We recount here some preliminary attempts to devise quantum analogues of certain aspects of Mather’s theory of minimizing measures [M1-2, M-F], augmented by the PDE theory from Fathi [F1,2] and from [E-G1]. This earlier work provides us with a Lipschitz continuous function $u$ solving the eikonal equation aėȧnd a probability measure $\sigma$ solving a related transport equation.We present some elementary formal identities relating certain quantum states $\psi$ and $u,\sigma$. We show also how to build out of $u,\sigma$ an approximate...