Generalized guidance equation for peaked quantum solitons: the single particle case

TL;DR

This paper explores non-linear Schr{"o}dinger equations with solitonic solutions, deriving a generalized guidance equation that aligns with de Broglie-Bohm theory, especially for peaked quantum solitons and spinorial electrons.

Contribution

It introduces a class of non-linear Schr{"o}dinger equations with static solitons and derives a generalized guidance equation consistent with de Broglie-Bohm theory for these solutions.

Findings

Peaked quantum solitons obey a generalized guidance equation.

In the non-relativistic limit, guidance reduces to de Broglie-Bohm velocity.

Spinorial electron barycenters follow the de Broglie-Bohm guidance.

Abstract

We study certain non-linear generalisations of the Schr{\"o}dinger equation which admit static solitonic 2 solutions in absence of external potential acting on the particle. We consider a class of solutions that can be written as a product of a solution of the linear Schr{\"o}dinger equation with a peaked quantum soliton, in a regime where the size of the soliton is quite smaller than the typical scale of variation of the linear wave. In the non-relativistic limit, the solitons obey a generalized de Broglie-Bohm (dB-B) guidance equation. In first approximation, this guidance equation reduces to the dB-B guidance equation according to which they move at the so-called de Broglie-Bohm velocity along the hydrodynamical flow lines of the linear Schr{\"o}dinger wave. If we consider a spinorial electronic wave function a la Dirac, its barycentre is predicted to move exactly in accordance with the dB-B guidance equation.