A Multipoint Method for Approximation of Wavefunctions at a Mesoscopic Scale

Abstract

We discuss the concept of a mesoscopic wavefunction, first in a general context, as the concept of wavefunction has evolved, and then in a more specific context of modeling. The paper concentrates on a simple, abstract one-dimensional situation. In this context, there are three problems to be considered. In the first problem, we consider the construction of a wavefunction as a problem of interpolation, with information content in a multipoint perspective at finitely many discrete points and complete uncertainty elsewhere. The wavefunction is conceived abstractly as our unified subjective picture of information content. Each point of information is essentially free and independent of all others. This is a wavefunction for a classical system at the mesoscopic threshold. In the second problem, we consider how, using the concept of scaling and renormalization, the classical system can be brought to represent a mesoscopic level of integrated information, with points still treated as free, but now with the need to consider each point as an extended region, with possible boundary overlaps. We then, in the section on the third problem, consider modeling this renormalized system as an Ising-like system of interacting spins. This is the final picture we develop for a mesoscopic wavefunction. This can be viewed from the perspective of the evolution of the concept of wavefunction at the microscopic level, and we briefly discuss the new point of view being developed here. Finally, we present a discussion concerning the bearing of this on Gibbs phenomenon.

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