Theoretical Physics on the Personal Computer

219 Pages·1988·9.68 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
With 152 Figures
Springer-Verlag Berlin Heidelberg New York
London Paris Tokyo
Professor Dr. Erich W. Schmid
Dr. Gerhard Spitz
Wolfgang Losch
Institut fOr Theoretische Physik, Universilal T Obingen , Auf der Morgenstelle 14,
0-7400 TGbingen, Fed. Rep. of Germany
Translator: A. H. Armstrong
"Everglades", B rimpton Common, Reading,
RG74 RY Berks, UK
Tille 01 the original German edition: Theoretische Physik mit dem Personal Computer
ISBN-13: 978-3-642-97090-0 e-ISBN- 13: 978-3-642-97088-7
001: 10.10071978-3-642·97088·7
C Springer-Verlag, Berlin, Heidelberg 1987
library 01 Congress Cataloging-in 'Publication Data. Schmid, Erich W., 1931-. Theoretical physics on the
personal computer. Translation of: Theoretische Physik mit dem Personal-Computer. Includes Index.
1. Mathematical physics- Data processing. 2. Microcomputers. l. Spilz. Gerhard, 1955-. II. l OSch, Wolf·
gang, 1956·.111. Tille. aC20.7.E4S3S13 1988 530.1 '028'5526 88·4638
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is
concerned, specil ically the rights 01 transtation. reprint ing, reuse of illustrations, recitation, broad-
casting, reproduction on microtUms or in other ways, and storage in deta bankS. Duplication 01 this
publicallon or parts thereof is only permitted under the provisions 01 the German Copyright l aw of
September 9, 1965, in tis version 01 June 24. 1985, and a copyright lee must always be paid. Violations
fail under the prosecution act 01 the German Copyright law.
e Springer-Verlag Berlin Heidelberg 1988
Softcover reprint of the hardcover 1st edition 1988
The use of registered names, trademarks, etc. in th is publication does not imply, even in the absence of
a speci l ic statement, that such names are exempt from the relevant protective laws and regulations and
therefore lree lor general use.
PI. llle nole: Before using the programs contained in this book. please consult for techriical advice the
manuals provided by the respective manulaclurer 0 1t he computer - and any additional plug-in-boards-
to be used. The authors and the publisher accept no el gal responsibility lor any damage caused by
improper use of instructions and programs contained wi th in. The programs appearing here have been
tested carefully. Nevertheless we can offer no guarantee for the correct lunctioning of the programs.
Preface to the English Edition
We would like to thank Mr. A.H. Armstrong, 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
v
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
VII
Contents
1. Introduction........................ ........ ....... ........ ... 1
1.1 Programming of the Numerical Portions of the Programs .... 3
1.2 Programming of the Input and Output ...................... 5
2. Numerical Differentiation and Introduction into Screen
Dialogue ...................................................... 10
2.1 Formulation of the Problem. ....... ........ ....... ........ .. 10
2.2 Mathematical Methods .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Programming ............................................... 12
2.4 Exercises ................................................... 17
2.5 Solutions to the Exercises ................................... 17
3. Numerical Integration ....................................... 20
3.1 Formulation of the Problem. ................................ 20
3.2 Numerical Methods ......................................... 21
3.2.1 The Trapezoidal Rule ................................. 21
3.2.2 The Simpson Rule .................................... 22
3.2.3 Newton-Cotes Integration ............................. 23
3.2.4 The Gauss-Legendre Integration.. ............... ...... 23
3.3 Programming............................................... 27
3.4 Exercises ................................................... 31
3.5 Solutions to the Exercises ................................... 31
4. Harmonic Oscillations with Sliding and Static Friction,
Graphical Output of Curves ................................ 33
4.1 Formulation of the Problem ................................. 33
4.2 Numerical Treatment ....................................... 34
4.2.1 Transformation of the Differential Equation..... ....... 34
4.2.2 The Euler Method.................................... 35
4.3 Programming ................................................ 35
4.4 Exercises ................................................... 39
4.5 Solutions to the Exercises ................................... 39
5. Anharmonic Free and Forced Oscillations ................. 41
5.1 Formulation of the Problem........ ..... ... . ...... ... ... . . .. 41
5.2 Numerical Treatment ....................................... 42
IX
5.2.1 Improvement of the Euler Method..................... 42
5.2.2 The Runge-Kutta Method............................. 44
5.3 Programming ............................................... 45
5.4 Exercises ................................................... 47
5.5 Solutions to the Exercises ................................... 48
6. Coupled Harmonic Oscillations ............................. 51
6.1 Formulation of the Problem................ ....... .......... 51
6.2 Numerical Method .......................................... 52
6.3 Programming ............................................... 53
6.4 Exercises ................................................... 55
6.5 Solutions to the Exercises ................................... 55
7. The Flight Path of a Space Craft as a Solution of the
Hamilton Equations .......................................... 57
7.1 Formulation of the Problem ............................... " 57
7.2 Mathematical Methods...................................... 61
7.2.1 Mesh Width Adaptation in the Runge-Kutta Method.. 61
7.2.2 Coordinate Transformation...... ...................... 64
7.3 Programming............................................... 65
7.3.1 Hamilton's Equations of Motion....... ................ 65
7.3.2 Automatic Mesh Width Adjustment in the
Runge-Kutta Method................................. 67
7.3.3 Coordinate Transformation...... ........... ........... 69
7.3.4 Main Program ........................................ 71
7.4 Exercises ................................... "................ 77
7.5 Solutions to the Exercises ................................... 77
8. The Celestial Mechanics Three-body Problem............ 79
8.1 Formulation of the Problem ................................. 79
8.2 Mathematical Method ...................................... 83
8.3 Programming............................................... 83
8.4 Exercises ................................................... 87
8.5 Solutions to the Exercises ................................... 87
9. Computation of Electric Fields by the Method of
Successive Over-relaxation .................................. 88
9.1 Formulation of the Problem........ ............... ........ .. 88
9.2 Numerical Method........................................... 90
9.2.1 Discretisation of Laplace's Equation..... ....... ....... 90
9.2.2 The Method of Successive Over-relaxation............. 91
9.3 Programming............................................... 93
