Questions about some analogies between heat and Shroedinger equations

Question

Let $Z(x,t)=(4\pi t)^{-1/2}e^{-x^2/(4t)}$ and $G(x,t)=(4\pi t)^{-1/2}e^{-x^2/(4it)}$ be fundamental solutions of the heat $u_t=u_{xx}$ and Shroedinger $iu_t=u_{xx}$ equations.

Function $Z(x,t)$ represents a temperature on the line for one point initial heat source (delta function). Being positive on the line for $t>0$ means that heat propagates with infinite speed in this model. I've seen an explanation that it's because the Fourier law used to derive the heat equation doesn't hold on the wave front of the heat expansion.

1. Is there an analogous explanation why solution $G$ of the Shroedinger equation also propagates with an infinite speed?

From the other hand the temperature expansion speed is effectively finite because function $e^{-x^2/(4t)}$ decays very fast as $x\to\infty$. Say, for a point source, at the moment $t>0$ 99% of the heat is contained in a ball of radius $3\sqrt{2t}$ as per $3\sigma$ rule.

2. What would be an analogous property for the Shroedinger equation?

Answer 1

The Schroedinger kernal describes the evolution of a wavefunction that starts off as a delta function at $t=0$. The delta-function state is an equally-weighted linear combination of of all momenta from $p=-\infty$ $p=+\infty$. As the speed of the particles in the non-relativistic Schroedinger syste is $v=p/m$, particles travel at all possible speeds, and the wavefunction gets everywhere on the real line immediately after $t=0$.