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

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  • Giovanni Modanese

    (Faculty of Science and Technology, Free University of Bozen-Bolzano, I-39100 Bolzano, Italy)

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.

Suggested Citation

  • Giovanni Modanese, 2018. "Time in Quantum Mechanics and the Local Non-Conservation of the Probability Current," Mathematics, MDPI, vol. 6(9), pages 1-14, September.
    • Handle: RePEc:gam:jmathe:v:6:y:2018:i:9:p:155-:d:167759
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    References listed on IDEAS

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    1. E. K. Lenzi & B. F. de Oliveira & N. G.C. Astrath & L. C. Malacarne & R. S. Mendes & M. L. Baesso & L. R. Evangelista, 2008. "Fractional approach, quantum statistics, and non-crystalline solids at very low temperatures," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 62(2), pages 155-158, March.
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    Cited by:

    1. Lenzi, E.K. & Evangelista, L.R. & Zola, R.S. & Scarfone, A.M., 2022. "Fractional Schrödinger equation for heterogeneous media and Lévy like distributions," Chaos, Solitons & Fractals, Elsevier, vol. 163(C).

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