Theoretical Physics on the Personal Computer

222 Pages·1990·5.865 MB·English

by  Professor Dr. Erich W. Schmid, Dr. Gerhard Spitz, Wolfgang Lösch (auth.)

E.W. Schmid
G. Spitz
W. Losch
Theoretical Physics
on the Personal Computer
Erich W Schmid Gerhard Spitz
Wolfgang Losch
Theoretical Physics
on the
Personal Computer
Second Edition
With 154 Figures
Springer-Verlag Berlin Heidelberg New York
London Paris Tokyo Hong Kong
Professor Dr. Erich W. Schmid
Institut fOr Theoretische Physik, Universitat TObingen, Auf der Morgenstelle 14,
0-7400 TObingen, Fed. Rep. of Germany
Dr. Gerhard Spitz
Siemens AG, 0-8000 MOnchen 70, Fed. Rep. of Germany
Wolfgang Losch
FB Physik, Universitat Essen, Universitatsstr. 5, 0-4300 Essen, Fed. Rep. of Germany
Tt-ans/ator: A. H. Armstrong
"Everglades", Brimpton Common, Reading, RG74 RY Berks, UK
Additional material to this book can be downloaded from http://extras.springer.com.
Title of the original German edition: Theoretische Physik mit dem Personal Computer
ISBN 3-540-18310-8 Springer-Verlag Berlin Heidelberg New York
ISBN 0-387-18310-8 Springer-Verlag New York Berlin Heidelberg
© Springer-Verlag, Berlin, Heidelberg 1987
ISBN-13:978-3-642-75473-9 e-ISBN-13:978-3-642-75471-5
001: 10.1007/978-3-642-75471-5
Library of Congress Cataloging-in-Publication Data. Schmid, Erich w., 1931-[Theoretische Physik mit dem Per
sonal-Computer. English] Theoretical physics on the personal computer 1 Erich W. Schmid, Gerhard Spitz,
Wolfgang Losch; [translator, A. H. Armstrong]. - 2nd ed. p. cm. Translation of: Theoretische Physik mit dem Per
sonal-Computer.lncludes bibliographical references(p.lSBN-13:978-3-642-75473-9(U.S.:alk.paper)1.Mathematical
physics-Data processing. 2. Microcomputers. I. Spitz, Gerhard, 1955-. II. Losch, Wolfgang, 1956-. III. Title.
aC20.7.E4S3513 1990 530.1'0285'526-dc20 90-9441
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned,
specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on
microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only per
mitted under the provisions of the German Copyright Law of September 9, 1965, in its version of June 24, 1985,
and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law.
© Springer-Verlag Berlin Heidelberg 1988 and 1990
Softcover reprint of the hardcover 2nd edition 1990
The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific
statement, that such names are exempt from the relevant protective laws and regulations and therefore free for
general use.
Please note: Before using the programs contained in this book, please consult for technical advice the manuals
provided by the respective manufacturer of the computer - and any additional plug-in-boards - to be used. The
authors and the publisher accept no legal responsibility for any damage caused by improper use of instructions
and programs contained within. The programs appearing here have been tested carefully. Nevertheless we can
offer no guarantee for the correct functioning of the programs.
2156/3150-543210 - Printed on acid-free paper
Preface to the Second English Edition
The second English edition has given us the opportunity to implement some
suggestions made by our readers.
A version of our plot package adapted to the Microsoft FORTRAN-77
compiler (on a diskette supplied by H. U. Zimmermann) had to be mailed to
many readers who have a computer without a coprocessor and who therefore
have to use a compiler which does not require one. For this reason we have
modified the diskette accompanying our book by offering the option to work
with Microsoft FORTRAN- 77 (version 4.0 and up). In a last-minute effort we
were also able to offer the option to use the new FORTRAN /2 compiler by
IBM; we thank Dr. von Dehn for his quick help.
The list of personal computers on which we have installed and tested the
programs given in the book has become longer. It now includes the IBM PS/2
series, Siemens PC-D and PC-D2, and many other IBM-compatible computers.
Chapter 1 of the book has been rewritten because of the new options. In
other chapters we have corrected errors; some paragraphs have been modified
to improve understanding.
Tiibingen, January 1990 E. W. Schmid, G. Spitz
v
Preface to the First English Edition
We would like to thank Mr. A.H. Annstrong, who translated this book, for his
many valuable suggestions and corrections. We also acknowledge a stimulating
response from our readers. Mr. J. Peeck sent us a diskette containing the pro
grams modified to run on an ATARI computer. Mr. H.U. Zimmermann sent us
diskettes, on which the graphics software of the book is adapted to the require
ments of the FORTRAN-77 compiler by MICROSOFT. Readers interested in
these adaptations should contact the authors.