9.4 Exercises ................................................... 98
9.5 Solutions to the Exercises ................................... 98
x
10. The Van der Waals Equation ............................... 100
10.1 Formulation of the Problem................................ 100
10.2 Numerical Method ......................................... 102
10.3 Programming .............................................. 104
10.4 Exercises .................................................. 110
10.5 Solutions to the Exercises .................................. 111
11. Solution of the Fourier Heat Conduction Equation
and the "Geo-Power Station" ............................... 113
11.1 Formulation of the Problem ................................ 113
11.2 Method of Solution ........................................ 115
11.3 Programming .............................................. 117
11.4 Exercises .................................................. 119
11.5 Solutions to the Exercises .................................. 120
12. Group and Phase Velocity in the Example of Water Waves 123
12.1 Formulation of the Problem................................ 123
12.2 Numerical Method ......................................... 127
12.3 Programming.............................................. 129
12.4 Exercises .................................................. 132
12.5 Solutions to the Exercises .................................. 132
13. Solution of the Radial Schrodinger Equation by
the Fox-Goodwin Method ................................... 134
13.1 Formulation of the Problem ................................ 134
13.2 Numerical Method of Solution ............................. 138
13.3 Programming .............................................. 140
13.4 Exercises .................................................. 142
13.5 Solutions to the Exercises .................................. 143
14. The Quantum Mechanical Harmonic Oscillator ........... 147
14.1 Formulation of the Problem................................ 147
14.2 Numerical Method ......................................... 148
14.3 Programming .............................................. 151
14.4 Exercises .................................................. 154
14.5 Solutions to the Exercises .................................. 154
15. Solution of the Schrodinger Equation in Harmonic
Oscillator Representation ................................... 156
15.1 Formulation of the Problem ................................ 156
15.2 Numerical Method ......................................... 157
15.3 Programming .............................................. 158
15.4 Exercises .................................................. 161
15.5 Solutions to the Exercises .................................. 161
XI
16. The Ground State of the Helium Atom by the
Hylleraas Method ............................................ 163
16.1 Formulation of the Problem................................ 163
16.2 Setting up the State Basis and the Matrix Equation........ 165
16.3 Programming .............................................. 170
16.4 Exercises .................................................. 178
16.5 Solutions to the Exercises .................................. 178
17. The Spherical Harmonics .................................... 179
17.1 Formulation of the Problem................................ 179
17.2 Numerical Method......................................... 182
17.3 Programming .............................................. 183
17.4 Exercises .................................................. 185
17.5 Solutions to the Exercises .................................. 186
18. The Spherical Bessel Functions ............................. 187
18.1 Formulation of the Problem................................ 187
18.2 Mathematical Method ..................................... 189
18.3 Programming .............................................. 190
18.4 Exercises .................................................. 193
18.5 Solutions to the Exercises .................................. 193
19. Scattering of an Uncharged Particle from a Spherically
Symmetric Potential ............................... '.' . . . . . . . . 195
19.1 Formulation ofthe Problem...................... ........... 195
19.2 Mathematical Treatment of the Scattering Problem ........ 198
19.3 Programming .............................................. 201
19.4 Exercises .................................................. 204
19.5 Solutions to the Exercises.................................. 205
References ......................................................... 207
Subject Index ..................................................... 209
XII
1. Introduction
The solution of numerical problems in theoretical physics was until a few years
ago the domain of large computers. In recent years there have appeared on the
market personal computers which are as powerful as the large computers of the
early sixties. Apart from their high computational performance, personal com-
puters offer interactive capabilities and the rapid graphical output of results,
which were not available twenty years ago. Personal computers accordingly offer
us a wide field of possibilities in education and research.
In education it used to be customary to convey an understanding of the
fundamentals of theoretical physics by deduction. Understanding could be con-
solidated and deepened by the solution of exercises, but the number of analyt-
ically soluble examples is unfortunately rather limited. The personal computer
now enables us to increase significantly the number of examples by the inclu-
sion of examples which can be solved numerically. The computed results can
be graphically displayed in an instructive way. Moreover, in interaction with
the computer one can easily vary the physical parameters and the boundary
or starting conditions arid so become familiar with a whole family of solutions.
This clarifies in particular the connection between theory and experiment. The
emphasis in teaching is today moving from the derivation of equations towards
application.
The personal computer is also a useful tool in theoretical physics research.
It performs valuable service in the development and testing of new computer
programs. Anybody who has worked for years with large computers, serving
many users simultaneously in so-called Time Sharing Mode, knows the joy of
having a computer entirely to himself, rather than having to wait in a queue. In
applying to complicated scientific studies a program developed on a personal
computer, one will still have recourse to large computers. It is most conve-
nient when there is a cable connection between personal computer and large
computer.
This book is an attempt to facilitate the introduction of the personal
computer into education and research. The programs of the book have been
tested on personal computers of the IBM PCI AT type, with mathematical co-
processor, 512KB main memory, hard disc, EGA board and Professional FOR-
TRAN compiler. The programs were also tried out on other IBM-compatible
machines (even without hard discs). In most cases no problems arose, provided
that the machines were equipped with a mathematical co-processor and one of
the usual graphics boards. A memory capacity of 512 KB is required for only