Quantum Mechanics on the Macintosh®

317 Pages·1994·13.14 MB·German

by  Siegmund Brandt, Hans Dieter Dahmen (auth.)

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Quantum Mechanics on the Macintosh®
Siegmund Brandt Hans Dieter Dahmen
Quantum Mechanics
on the Macintosh
®
Second Edition
Springer Science+Business Media, LLC
Siegmund Brandt
Hans Dieter Dahmen
Physics Department
Siegen University
57068 Siegen
Germany
Library of Congress Cataloging-in-Publication Data
Brandt, Siegmund.
Quantum mechanics on the Macintosh/Siegmund Brandt, Hans Dieter
Dahmen. - Second ed.
p. cm.
Includes bibliographical references and index.
paper)
1. Quantum theory-Computer programs. 2. Quantum theory-Computer
assisted instruction. 3. Macintosh (Computer). 4. Interactive
computer systems. 5. Computer graphics. I. Dahmen, Hans Dieter,
1936- II. Title.
QCI74.17.D37B73 1994
530.1 '2 '02855369-dc20 94-19963
Printed on acid-free paper.
© 1991, 1995 Springer Science+Business Media New York
Originally published by Springer-Verlag New York, Inc in 1995.
Softcover reprint of the hardcover 1st edition 1995
All rights reserved. This work may not be translated or copied in whole or in part without the
written permission of the publisher ( Springer Science+Business Media, LLC), except for brief
excerpts in connection with reviews or scholarly analysis.
Use in connection with any form of information storage and retrieval, electronic
adaptation, computer software, or by similar or dissimilar methodology now known or hereaf
ter developed is forbidden.
The use of general descriptive names, trade names, trademarks, etc., in this publication, even
if the former are not especially identified, is not to be taken as a sign that such names, as
understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely
by anyone.
The program on the enclosed diskette is under copyright protection and may not be reproduced
without written permission from Springer-Verlag. One copy of the program may be made as a
backup, but all other copies offend copyright law.
The program was written in the programming language FORTRAN 77 and compiled using a
FORTRAN compiler by Language Systems Corp. This is acknowledged by the following
copyright notice: Certain portions of this software are copyrighted by Language Systems
Corp. © 1988-1990 Language Systems Corp.
Before using the program, please consult the technical manuals provided by the manufacturer
of the computer - and of any additional plug-in boards - to be used. The authors and publisher
accept no legal responsibility for any damage caused by improper use of the instructions and
program contained herein. Although the program has been tested with extreme care, we can
offer no formal guarantee that it will function correctly.
Apple, Macintosh, ImageWriter, and LaserWriter are registered trademarks of Apple Com
puter, Inc.
POSTSCRIPT is a registered trademark of Adobe Systems, Inc.
Production managed by Francine McNeill; manufacturing supervised by Jacqui Ashri.
Camera-ready copy prepared by the authors using TeX.
Printed and bound by Edwards Brothers, Inc., Ann Arbor, MI.
987654321
ISBN 978-3-540-94272-6 ISBN 978-3-662-25104-1 (eBook)
DOI 10.1007/978-3-662-25104-1
Preface to the Second Edition
Ever since we published our Picture Book of Quantum Mechanicsl we have
been asked to make available the programs we wrote to generate the computer
graphics that illustrate the book. We have called the result INTERQUANTA
(the Interactive Program ofQ uantum Mechanics), which we like to abbreviate
further by IQ.
This book is essentially an extensive user's guide for INTERQUANTA.
The book also contains a short summary of the different subjects of quantum
mechanics treated by IQ as well as a large number of exercises. The program
can be used in two ways. By working through (at least a part of) these exer
cises, the user of IQ explores a computer laboratory in quantum mechanics
gaining experience in this abstract field by performing computer experiments.
