Mean position of a particle submitted to a potential barrier

TL;DR

This paper numerically investigates the mean position and standard deviation of a quantum particle described by the Klein-Gordon equation as it interacts with a potential barrier, focusing on their temporal evolution.

Contribution

It introduces a variational formulation and a Newmark numerical method to compute and analyze the particle's mean position and standard deviation over time.

Findings

Computed the time evolution of mean position and standard deviation.

Demonstrated the effectiveness of the variational and Newmark methods for this quantum problem.

Provided numerical insights into particle behavior near potential barriers.

Abstract

A one-dimensional Klein-Gordon problem, which is a physical model for a quantum particle submitted to a potential barrier, is studied numerically : using a variational formulation and a Newmark numerical method, we compute the mean position and standard deviation of the particle as well as their time evolution.