Duality of $\mathit{x}$ and $\mathit{\psi }$ and a Statistical Interpretation of Space in Quantum Mechanics

We introduce a “prepotential” $F$ in quantum mechanics and show that the coordinate $x$ is proportional to the Legendre transform of $F$ with respect to the probability density. Inversion of the Schrödinger equation leads us to consider an $x$– $\psi$ duality related to a modular symmetry. The scaling of $x$ is determined by the “beta function,” suggesting that in quantum mechanics the space coordinate is a macroscopic variable of a statistical system with $ħ$ playing the role of scale. The formalism is extended to higher dimensions and to the Klein-Gordon equation.