Quantum Mechanics on the Personal Computer

276 Pages·1992·9.602 MB·English

by  Professor Dr. Siegmund Brandt, Professor Dr. Hans Dieter Dahmen (auth.)

S. Brandt
H.D. Dahmen
Quantum Mechanics on the
Personal Computer
s.
Brandt H. D. Dahmen
Quantum Mechanics
on the
Personal Computer
Second Edition
With a Program Diskette,
69 Figures and 284 Exercises
Springer-Verlag
Berlin Heidelberg New York
London Paris Tokyo
Hong Kong Barcelona
Budapest
Professor Dr. Siegmund Brandt
Professor Dr. Hans Dieter Dahmen
Physics Department, Siegen University, P. O. Box 101240
W-5900 Siegen, FRG
2nd Edition 1992
ISBN-13: 978-3-642-97420-5 e-ISBN-13: 978-3-642-97418-2
DOl: 10.1007/978-3-642-97418-2
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 any other way, and storage in data banks. Duplication of this publication or parts thereof is
permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and
permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the
German Copyright Law.
© Springer-Verlag Berlin Heidelberg 1989, 1992
Softcover reprint of the hardcover 2nd edition 1992
The use of general descriptive names, 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 in this book (VerSion 2.0), 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 the publisher
accept no legal responsibility for any damage caused by improper use of the instructions and programs contained herein.
Although these programs have been tested with extreme care, we can offer no formal guarantee that they will function
correctly.
The program on the enclosed disc is under copyright-protection and may not be reproduced without written permission
by Springer-Verlag. One copy of the program may be made as a back-up, but all further copies offend copyright law.
56/3140 -5 4 3 2 1 0 -Printed on acid-free paper
Preface to the Second Edition
For this new edition the program INTERQUANTA has been improved in several ways.
A number of numerical procedures has been made more stable and/or speeded up ap
preciably. This applies in particular to the treatment of wave packets in Chapter 4 and
three-dimensional problems in Chapters 6 and 8.
Because of the increase in speed, acceptable computing times may also be reached
on personal computers that do not have a mathematical coprocessor. We therefore now
provide two versions of the program on the diskette accompanying this book. One, as in
the first edition, needs a coprocessor. The other is a version of INTERQUANTA that
does not need a coprocessor. This version should be particularly useful to students.
There are only a few changes in the text of the book. Most of them are in Appendices B
and C. The only user command that has changed slightly is the command CS, see Section
A.5.5.5.
We are grateful for the friendly resonance from many students and colleagues.
Owing to this interest, an edition of this book for Macintosh computers has been published
by Springer-Verlag New York and a Japanese edition for NEC computers by Springer
Verlag Tokyo. We thank Dr. Tadashi Ishikawa and Professor Kunio Hirata for adapting
INTERQUANTA to the NEC computer and for the translation of this book. Our partic
ular thanks go to Tilo Stroh who devised and implemented most of the improvements in
version 2.0 of INTERQUANTA.
Siegen, Germany Siegmund Brandt
July 1992 Hans Dieter Dahmen
Preface to the First Edition
Ever since we published our Picture Book of Quantum Mechanics we have been asked
to make available the programs we wrote to generate the computer graphics that illus
trate the book. Supported by a study contract with IBM Germany we were able to im
prove and generalize the original set of programs considerably. We have called the result
INTERQUANTA (the Interactive Picture Program of Quantum Mechanics), which we
like to abbreviate further by IQ.
IQ consists of the program proper and various additional files. These allow classroom
demonstrations of prerecorded problems with explanatory text and are an essential help in
the organization of coursework. This book contains exercises worked out for a complete
course we call A Computer Laboratory on Quantum Mechanics.
We have tried out the course with groups of students at Siegen University. Note that
the students are not expected to have any knowledge of computer programming. The
Laboratory is primarily meant to help students in their first encounter with quantum me
chanics, but we have also found it useful for students who are already familiar with the
basics of the theory. In our experience the Laboratory is best offered in parallel with a
lecture course such as Quantum Physics or Introductory Quantum Mechanics.
It is a pleasure to acknowledge the generous help provided by IBM Germany for the
preparation of INTERQUANTA. In particular we would like to thank Dr. U. Groh for his
competent help with the computer hardware and the systems software.
The program was written mostly in FORTRAN 77, with a few routines in C, developed
on an IBM 6150 RT PC computer and later adapted to other systems. The version pub
lished with this book runs on the IBM PC and PS/2 or compatible systems; see Appendix B
for details. We have also tested versions for the IBM 6150 RT PC, VAX and Atari comput
ers and are willing to make them available for use by universities on request. We should
be grateful for comments on the use of the program and suggestions for improvements.
At various stages of the project we were helped considerably by friends and students
in Siegen. We would particularly like to thank Martin S. Brandt, Karin Dahmen, Helge
Meinhard, Martin Schmidt, Tilo Stroh, Clemens Stupperich and Dieter Walmer for their
excellent work.
