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.
Highlights
In most physical models, including those of elementary particles and condensed matter, the continuity equation, ∂t ρ + ∇ · j = 0, is of fundamental importance, expressing the basic concept that the variation in time of the number of particles inside a certain region is opposite to the number of those crossing the boundary of that region in the same time
To visualize the essence of this statement it is sufficient to think of a 1D system, like a fluid flowing in a pipe or charge carriers flowing in a wire, in which case the equation reads ∂t ρ + ∂ x j = 0, where j = ρv
We recall that the current density in fractional quantum mechanics is given by: h i α jα = −ihα−1 Dα Ψ∗ (−∇2 ) 2 −1 ∇Ψ − c.c
Summary
Some useful quantum mechanical models exist, which are explicitly non-local One of these is the fractional Schrödinger equation [10,11], where the kinetic energy operator is proportional to |p|α ,. Wei [12] has recently shown that its continuity equation contains an anomalous source term, taking the form ∂t ρ + ∇ · j = Iα , where Iα is called the “extra-current” This property is related to the non-locality of the fractional. While in our present calculations only free wave packets are considered, results in this field have potential applications in diffraction-free and self-healing optoelectronic devices Another well-known non-local wave equation of the Schrödinger kind is the equation: ih.
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