Suppose I know an initial state [tex]\Phi(x)= \exp(-x^2)[/tex], the Hamiltonian is [tex]H = p^2/2m + x^2/2[/tex] where p is the mometum operator. If I want to find the time evolution of the state [tex]\Phi(x)[/tex], should I write it as the following? [tex]\Psi(x, t) = \exp(-i H t/\hbar)\Phi(x)[/tex] However, since [tex]H[/tex] contains an operator, I don't know how to find the close form of the time-dependent state. Should I expand it as a series and then operate it on [tex]\Phi(x)[/tex] term by term? But in this way, it seems not easy to combine the result to get the close form!?