Technical report | Rapid solution of the Schrödinger equation: Towards a study of the utility of the Bohm filter

Abstract

The second project report for the Efficient Generation and Evolution of Probability Density Maps project is reproduced as a Defence Science and Technology Group technical report. Here we focus on solving the Schrödinger equation numerically for several simple potentials using Fourier and Chebyshev pseudo-spectral methods. The report has been written in such way to be more pedagogical rather than complete.

Executive Summary

The Bohm filter as proposed by Drake [1] is an attempt to confront some of the shortcomings of the well-known Kalman filter and its variants. The underlying idea is to use a Schrödinger equation to model the motion of an object we wish to track. This report is a first step towards developing a filter based on such an approach. Analytic solutions to the Schrödinger equation are only known for a few well known potentials. As a prerequisite to constructing a filter based on this approach we will require accurate and efficient methods to solve the second order differential equation for potentials relevant to tracking problems. This aspect of implementing a Bohm filter is the focus of this report. Here we concentrate on finding numerical solutions to the Schrödinger equation for several time-independent potentials using Fourier and Chebyshev pseudo-spectral methods.

The report begins with a brief introduction to several relevant topics such as the reduction of the time-dependent Schrödinger equation to the time-independent equation for time independent potentials, collocation interpolation and numerical quadrature. We then proceed to introduce Fourier and Chebyshev pseudo-spectral methods. We then apply them to several time-independent Schrödinger equations. We then conclude and discuss future research directions.

The report has been written in a pedagogical style with many illustrative examples given. For this reason it may also be of use to a wider audience interested in solving other unrelated differential equations. In the appendices we have provided a few additional pieces of information. In particular, we have written several simple MATLAB scripts which illustrate various ideas presented in the report. These scripts are stored in Govdex and are available upon request. All figures found herein can be obtained using these scripts with either no or only minimal modifications required.