Non-dispersive wavepacket solutions of the Schrodinger equation

TL;DR

This paper investigates non-dispersive wavepacket solutions of the Schrödinger equation, examining their properties and potential as basis states, contrasting with traditional plane wave solutions.

Contribution

It introduces and analyzes constant velocity wavepacket solutions of the Schrödinger equation and explores their suitability as basis states.

Findings

Wavepacket solutions are eigenvectors of a symmetric momentum operator.

These solutions are non-dispersive and maintain shape over time.

Potential to use wavepackets as basis states instead of plane waves.

Abstract

The free Schrodinger equation has constant velocity wavepacket solutions \psi_{\bf v} of the form \psi= f({\bf r} - {\bf v}t) e^{- i m c^2 t / 2}. These solutions are eigenvectors of a momentum operator {\bf \tilde p} which is symmetric in a positive definite scalar product space. We discuss whether these \psi_{\bf v} can act as basis states rather than the usual plane wave solutions.