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arXiv · 2512.06940

Optimal Transport of a Free Quantum Particle and its Shape Space Interpretation

Bernadette Lessel
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Abstract

A solution of the free Schr\"odinger equation is investigated by means of Optimal transport. The curve of probability measures $\mu_t$ this solution defines is shown to be an absolutely continuous curve in the Wasserstein space $W_2(\mathbb{R}^3)$. The optimal transport map from $\mu_t$ to $\mu_s$, the cost for this transport (i.e. the Wasserstein distance) and the value of the Fisher information along $\mu_t$ 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, $\mu_t$ continues to be a shortest path geodesic.

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Bernadette Lessel. 2025-12-07. Optimal Transport of a Free Quantum Particle and its Shape Space Interpretation. https://arxiv.org/abs/2512.06940

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