Tiibingen, January 1988 E. W. Schmid, G. Spitz
VII
Preface to the German Edition
This book is based on the lecture course "Computer applications in Theo
retical Physics", which has been offered at the University of Tiibingen since
1979. This course had as its original aim the preparation of students for a nu
merical diploma course in theoretical physics. It soon became clear, however,
that the course provides a valuable supplement to the fundamental lectures in
theoretical physics. Whereas teaching in this field had previously been prin
cipally characterised by the derivation of equations, it is now possible to give
deeper understanding by means of application examples. A graphical presen
tation of numerical results proves to be important in emphasizing the physics.
Interaction with the machine is also valuable. At the end of each calculation
the computer should ask the question: "Repeat the calculation with new data
(yes/no )?". The student can then answer "yes" and input the new data, e.g.
new starting values for position and velocity in solving an equation of motion.
The programming of a user-friendly dialogue is not really difficult, but
time consuming. At the beginning of the course the student therefore constructs
only the numerical parts of the programs. The numerical parts are therefore
deleted from the programs under consideration, and newly programmed by the
student. Later on, the programming of the graphical output and of the dialogue
is taught.
In the initial phase of the course several assistants contributed in the prepa
ration of the teaching schedule. We are particularly grateful to Dr. K. Hahn,
Dr. R. Kircher, Dr. H. Leeb, Dr. M. Orlowski and Dr. H. Seichter. For suggest
ing and discussing the example of "the electrostatic lens", we thank Prof. Dr. F.
Lenz and Prof. Dr. E. Kasper.
Until 1985 this course had been held at the terminals of the Computer
Centre of the University of Tiibingen. After the arrival of powerful personal
computers the students themselves showed a desire to have the course adapted
to these machines. With the welcome cooperation of IBM the FORTRAN pro
grams for all the chapters were rewritten for the personal computer as a diploma
task (W. Losch). In order to make the programs accessible to a wider circle of
users this book was produced. We wish you great fun on the personal computer!
Tiibingen, October 1987 E. W. Schmid, G. Spitz
IX
Contents
1. Introduction......... ........ .............................. ... 1
1.1 Programming of the Numerical Portions of the Programs.... 3
1.2 Programming of the Input and Output ...................... 6
2. Numerical Differentiation and Introduction into Screen
Dialogue ...................................................... 12
2.1 Formulation of the Problem................................. 12
2.2 Mathematical Methods ...................................... 13
2.3 Programming ............................................... 14
2.4 Exercises ................................................... 19
2.5 Solutions to the Exercises ................................... 19
3. Numerical Integration ....................................... 22
3.1 Formulation of the Problem ................................. 22
3.2 Numerical Methods... .............................. ........ 23
3.2.1 The Trapezoidal Rule ................................. 23
3.2.2 The Simpson Rule .................................... 24
3.2.3 Newton-Cotes Integration ............................. 25
3.2.4 The Gauss-Legendre Integration. . . . . . . . . . . . . . . . . . . . . . . 25
3.3 Programming............................................... 29
3.4 Exercises ................................................... 33
3.5 Solutions to the Exercises ................................... 33
4. Harmonic Oscillations with Sliding and Static Friction,
Graphical Output of Curves ................................ 35
4.1 Formulation of the Problem ................................. 35
4.2 Numerical Treatment ....................................... 36
4.2.1 Transformation of the Differential Equation ............ 36
4.2.2 The Euler Method .................................... 37
4.3 Programming ............................................... 37
4.4 Exercises ................................................... 41
4.5 Solutions to the Exercises ................................... 41
5. Anharmonic Free and Forced Oscillations ................. 43
5.1 Formulation of the Problem ................................. 43
5.2 Numerical Treatment ....................................... 44
XI
5.2.1 Improvement of the Euler Method.... ............... .. 44
5.2.2 The Runge-Kutta Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.3 Programming............................................... 47
5.4 Exercises ................................................... 49
5.5 Solutions to the Exercises ................................... 50
6. Coupled Harmonic Oscillations ............................. 53
6.1 Formulation of the Problem ................................. 53
6.2 Numerical Method.......................................... 54
6.3 Programming ............................................... 55
6.4 Exercises ................................................... 57
6.5 Solutions to the Exercises ................................... 57
7. The Flight Path of a Space Craft as a Solution of the
Hamilton Equations .......................................... 59
7.1 Formulation of the Problem............................... .. 59
7.2 Mathematical Methods...................................... 63
7.2.1 Mesh Width Adaptation in the Runge-Kutta Method.. 63
7.2.2 Coordinate Transformation............................ 66
7.3 Programming............................................... 67
7.3.1 Hamilton's Equations of Motion....... ............... . 67
7.3.2 Automatic Mesh Width Adjustment in the
