Soliton-like solutions to the ordinary Schroedinger equation

TL;DR

This paper demonstrates the existence of soliton-like, localized wave solutions for the ordinary Schrödinger equation, providing both approximate and exact solutions, and discusses methods to obtain finite-energy solutions relevant to quantum mechanics.

Contribution

It shows that localized, soliton-like solutions exist for the Schrödinger equation, extending the concept of non-diffracting waves within standard quantum mechanics.

Findings

Existence of localized solutions for Schrödinger equation

Methods to obtain finite-energy solutions

Examples of solutions with and without potential

Abstract

In recent times it has been paid attention to the fact that (linear) wave equations admit of "soliton-like" solutions, known as Localized Waves or Non-diffracting Waves, which propagate without distortion in one direction. Such Localized Solutions (existing also for K-G and Dirac equations) are a priori suitable, more than Gaussian's, for describing elementary particle motion. In this paper we show that, mutatis mutandis, Localized Solutions exist even for the ordinary Schroedinger equation, within standard Quantum Mechanics; and we obtain both approximate and exact solutions, setting forth particular examples for them. In the ideal case such solutions bear infinite energy, as well as plane or spherical waves: we also demonstrate, therefore, how to obtain finite-energy solutions. At last, we briefly consider solutions for a particle moving in the presence of a potential. Some physical comments are added.