Quantum Mechanics on the Macintosh®: With two Program Diskettes

317 Pages·1991·21.128 MB·English

by  Siegmund Brandt, Hans Dieter Dahmen

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Quantum Mechanics on the Macintosh"
Siegmund Brandt
Hans Dieter Dahmen
Quantum Mechanics
on the Macintosh"
With two Program Diskettes,
69Figures, and 284Exercises
Springer Science+Business Media, LLC
Siegmund Brandt Hans DieterDahmen
Physics Department Physics Department
Siegen University Siegen University
P.O. Box 101250 P.O. Box 101250
W-5900Siegen W-5900Siegen
Germany Germany
Library ofCongress Cataloging-in-PublicationData
Brandt,Siegmund.
QuantummechanicsontheMacintosh/S. Brandtand H.D.Dahmen.
p. cm.
Includes bibliographicalreferencesand indexes.
I. Quantumtheory-Dataprocessing. 2. Interquanta(Computer
program) 3. Macintosh(Computer)-Programming. I. Dahmen,Hans
Dieter,1936- . II. Title.
QCI74.17.D37B72 1991
530.1'2'02855369- dc20 91-17793
Printedonacid-free paper.
©1991 SpringerScience+BusinessMediaNewYork
OriginallypublishedbySpringer-VerlagNewYork,Inc.in1991.
SoftcoverreprintofthehardcoverIstedition1991
Allrights reserved.This work maynot betranslatedorcopied inwholeor inpartwithout the
written permission ofthe publisherSpringerScience+BusinessMedia,LLC, except for brief
excerpts in connection with reviews or scholarlyanalysis. Use in connection with anyform
of inforrnation storage and retrieval, electronic adaptation,computersoftware, or bysimilar
ordissimilar methodologynowknown orhereafterdevelopedisforbidden.
The useofgeneraldescriptivenames, trade names,tradernarks,etc., inthispublication,evenif
the former are not especially identified, is not to be taken as a sign that such names, as
understoodbytheTradeMarksand MerchandiseMarksAct, mayaccordinglybeusedfreelyby
anyone.
The program on the enclosed diskettes is under copyright protection and may not be repro
ducedwithoutwritten permissionfromSpringer-Verlag.Onecopyoftheprogrammaybemade
asaback-up, but allothercopiesoffend copyrightlaw,
The program waswritten in the programming language FORTRAN 77and compiled using a
FORTRAN compilerbyLangnage SystemsCorp.Thisisacknowledgedbythe followingcopy
right notice:Certainportionsofthissoftware are copyrighted byLanguage SystemsCorp.©
1988-1990LanguageSystemsCorp.
Beforeusingtheprogram pleaseconsultthetechnical manualsprovidedbythemanufacturerof
the computer-and ofany additional plug-in boards-to be used.The authorsand publisher
accept no legal responsibilityfor any damage caused byimproper useofthe instructionsand
program contained herein. Although the program has been tested with extreme care, wecan
offernoformal guaranteethat itwillfunction correctly.
Apple, Macintosh,ImageWriterand LaserWriterareregistered trademarksofApple Cornput
er,lnc.
POSTSCRIPTisaregisteredtrademarkofAdobe Systems, Inc.
Camera-readycopyprepared bytheauthorsusingTEX.
987654321
ISBN 978-3-540-97627-1 ISBN 978-3-662-25102-7 (eBook)
DOI 10.1007/978-3-662-25102-7
Preface
Ever since we published our Picture Book ofQuantum Mechanics we have
been askedtomake availabletheprogramswewrotetogeneratethecomputer
graphicsthat illustratethe book. Wehave calledthe result INTERQUANTA
(the Interactive Picture Program ofQuantum Mechanics), which we like to
abbreviatefurther byIQ.
IQconsistsoftheprogramproperandvariousadditionalfiles.Theseallow
classroomdemonstrationsofprerecordedproblems withexplanatorytext and
are an essential help in the organization of coursework. This book contains
exercises worked out for a complete course we callA Computer Laboratory
onQuantum Mechanics.
Wehave tried outthecourse withgroups ofstudentsatSiegen 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
firstencounterwith quantummechanics, but wehave alsofound ituseful for
students who are already familiar with the basics of the theory. In our expe
rience theLaboratoryis best offered in parallel with alecture course such as
Quantum Physics orIntroductory Quantum Mechanics.
