Phase-retrieval from Bohm's equations

Abstract: We present an analytic method, based on the Bohmian equations for quantum mechanics, for approaching the phase-retrieval problem in the following formulation: By knowing the probability density $\left\vert \psi\left(\overrightarrow{r},t\right)\right\vert ^{2}$ and the energy potential $V\left(\overrightarrow{r},t\right)$ of a system, how can one determine the complex state $\psi\left(\overrightarrow{r},t\right)$? We illustrate our method with three classic examples involving Gaussian states, suggesting applications to quantum state and Hamiltonian engineering.