A Class of Self-Trapped and Self-Focusing Wave Functions in Madelung Fluid Picture of A Single Free Particle Quantum System

TL;DR

This paper explores a class of wave functions in the Madelung fluid framework that are self-trapped and self-focusing, demonstrating how the quantum potential they generate leads to localization of the quantum probability density over time.

Contribution

It introduces a specific class of self-trapped wave functions in the Madelung picture and analyzes their self-focusing behavior and localization effects.

Findings

Quantum probability density can be self-trapped by its own quantum potential.

The quantum potential's convexity enhances localization over a finite time.

Self-generated focusing leads to finite-time localization of the wave function.

Abstract

Using the Madelung fluid picture of Schr\"odinger equation for a single free particle moving in one spatial dimension, we shall specify a class of wave functions whose quantum probability density is being trapped by the quantum potential it itself generates. The global convexity of the quantum potential generated by the initial self-trapped wave function will then be shown to further localize the quantum probability density through a self-generated focusing equation for a finite interval of time.