No knowledge of computer programming is required. The user only has to
learn how to use some ofthe simple IQ commands and, in particular, the exten
sive HELP facilities. A simpler way to use IQ is to study one or several of the
ready-made demonstrations. In each demonstration the user is taken through
one chapter of quantum mechanics. Short explanatory texts displayed on the
screen interchange with graphics illustrating quantum-mechanical problems
which are solved by the program.
The program was originally published2 for use on Personal Computers
(mM PC or PS/2). The program is coded in a machine-independent way in
FORTRAN 77. This makes it easy to adapt it to other computers but difficult
to take advantage of special machine-dependent features. For this reason no
use was made of the mouse.
In the present edition there are only a few changes in the text of the book,
but a number of misprints and errors was corrected. The installation proce
dure was simplified and Appendixes Band C were changed accordingly. The
diskette contains two versions of the program. One version requires the pres
ence of a mathematical coprocessor. The other does not. Both versions were
thoroughly tested on several Macintosh models including the Quadra.
I S. Brandt and H. D. Dahmen, The Picture Book of Quantum Mechanics, 2nd edition,
Springer-Verlag, New York 1994
2S. Brandt and H. D. Dahmen, Quantum Mechanics on the Personal Computer, 3rd edition,
Springer-Verlag, Berlin 1994
vi Preface to the Second Edition
It is a pleasure to acknowledge the generous help provided by mM Ger
many in the development of IQ. In particular we want to thank Dr. U. Groh
for his competent help in the early phase of the work.
At various stages of the project we were helped considerably. by friends
and students in Siegen. We would particularly like to thank Tilo Stroh for his
many valuable contributions.
Last but not least, we thank Drs. H.-U. Daniel, T. von Foerster and H. J.
KOlsch of Springer-Verlag for their constant interest and support.
Siegen, Germany Siegmund Brandt
May 1994 Hans Dieter Dahmen
Contents
Preface v
1 Introduction 1
1.1 Interquanta........ 1
1.2 The Structure of this Book 2
1.3 The Computer Laboratory 3
1.4 The Classroom Demonstrations 3
1.5 Literature............ 4
2 Free Particle Motion in One Dimension 5
2.1 Physical Concepts . . . . . . . . . . . . . . . . . 5
2.2 A First Session with the Computer . . . . . . . . . 9
2.3 The Time Development of a Gaussian Wave Packet 13
2.4 The Spectral Function of a Gaussian Wave Packet 15
2.5 The Wave Packet as a Sum of Harmonic Waves 16
2.6 Exercises..................... 19
3 Bound States in One Dimension 22
3.1 Physical Concepts . . . . . . . . . . . . . . . . . . . . .. 22
3.2 Eigenstates in the Infinitely Deep Square-Well Potential and
in the Harmonic-Oscillator Potential . . . . . . . . . . . .. 29
3.3 Eigenstates in the Step Potential . . . . . . . . . . . . . .. 32
3.4 Harmonic Particle Motion . . . . . . . . . . . . . . . . .. 35
3.5 Particle Motion in the Infinitely Deep Square-Well Potential. 36
3.6 Exercises........................... 38
4 Scattering in One Dimension 44
4.1 Physical Concepts . . . . . . . . . . . . . . . . . . 44
4.2 Stationary Scattering States in the Step Potential. . . 58
4.3 Scattering of a Harmonic Wave by the Step Potential 60
4.4 Scattering of a Wave Packet by the Step Potential . 61
4.5 Transmission and Reflection. The Argand Diagram 64
4.6 Exercises...... 66
4.7 Analogies in Optics ................ 76
vii
viii Contents
