansion. Such functions are not sufficient.
