A method for obtaining nonspreading solutions of the Schr\"odinger equation

TL;DR

This paper introduces a straightforward analytical approach to derive nonspreading solutions of the time-dependent Schrödinger equation, utilizing Airy functions and phase restrictions, applicable in various dimensions and potential scenarios.

Contribution

The paper presents a novel, simple method for obtaining nonspreading Schrödinger solutions, extending to linear potentials and multiple dimensions, with potential for nonlinear cases.

Findings

Nonspreading solutions are derived using Airy functions.

Method applies to free particles and linear potentials.

Works in one and two dimensions, extendable further.

Abstract

In this paper, we present a simple analytical method for obtaining a nonspreading solution of the time-dependent Schr\"odinger equation, which is given by the Airy function. The solution is derived by imposing a restriction on the phase factor of the ansatz that is taken to solve the differential equation. Considering at first the free particle, we show that nonspreading solutions can also be obtained for a time-dependent linear potential. The method is shown to work in both one and two dimensions, and can be easily extended if required. The applicability of the method is discussed in relation to the nonlinear case.