On the unique mapping relationship between initial and final quantum states

TL;DR

This paper demonstrates how the Bohmian formulation of quantum mechanics uniquely links initial and final states through probability tubes, enabling direct calculation of final probabilities from initial conditions, illustrated by tunneling and diffraction examples.

Contribution

It introduces probability tubes in Bohmian mechanics to establish a unique mapping between initial and final quantum states, resolving ambiguity in standard quantum mechanics.

Findings

Probability tubes maintain constant enclosed probability over time.

Final probabilities can be derived from initial localized conditions.

Numerical examples confirm the method's effectiveness in tunneling and diffraction.

Abstract

In its standard formulation, quantum mechanics presents a very serious inconvenience: given a quantum system, there is no possibility at all to unambiguously (causally) connect a particular feature of its final state with some specific section of its initial state. This constitutes a practical limitation, for example, in numerical analyses of quantum systems, which often make necessary the use of some extra assistance from classical methodologies. Here it is shown how the Bohmian formulation of quantum mechanics removes the ambiguity of quantum mechanics, providing a consistent and clear answer to such a question without abandoning the quantum framework. More specifically, this formulation allows to define probability tubes, along which the enclosed probability keeps constant in time all the way through as the system evolves in configuration space. These tubes have the interesting property that once their boundary is defined at a given time, they are uniquely defined at any time. As a consequence, it is possible to determine final restricted (or partial) probabilities directly from localized sets of (Bohmian) initial conditions on the system initial state. Here, these facts are illustrated by means of two simple yet physically insightful numerical examples: tunneling transmission and grating diffraction.