I'm trying to transform the time evolution operator from momentum space to position space. I know that
$$ U(t) = e^{-iHt/h} = \int_{-\infty}^\infty e^{-ip^2t/2uh} | p \rangle \langle p | dp $$
and I'm trying to find the form of
$$ \langle x | U(t) | x' \rangle $$
I'm given the hint (paraphrased):
To evaluate this explicitly, use the Fourier transform of a Gaussian function, with imaginary $a$
I'm trying to apply the time operator to a momentum space wave function:
$$ U(t)|\psi (p,0) \rangle = \int_{-\infty}^\infty e^{-ip^2t/2uh} \psi (p,0) | p \rangle dp $$
But I'm not sure how to simplify to a form where a fourier transform would be straightforward