Time in quantum mechanics and the local nonconservation of the probability current
Subject
time in quantum mechanics; local conservation laws; Fractional Schrödinger equation; Schrödinger equation with nonlocal potential
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 nonrelativistic 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 nonlocal terms have been successfully employed in several applications. The nonlocality 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 nonlocal 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 nonlocal Gorkov equation for superconductors and their proximity effect is also outlined.
URI
http://dx.doi.org/10.3390/math6090155https://www.mdpi.com/22277390/6/9/155
https://bia.unibz.it/handle/10863/12228
Collections
Related items
Showing items related by title, author, creator and subject.

Electromagnetic coupling of strongly nonlocal quantum mechanics
Modanese G (2017)Although standard quantum mechanics has some nonlocal features, the probability current of the Schrodinger equation is locally conserved, and this allows minimal electromagnetic coupling. For some important extensions of ... 
Generalized Maxwell equations and charge conservation censorship
Modanese G (2017)The AharonovBohm electrodynamics is a generalization of Maxwell theory with reduced gauge invariance. It allows to couple the electromagnetic field to a charge which is not locally conserved, and has an additional degree ... 
A comparative study of iterative solutions to linear systems arising in quantum mechanics
Jing YF; Huang TZ; Duan Y; Carpentieri B (2010)This study is mainly focused on iterative solutions with simple diagonal preconditioning to two complexvalued nonsymmetric systems of linear equations arising from a computational chemistry model problem proposed by Sherry ...