The program was originally publishedI for use on Personal Computers
(IBM PC or PS/2). The program is coded in a machine-independent way in
FORTRAN77. This makes iteasy to adapt ittoother computers butdifficult
to take advantageofspecial machine-dependentfeatures. For this reason no
use was made ofthe mouse.
The original developmentofINTERQUANTA,supportedbyastudycon
tract with IBM Germany, was done on an IBM 6150 RTPC computer. Itis
a pleasure to acknowledge the generous help provided by IBM Germany, in
particularwewould liketothank Dr.U.Groh forhiscompetenthelp with the
computerhardwareand the systems software.
At various stages ofthe project we were helped considerably by friends
andstudents inSiegen. WewouldparticularlyliketothankMartinS.Brandt,
Karin Dahmen, Helge Meinhard,Martin Schmidt, Tilo Stroh,Clemens Stup
perich and Dieter Wiihner for their excellent work. We thank Wolfgang
IS. Brandl and H. D. Dahmen, (1989): Quantum Mechanics on the Personal Com
puter, Springer-Verlag (Berlin,Heidelberg, NewYork,London, Paris,Tokyo, HongKong,
Barcelona),ISBN0-387-51541-0
v
vi Preface
Merzenich (Informatics department of Siegen University) for letting us use
his system ofMacintoshcomputers and Dirk Diillmann (DESY) and Marcel
Luis (Siegen) for help and advice. Inparticularwe want to thank Tilo Stroh
who did most ofthe workneededto make this version ofIQrun.
This book was typesetin our group with the typesetting programT}3X by
Donald E. Knuth. Ourcomputergraphics coded asPOSTSCRIPT files were
integratedintheT}3XfileusingaT}3X\specialcommand.The completefile
wasthen printedby aPOSTSCRIPTdriver.
Siegen,Germany SiegmundBrandt
June 1991 Hans DieterDahmen
Contents
Preface v
1 Introduction 1
1.1 Interquanta ... .. ... 1
1.2 TheStructureofthisBook 2
1.3 TheComputerLaboratory 3
1.4 TheClassroomDemonstrations 3
1.5 Literature... . . .. . . . .. 4
2 Free ParticleMotion in One Dimension 5
2.1 PhysicalConcepts . . . . . . . . . . . . . . . . . 5
2.2 AFirstSessionwiththeComputer . . . . . . . . . 9
2.3 TheTimeDevelopmentofaGaussianWavePacket 13
2.4 TheSpectralFunctionofaGaussianWavePacket 15
2.5 TheWavePacketasaSumofHarmonicWaves 16
2.6 Exercises .... . . . . . . . . ... . .... . 19
3 BoundStatesinOne Dimension 22
3.1 PhysicalConcepts . . . . . . . . . . . . . . . . . . . . .. 22
3.2 EigenstatesintheInfinitelyDeepSquare-WellPotentialand
intheHarmonic-OscillatorPotential . . . . . . . . . . . . . 29
3.3 EigenstatesintheStepPotential . . . . . . . . . . . . . .. 32
3.4 HarmonicParticleMotion . . . . . . . . . . . . . . . . . . 35
3.5 ParticleMotionintheInfinitelyDeepSquare-WellPotential . 36
3.6 Exercises .. .. . . .. . . . ....... .... ... . . 38
4 ScatteringinOne Dimension 44
4.1 PhysicalConcepts . . . . . . . . . . . . . . . . . . 44
4.2 StationaryScatteringStatesintheStepPotential . . . 58
4.3 ScatteringofaHarmonicWavebytheStepPotential 60
4.4 ScatteringofaWavePacketbytheStepPotential . 61
4.5 TransmissionandReflection.TheArgandDiagram 64
4.6 Exercises.. . . .. 66
4.7 AnalogiesinOptics 76
vii
viii Contents
4.8 ReflectionandRefractionofStationaryElectromagneticWaves 80
4.9 ReflectionandRefractionofaHarmonicLightWave . ... 82
4.10 ScatteringofaWavePacketofLight . . . . . . . . . . . . . 83
4.11 Transmission,ReflectionandArgandDiagramforaLightWave 86
