Time evolution of a free particle with a given initial state [closed]

My homework problem reads:

Consider a free particle in one dimension. Write an expression for the wavefunction $$\psi(x, t)$$ given an initial state $$\psi_0(x) = Ae^{-ax^2}$$ at $$t = 0$$, where $$A$$ is a normalization constant (that you do not need to calculate).

This is my attempt:

\begin{align} \left &= \int \left\left dx\\\\ &= \frac{A}{\sqrt{2\pi\hbar}}\int \exp\left(-ipx/\hbar - ax^2\right) dx, \end{align}

and

\begin{align} \psi\left(x, t\right) = \left &= \int \exp\left(-iEt/\hbar\right) \left\left dp\\\\ &= \frac{A}{2\pi\hbar}\int\exp\left(ipx/\hbar-iEt/\hbar\right)\int \exp\left(-ipx'/\hbar - ax'^2\right) dx'dp. \end{align}

I'm not sure I'm moving in the right direction and if I am, I don't know how to proceed. I'd like some guidance please.

closed as off-topic by ZeroTheHero, user191954, Kyle Kanos, PiKindOfGuy, Jon CusterNov 30 '18 at 0:09

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You are moving in the right direction. You only need to take into account p-dependence of E: $$E = \frac{p^2}{2m}.$$ After this you should also be able to find expression for the 2d Gauss integral.