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Mathematics 2018, 6(9), 155; https://doi.org/10.3390/math6090155

Time in Quantum Mechanics and the Local Non-Conservation of the Probability Current

Faculty of Science and Technology, Free University of Bozen-Bolzano, I-39100 Bolzano, Italy
Received: 19 June 2018 / Revised: 17 August 2018 / Accepted: 3 September 2018 / Published: 4 September 2018
(This article belongs to the Special Issue Time and Time Dependence in Quantum Mechanics)
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Abstract

In relativistic quantum field theory with local interactions, charge is locally conserved. This implies local conservation of probability for the Dirac and Klein–Gordon wavefunctions, as special cases; and in turn for non-relativistic quantum field theory and for the Schrödinger and Ginzburg–Landau equations, regarded as low energy limits. Quantum mechanics, however, is wider than quantum field theory, as an effective model of reality. For instance, fractional quantum mechanics and Schrödinger equations with non-local terms have been successfully employed in several applications. The non-locality of these formalisms is strictly related to the problem of time in quantum mechanics. We explicitly compute, for continuum wave packets, the terms of the fractional Schrödinger equation and the non-local Schrödinger equation by Lenzi et al. that break local current conservation. Additionally, we discuss the physical significance of these terms. The results are especially relevant for the electromagnetic coupling of these wavefunctions. A connection with the non-local Gorkov equation for superconductors and their proximity effect is also outlined. View Full-Text
Keywords: time in quantum mechanics; local conservation laws; Fractional Schrödinger equation; Schrödinger equation with non-local potential time in quantum mechanics; local conservation laws; Fractional Schrödinger equation; Schrödinger equation with non-local potential
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).
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Modanese, G. Time in Quantum Mechanics and the Local Non-Conservation of the Probability Current. Mathematics 2018, 6, 155.

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