4.12 Exercises . . . . . . . . . . . . . . . . . . . . . . . 88
5 ATwo-ParticleSystem: CoupledHarmonicOscillators 91
5.1 PhysicalConcepts . . . . . . . . . . . 91
5.2 StationaryStates . . . . . . . . . . . . 97
5.3 TimeDependenceofGlobalQuantities 97
5.4 JointProbabilityDensities 100
5.5 MarginalDistributions 102
5.6 Exercises .. .. . .... 103
6 Free ParticleMotioninThree Dimensions 109
6.1 PhysicalConcepts . . . . . . . . . . . . . . . . . 109
6.2 The3DHarmonicPlaneWave . . . . . . . . . . . 119
6.3 ThePlaneWaveDecomposedintoSphericalWaves 122
6.4 The3DGaussianWavePacket. . . . . . . . . . . 123
6.5 TheProbabilityEllipsoid . . . . . . . . . . . . . . 125
6.6 Angular-MomentumDecompositionofaWavePacket . 126
6.7 Exercises.. . .. .. . . . . .. ... . . . . .. .. 128
7 BoundStates inThree Dimensions 131
7.1 PhysicalConcepts . . . . . . . . . . . . . 131
7.2 RadialWaveFunctionsinSimplePotentials 141
7.3 RadialWaveFunctionsintheStepPotential 145
7.4 ProbabilityDensities. . . 147
7.5 HarmonicParticleMotion 151
7.6 Exercises . . .. ... .. 153
8 ScatteringinThree Dimensions 157
8.1 PhysicalConcepts . . . . . . . . . . . . . . . . 157
8.2 RadialWaveFunctions . . . . . . . . . . . . . . 165
8.3 StationaryWaveFunctionsandScatteredWaves . 168
8.4 DifferentialCrossSections . . . . . . . . . . . . 170
8.5 ScatteringAmplitude. PhaseShift. PartialandTotalCross
Sections . . . . . . . . . . . . . . . . . . . . . . . . . .. 171
8.6 Exercises. . . .. . . .... .. . . .. . ... .. .... 175
9 SpecialFunctions ofMathematicalPhysics 180
9.1 BasicFormulae. . . . 180
9.2 HermitePolynomials . . . . . . . . . . 187
Conrenffi ix
9.3 EigenfunctionsoftheOne-DimensionalHarmonicOscillator 187
904 LegendrePolynomialsandAssociatedLegendreFunctions 188
9.5 SphericalHarmonics . . . . 192
9.6 BesselFunctions . . . . . . 192
9.7 SphericalBesselFunctions . 195
9.8 LaguerrePolynomials . . . 196
9.9 RadialEigenfunctionsoftheHarmonicOscillator 198
9.10 RadialEigenfunctionsoftheHydrogenAtom 200
9.11 SimpleFunctionsofaComplexVariable. 200
9.12 Exercises. . . . . . . . . . . . . . . . . . . 202
10 AdditionalMaterial and Hints for the Solution ofExercises 204
10.1 UnitsandOrdersofMagnitude. . . . . . . . . . . . . . 204
10.2 ArgandDiagramsandUnitarityforOne-DimensionalProblems211
10.3 HintsandAnswerstotheExercises 220
Appendix
A ASystematicGuide toIQ 246
A.1 DialogBetweentheUserandIQ . 246
A.1.1 ASimpleExample . . . . 246
A.1.2 TheGeneralFormofCommands 249
A.1.3 TheDescriptorFile . . 250
A.1A TheDescriptor(Record) 253
A.I.5 ThePLOTCommand . 255
A.I.6 TheSTOPCommand . 256
A.I.7 HELP:TheCommandsHEandPH 256
A.2 CoordinateSystemsandTransformations 257
A.2.1 TheDifferentCoordinateSystems 257
A.2.2 DefiningtheTransformations . . 258
A.3 TheDifferentTypesofPlot 263
A.3.1 ChoosingaPlotType:TheCommandCH . 263
A.3.2 Cartesian3DPlots(Type0Plots) 263
A.3.3 Polar3DPlots(Type1Plots) . 264
A.304 2DPlots(Type2Plots) . . . . . 266
A.3.5 3DColumnPlots(Type3Plots) . 269
A.3.6 Special3DPlots(Type10Plots) . 270
Ao4 TheBackgroundinthePlots . . . . . . . 270
Ao4.1 BoxesandCoordinateAxes:TheCommandBO . 270
Ao4.2 Scales . . . . . . 271
Ao4.3 Arrows. . ... . .. ... . . . . . 273
Ao404 TextandNumbers . . . . . . . . . . 275
A.4.5 MathematicalSymbolsandFormulae 277