Property of Zero-Energy Flows and Creations and Annihilations of Vortices in Quantum Mechanics

TL;DR

This paper investigates vortex creation and annihilation in quantum mechanics using Gel'fand triplet eigenstates, revealing zero-energy flows' stability, informational richness, and energy efficiency.

Contribution

It introduces a novel analysis of vortex dynamics via conjugate space eigenstates, highlighting zero-energy flows' properties in quantum systems.

Findings

Zero-energy flows are energy-efficient mechanisms.

Infinite variety of zero-energy flows carry extensive information.

Zero-energy flow patterns are stable against disturbances.

Abstract

Time-dependent processes accompanied by vortex creations and annihilations are investigated in terms of the eigenstates in conjugate spaces of Gel'fand triplets in 2-dimensions. Creations and annihilations of vortices are described by the insertions of unstable eigenstates with complex-energy eigenvalues into stable states written by the superposition of eigenstates with zero-energy eigenvalues. Some concrete examples are presented in terms of the eigenfunctions of the 2-dimensional parabolic potential barrier, i.e., $-m \gamma^2 (x^2+y^2)/2$. We show that the processes accompanied by vortex creations and annihilations can be analyzed in terms of the eigenfunctions in the conjugate spaces of Gel'fand triplets. Throughout these examinations we point out three interesting properties of the zero-energy flows. (i) Mechanisms using the zero-energy flows are absolutely economical from the viewpoint of energy consumption. (ii) An enormous amount of informations can be discriminated in terms of the infinite variety of the zero-energy flows. (iii) The zero-energy flow patterns are absolutely stable in any disturbance by inserting arbitrary decaying flows with complex-energy eigenvalues.