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.

Support from the Ministerio de Econom´ıa y Competitividad (Spain) under Projects and FIS2010-22082 and FIS2011-29596-C02-01, as well as from the COST Action MP1006 (Fundamental Problems in Quantum Physics) is acknowledged. The authors are grateful to anonymous referees for useful comments. A. S. Sanz thanks the Ministerio de Econom´ıa y Competitividad for a “Ram´on y Cajal” Research Fellowship and the University College London for its kind hospitality.

Peer Reviewed