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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:

$\begin{align} \left<p|\psi_0\right> &= \int \left<p|x\right>\left<x|\psi_0\right> dx\\\\ &= \frac{A}{\sqrt{2\pi\hbar}}\int \exp\left(-ipx/\hbar - ax^2\right) dx, \end{align}$

and

$\begin{align} \psi\left(x, t\right) = \left<x|\psi\right> &= \int \exp\left(-iEt/\hbar\right) \left<x|p\right>\left<p|\psi_0\right> dp\\\\ &= \frac{A}{2\pi\hbar}\int\exp\left(ipx/\hbar-iEt/\hbar\right)\int \exp\left(-ipx'/\hbar - ax'^2\right) dx'dp. \end{align}$

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.

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closed as off-topic by ZeroTheHero, user191954, Kyle Kanos, PiKindOfGuy, Jon Custer Nov 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.

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