4.8 Reflection and Refraction of Stationary Electromagnetic Waves 80
4.9 Reflection and Refraction ora Hannonic Light Wave . . .. 82
4.10 Scattering of a Wave Packet of Light . . . . . . . . . . . .. 83
4.11 Transmission, Reflection and Argand Diagram for a Light Wave 86
4.12 Exercises. . . . . . . . . . . . . . . . . . . . . . . 88
5 A Two-Particle System: Coupled Harmonic Oscillators 91
5.1 Physical Concepts . . . . . . . . . . . 91
5.2 Stationary States . . . . . . . . . . . . 97
5.3 Time Dependence of Global Quantities 97
5.4 Joint Probability Densities 100
5.5 Marginal Distributions 102
5.6 Exercises......... 103
6 Free Particle Motion in Three Dimensions 109
6.1 Physical Concepts . . . . . . . . . . . . . . . . . 109
6.2 The 3D Hannonic Plane Wave . . . . . . . . . . . 119
6.3 The Plane Wave Decomposed into Spherical Waves 122
6.4 The 3D Gaussian Wave Packet . . . . . . . . . . . 123
6.5 The Probability Ellipsoid. . . . . . . . . . . . . . 125
6.6 Angular-Momentum Decomposition of a Wave Packet. 126
6.7 Exercises........................ 128
7 Bound States in Three Dimensions 131
7.1 Physical Concepts . . . . . . . . . . . . . 131
7.2 Radial Wave Functions in Simple Potentials 141
7.3 Radial Wave Functions in the Step Potential 145
7.4 Probability Densities . . . 147
7.5 Harmonic Particle Motion 151
7.6 Exercises......... 153
8 Scattering in Three Dimensions 157
8.1 Physical Concepts . . . . . . . . . . . . . . . . 157
8.2 Radial Wave Functions. . . . . . . . . . . . . . 165
8.3 Stationary Wave Functions and Scattered Waves. 168
8.4 Differential Cross Sections . . . . . . . . . . . . 170
8.5 Scattering Amplitude. Phase Shift. Partial and Total Cross
Sections . . . . . . . . . . . . . . . . . . . . . . . . . .. 171
8.6 Exercises................ 175
9 Special Functions of Mathematical Physics 180
9.1 Basic Formulae. . . . 180
9.2 Hermite Polynomials ......... . 187
Contents IX
9.3 Eigenfunctions of the One-Dimensional Harmonic Oscillator 187
9.4 Legendre Polynomials and Associated Legendre Functions 188
9.5 Spherical Harmonics. . . . 192
9.6 Bessel Functions. . . . . . 192
9.7 Spherical Bessel Functions. 195
9.8 Laguerre Polynomials ... 196
9.9 Radial Eigenfunctions of the Harmonic Oscillator 198
9.10 Radial Eigenfunctions of the Hydrogen Atom 200
9.11 Simple Functions of a Complex Variable . 200
9.12 Exercises .................. . 202
10. Additional Material and Hints for the Solution of Exercises 204
10.1 Units and Orders of Magnitude. . . . . . . . . . . . 204
10.2 Argand Diagrams and Unitarity for One-Dimensional
Problems ............. . 211
10.3 Hints and Answers to the Exercises 220
Appendix
Appendix A. A Systematic Guide to IQ 246
A.I Dialog Between the User and IQ . 246
A.I.I A Simple Example . . . . 246
A.I.2 The General Form of Commands 249
A. 1.3 The Descriptor File .. 250
A.I.4 The Descriptor (Record) 253
A.I.5 The PLOT Command . 255
A.I.6 The STOP Command . 256
A.I.7 HELP: The Commands HE and PH 256
A.2 Coordinate Systems and Transformations . 257
A.2.1 The Different Coordinate Systems . 257
A.2.2 Defining the Transformations ... 258
A.3 The Different Types of Plot ........ 263
A.3.1 Choosing a Plot Type: The Command CH 263
A.3.2 Cartesian 3D Plots (Type 0 Plots) 263
A.3.3 Polar 3D Plots (Type 1 Plots) . 264
A.3.4 2D Plots (Type 2 Plots) ..... 266
A.3.5 3D Column Plots (Type 3 Plots) . 269
A.3.6 Special 3D Plots (Type 10 Plots) . 270
A.4 The Background in the Plots . . . . . . . 270
AA.I Boxes and Coordinate Axes: The Command BO . 270
A.4.2 Scales . . . . . . 271
A.4.3 Arrows...... 273
A.4.4 Text and Numbers 275