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## Main Question or Discussion Point

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!?

[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!?