This book was typeset in our group with the typesetting program TEX by Donald E.
Knuth. Our computer graphics coded as POSTSCRIPT files were integrated in the TEX file
using a TEX \special command. The complete file was then printed by a POSTSCRIPT
driver.
Siegen, Germany Siegmund Brandt
July 1989 Hans Dieter Dahmen
Contents
1 Introduction 1
1.1 Interquanta........ 1
1.2 The Structure of this Book 2
1.3 The Computer Laboratory 2
1.4 The Classroom Demonstrations 3
1.5 Literature ........... . 3
2 Free Particle Motion in One Dimension 5
2.1 Physical Concepts . . . . . . . . . . . . . . . . . 5
2.2 A First Session with the Computer . . . . . . . . . 8
2.3 The Time Development of a Gaussian Wave Packet 12
2.4 The Spectral Function of a Gaussian Wave Packet 14
2.5 The Wave Packet as a Sum of Harmonic Waves 15
2.6 Exercises..................... 17
3 Bound States in One Dimension 20
3.1 Physical Concepts . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2 Eigenstates in the Infinitely Deep Square-Well Potential and in the
Harmonic-Oscillator Potential . 26
3.3 Eigenstates in the Step Potential . . . . . . . . . . . . . . . 29
3.4 Harmonic Particle Motion . . . . . . . . . . . . . . . . . . 32
3.5 Particle Motion in the Infinitely Deep Square-Well Potential. 33
3.6 Exercises........................... 35
4 Scattering in One Dimension 40
4.1 Physical Concepts . . . . . . . . . . . . . . . . . . 40
4.2 Stationary Scattering States in the Step Potential . . . 52
4.3 Scattering of a Harmonic Wave by the Step Potential 54
4.4 Scattering of a Wave Packet by the Step Potential . 55
4.5 Transmission and Reflection. The Argand Diagram 57
4.6 Exercises...................... 59
4.7 Analogies in Optics ................ 68
4.8 Reflection and Refraction of Stationary Electromagnetic Waves 72
4.9 Reflection and Refraction of a Harmonic Light Wave 73
4.10 Scattering of a Wave Packet of Light . . . . . . . . . . . . . . 75
x
Contents
4.11 Transmission, Reflection and Argand Diagram for a Light Wave . 77
4.12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5 A Two-Particle System: Coupled Harmonic Oscillators 82
5.1 Physical Concepts . . . . . . . . . . . 82
5.2 Stationary States . . . . . . . . . ... 87
5.3 Time Dependence of Global Quantities 88
5.4 Joint Probability Densities 90
5.5 Marginal Distributions 91
5.6 Exercises ........ . 93
6 Free Particle Motion in Three Dimensions 97
6.1 Physical concepts ................ . 97
6.2 The 3D Harmonic Plane Wave . . . . . . . . . . . 106
6.3 The Plane Wave Decomposed into Spherical Waves 108
6.4 The 3D Gaussian Wave Packet . . . . . . . . . . . 109
6.5 The Probability Ellipsoid. . . . . . . . . . . . . . 111
6.6 Angular-Momentum Decomposition of a Wave Packet. 112
6.7 Exercises ....................... . 114
7 Bound States in Three Dimensions 116
7.1 Physical Concepts . . . . . . . . . . . . . 116
7.2 Radial Wave Functions in Simple Potentials 124
7.3 Radial Wave Functions in the Step Potential 129
7.4 Probability Densities . . . 131
7.5 Harmonic Particle Motion 134
7.6 Exercises ........ . 136
8 Scattering in Three Dimensions 139
8.1 Physical Concepts . . . . . . . . . . . . . . . . 139
8.2 Radial Wave Functions ............. . 145
8.3 Stationary Wave Functions and Scattered Waves . 148
8.4 Differential Cross Sections . . . . . . . . . . . . 150
8.5 Scattering Amplitude. Phase Shift. Partial and Total Cross Sections 152
8.6 Exercises ............................. . 155
9 Special Functions of Mathematical Physics 159
9.1 Basic Formulae ............ . 159
9.2 Hermite Polynomials ..................... . 165
9.3 Eigenfunctions of the One-Dimensional Harmonic Oscillator 166
9.4 Legendre Polynomials and Associated Legendre Functions 167
9.5 Spherical Harmonics . . . . 170
9.6 Bessel Functions ..... . 171
9.7 Spherical Bessel Functions. 173
9.8 Laguerre Polynomials . . . 174
9.9 Radial Eigenfunctions of the Harmonic Oscillator 176
Contents XI
9.10 Radial Eigenfunctions of the Hydrogen Atom 177
9.11 Simple Functions of a Complex Variable . 178
9.12 Exercises ................... . 180
10 Additional Material and Hints for the Solution of Exercises 182
10.1 Units and Orders of Magnitude. . . . . . . . . . . . . . 182
10.2 Argand Diagrams and Unitarity for One-Dimensional Problems 188
10.3 Hints and Answers to the Exercises ............. . 195
Appendix
A A Systematic Guide to IQ 217
A.1 Dialog Between the User and IQ ..... 217
A.1.1 A Simple Example . . . . . . . . 217
A.1.2 The General Form of Commands 220
