Abstract
A solution of the free Schr\"odinger equation is investigated by means of Optimal transport. The curve of probability measures μ_t this solution defines is shown to be an absolutely continuous curve in the Wasserstein space W₂(R³). The optimal transport map from μ_t to μ_s, the cost for this transport (i.e. the Wasserstein distance) and the value of the Fisher information along μ_t are being calculated. It is finally shown that this solution of the free Schr\"odinger equation can naturally be interpreted as a curve in so-called Shape space, which forgets any positioning in space but only describes properties of shapes. In Shape space, μ_t continues to be a shortest path geodesic.