We consider the Erdős-Rényi graph G in its critical regime when its expected degree d scales like the logarithm of its number of vertices. On this critical scale, G undergoes a connectivity transition through the formation of isolated vertices. Moreover, localized eigenvectors emerge. The time evolution of a free quantum particle on G is governed by the adjacency matrix A of G through the Schrödinger equation. We determine the solution to this Schrödinger equation by comparison to an infinite tree. As A possesses localized and delocalized eigenvectors, the solution is in general a mixture of localized and scattering waves.