Reconciling Semiclassical and Bohmian Mechanics: I. Stationary states

TL;DR

This paper proposes a unified approach combining semiclassical and Bohmian mechanics to better describe stationary quantum states, capturing the smooth trajectories of semiclassical methods and the well-behaved functions of Bohmian mechanics.

Contribution

It introduces a novel method that unifies semiclassical and Bohmian mechanics, improving the description of stationary states while satisfying the correspondence principle.

Findings

Unified method captures features of both approaches

Improves representation of stationary eigenstates

Anticipates broader applications beyond stationary states

Abstract

The semiclassical method is characterized by finite forces and smooth, well-behaved trajectories, but also by multivalued representational functions that are ill-behaved at turning points. In contrast, quantum trajectory methods--based on Bohmian mechanics (quantum hydrodynamics)--are characterized by infinite 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.