Reconciling semiclassical and Bohmian mechanics. I. Stationary states

The semiclassical method is characterized by finite forces and smooth, well-behaved trajectories, but also by multivalued representational functions that are ill behaved at caustics. In contrast, quantum trajectory methods - based on Bohmian mechanics (quantum hydrodynamics) - are characterized by divergent forces and erratic trajectories near nodes, but also well-behaved, single-valued representational functions. In this paper, we unify these two approaches into a single method that captures the best features of both, and in addition, satisfies the correspondence principle. Stationary eigenstates in one degree of freedom are the primary focus, but more general applications are also anticipated