Runge-Kutta Method................................. 69
7.3.3 Coordinate Transformation............................ 71
7.3.4 Main Program ........................................ 73
7.4 Exercises ................................................... 79
7.5 Solutions to the Exercises ................................... 79
8. The Celestial Mechanics Three-body Problem ............ 81
8.1 Formulation of the Problem ................................. 81
8.2 Mathematical Method ...................................... 85
8.3 Programming ............................................... 85
8.4 Exercises ................................................... 89
8.5 Solutions to the Exercises ................................... 89
9. Computation of Electric Fields by the Method of
Successive Over-relaxation .................................. 90
9.1 Formulation of the Problem................................. 90
9.2 Numerical Method........................................... 92
9.2.1 Discretisation of Laplace's Equation................... 92
9.2.2 The Method of Successive Over-relaxation ............. 93
9.3 Programming............................................... 95
9.4 Exercises ................................................... 100
9.5 Solutions to the Exercises ................................... 100
XII
10. The Van der Waals Equation ............................... 102
10.1 Formulation of the Problem................................ 102
10.2 Numerical Method ......................................... 104
10.3 Programming .............................................. 106
10.4 Exercises.................................................. 112
10.5 Solutions to the Exercises .................................. 113
11. Solution of the Fourier Heat Conduction Equation
and the "Geo-Power Station" ............................... 115
11.1 Formulation of the Problem ................................ 115
11.2 Method of Solution ........................................ 117
11.3 Programming.............................................. 119
11.4 Exercises .................................................. 121
11.5 Solutions to the Exercises .................................. 122
12. Group and Phase Velocity in the Example of Water Waves 125
12.1 Formulation of the Problem................................ 125
12.2 Numerical Method ......................................... 129
12.3 Programming .............................................. 131
12.4 Exercises .................................................. 134
12.5 Solutions to the Exercises .................................. 134
13. Solution of the Radial Schrodinger Equation by
the Fox-Goodwin Method ................................... 136
13.1 Formulation of the Problem ................................ 136
13.2 Numerical Method of Solution ............................. 140
13.3 Programming .............................................. 142
13.4 Exercises .................................................. 144
13.5 Solutions to the Exercises .................................. 145
14. The Quantum Mechanical Harmonic Oscillator........... 149
14.1 Formulation of the Problem................................ 149
14.2 Numerical Method ......................................... 150
14.3 Programming .............................................. 153
14.4 Exercises .................................................. 156
14.5 Solutions to the Exercises .................................. 156
15. Solution of the Schrodinger Equation in Harmonic
Oscillator Representation ................................... 158
15.1 Formulation of the Problem ................................ 158
15.2 Numerical Method ......................................... 159
15.3 Programming .............................................. 160
15.4 Exercises .................................................. 163
15.5 Solutions to the Exercises .................................. 163
XIII
16. The Ground State of the Helium Atom by the
Hylleraas Method ............................................ 165
16.1 Formulation of the Problem................... ............. 165
16.2 Setting up the State Basis and the Matrix Equation ........ 167
16.3 Programming .............................................. 172
16.4 Exercises .................................................. 180
16.5 Solutions to the Exercises .................................. 180
17. The Spherical Harmonics.................................... 181
17.1 Formulation of the Problem................... ............. 181
17.2 Numerical Method......................................... 184
17.3 Programming .............................................. 185
17.4 Exercises .................................................. 187
17.5 Solutions to the Exercises .................................. 188
18. The Spherical Bessel Functions ............................. 189
18.1 Formulation of the Problem ................................ 189
18.2 Mathematical Method ..................................... 191
18.3 Programming .............................................. 192
18.4 Exercises .................................................. 195
18.5 Solutions to the Exercises .................................. 195
19. Scattering of an Uncharged Particle from a Spherically
Symmetric Potential ......................................... 197
19.1 Formulation of the Problem................................ 197
19.2 Mathematical Treatment of the Scattering Problem ........ 200
19.3 Programming.............................................. 203
19.4 Exercises .................................................. 206
19.5 Solutions to the Exercises .................................. 207
References ......................................................... 209
Subject Index ..................................................... 211
XIV