 Solution

Take the spatial Fourier transform with the same convention. The partial differential equation becomes $i{\partial }_t\hat{f}-k^2\hat{f}=0$, so $\hat{f}(k,t)=e^{-i{k}^{2}t}{\hat{f}}_0(k)$. Inverting and substituting the transform of the initial function yields the convolutionComplete the square, $iks-it{k}^2=i{s}^2/(4t)-it[k-s/(2t){]}^2$, and put $\ell =\sqrt{t}[k-s/(2t)]$. This givesThe oscillatory Gaussian is a Fresnel integral, understood as the limit with a positive Gaussian damping factor. That prescription also justifies the transform manipulations, first for rapidly decreasing smooth data and then by L2 space extension. The resulting kernel is the free-particle propagator in units $\hslash =1$, $m=1/2$; its phase and normalization agree with the multiplier $e^{-i{k}^{2}t}$.