document: In exploring the WKB limit of quantum theory, Bohm [2] was the first to notice that although one starts with all the ambiguities about the nature of a quantum system, the first order approximation fits the ordinary classical ontology. By that we mean that the real part of the Schrödinger equation under polar decomposition of the wave function becomes the classical Hamilton–Jacobi equation in the limit where terms involving ℏ are neglected. In contrast to this approach, in this Letter we show that the classical trajectories arise from a short-time quantum propagator when terms of O(Δt2) can be neglected. This fact was actually already observed by Holland some twenty years ago: In page 269 of his book [6] infinitesimal time intervals are considered whose sequence constructs a finite path. It is shown that along each segment the motion is classical (negligible quantum potential), and that it follows that the quantum path may be decomposed into a sequence of segments along each of which the classical action is a minimum. The novel contribution of the present Letter is an improved proof of Hollandʼs result using an improved version of the propagator due to Makri and Miller [9,10]. (See also de Gosson [3] for a further discussion.)
doc_bio_tags: O O O B I I I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I I I I I I I O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O
extractive_keyphrases: hamilton–jacobi equation infinitesimal time intervals propagator schrödinger equation under polar decomposition of the wave function wkb limit of quantum theory
abstractive_keyphrases: short-time quantum propagator