Using Lie theory to understand $U=e^{iHt}$ (Quantum Mechanics)

Question

Can we use the exponential map (lie theory) to understand how the Hamiltonian ($H$) gives rise to the unitary, and therefore compliments an essential property of the unitary operator (ie to preserve inner products)?

Answer 1

With fixed H, you are just doing a single one parameter group $U(t)$ sitting inside $U(n)$ for some $n$. Where it gets more interesting is if you have $H_1+\lambda H_2$. Then you will have to actually use some Lie brackets when doing the exponential.