A.1.3 The Descriptor File .. 220
A.1.4 The Descriptor (Record) 223
A.l.5 The PLOT Command . 225
A.1.6 The STOP Command . 226
A.1.7 HELP: The Commands HE and PH 226
A.2 Coordinate Systems and Transformations 227
A.2.1 The Different Coordinate Systems 227
A.2.2 Defining the Transformations .. 228
A.3 The Different Types of Plot ...... . 232
A.3.1 Choosing a Plot Type: The Command CH 232
A.3.2 Cartesian 3D Plots (Type 0 Plots) 232
A.3.3 Polar 3D Plots (Type 1 Plots) . 233
A.3.4 2D Plots (Type 2 Plots) ..... 234
A.3.5 3D Column Plots (Type 3 Plots) . 237
A.3.6 Special 3D Plots (Type 10 Plots) . 238
A.4 The Background in the Plots . . . . . . . 238
A.4.1 Boxes and Coordinate Axes: The Command BO . 238
A.4.2 Scales . . . . . . 239
A.4.3 Arrows................ 241
A.4.4 Text and Numbers . . . . . . . . . . 243
A.4.5 Mathematical Symbols and Formulae 245
A.5 Further Commands . . 245
A.5.1 Line Styles ... 245
A.5.2 Multiple Plots . 248
A.5.3 Combined Plots 249
A.5.4 Using Different Plotting Devices. 249
A.5.5 The Different Running Modes . . 250
A.5.6 Introducing Physical Variables: The Commands VO to V9 . 253
A.5.7 Reserved Commands. . . . . . . . . . . . . . . . . . . . 253
XII Contents
B How to Install IQ 254
B.l Hardware Requirements . . . . . 254
B.2 Operating-System Requirements. 254
B.3 Diskette Format ........ . 254
B.4 Installation ........... . 255
B.5 Reformatting IQ for Different Types of Diskette 255
C Lists of All Provided Files 256
C.l Command Files ............... 256
C.2 Program File . . . . . . . . . . . . . . . . . 256
C.3 Descriptor Files for Examples and Exercises . 256
C.4 Command Input Files and Associated Descriptor Files for Demonstrations 257
C.5 Data Files 257
C.6 Help Files ................................ 257
D Graphics Devices and Metafiles 258
Index of IQ Commands 262
Subject Index 263
1 Introduction
1.1 Interquanta
The language of quantum mechanics is needed to describe nature at the atomic or sub
atomic scale, e.g., the phenomena of atomic, nuclear, or particle physics. But there are
many other fields of modem science and engineering in which important phenomena can
be explained only by quantum mechanics, for example chemical bonds or the function
ing of semiconductor circuits in computers. It is therefore very important for students of
physics, chemistry, and electrical engineering to become familiar with the concepts and
methods of quantum mechanics.
It is a fact, however, that most students find quantum mechanics difficult and abstract,
much more so than classical point mechanics. One easily detects the reason for this by
recalling how students learn classical mechanics. Besides learning from lectures they
draw on their own experience, on experiments they perform in the laboratory, and on
problems they solve on paper. The important concept of a mass point is nothing but that
of a very small stone. The experience with throwing stones helps to understand mechanics.
Additional experiments are very direct and simple and there is a wealth of problems which
are easily solved.
All this is different with quantum mechanics. Although - for all we know - elemen
tary particles are point-like, the concept of the trajectory of a mass point breaks down and
has to be replaced by a complex probability amplitude. This function cannot be measured
directly; its properties have to be inferred indirectly from experiments involving optical
spectra or counting rates, etc. Finally, nearly all nontrivial problems pose severe compu
tational difficulties and require approximative or numerical methods. Thus students can
do only a few problems.
Many quantum-mechanical problems can, however, be quite quickly solved numer
ically by computer. The answer is often very easy to analyze if presented in graphi
cal form. We have written an interactive program taking alphanumeric input defining
quantum-mechanical problems and yielding graphical output to produce a large number
of illustrations for an introductory text book on quantum mechanics I.
Here we present an improved and generalized version of this program which we call
INTERQUANTA (IQ for short) - the interactive picture program of quantum mechan
ics. IQ can be used in a computer laboratory on quantum mechanics and for classroom
demonstrations.
In the laboratory course students define and solve a quantum-mechanical problem,
examine the results, change some parameters in their problem, analyze the new answer
IS. Brandt and H.D. Dahmen (1985): The Picture Book o/Quantum Mechanics (Wiley, New York)