Consider a particle moving in a potential field考虑粒子在势场.docVIP

NUMERICAL ANALYSIS FOR ENGINEERING TERM PAPER: Solving the Schrodinger Equation for a Particle in an Infinite Potential Well Eleanor Kaufman December 9, 1999 CONTENTS: Introduction Discussion of Analytical Solution Discussion of Numerical Techniques Discussion of Numerical Normalization Techniques List of Symbols Used Appendix A: Algorithms to find Eigenvalues Appendix B: Algorithms to Solve Initial Value Problems Appendix C: Numerically Determined Eigenfunctions Appendix D: Selected Normalized Eigenfunctions Appendix E: Error in Normalized Eigenfunctions for 1st Energy Eigenvalue Appendix F: Error in Normalized Eigenfunctions for 2nd Energy Eigenvalue INTRODUCTION Consider a particle moving in a three dimensional potential field. The wave function, ((r,t), is a description of the probability that the particle will be within a given spatial volume at a given time. The position probability density is given by:. Introducing the concept of wave-particle duality, a particle of mass m, well defined momentum p and energy E is described by this wave function . The Schr?dinger equation is a second order differential equation describing the wave function ( that can be derived by differentiating first with respect to time and then twice with respect to the spatial coordinates. Then the classical relation can be used to equate the two results. (For a more complete description of this derivation, please see Reference 2). Most generally, the Schr?dinger equation involves a 3-dimensional time dependent wave function ((x,y,z,t)= ((r,t): The 1 dimensional, time-independent Schr?dinger equation is Since ( depends only on x, the derivatives are no longer partial. The objective of this paper is to explore the 1-dimensional time-independent Schr?dinger equation as it applies to a particle inside of an infinite square potential well. The potential inside of the well is zero, while the potential outside of the well is infinite. For convenience, the wel