Interactive Quantum Mechanics

Quantum Experiments on the Computer

Second Edition

Interactive Quantum Mechanics

With CD-ROM, 128 Figures, and 344 Exercises

S. Brandt • H.D. Dahmen • T. Stroh

Springer New York Dordrecht Heidelberg London

Printed on acid-free paper

Springer is part of Springer Science+Business Media (www.springer.com)

DOI 10.1007/978-1-4419-7424-2

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Siegmund Brandt Physics Department Siegen University 57068 Siegen Germany brandt@physik.uni-siegen.de

Hans Dieter Dahmen Physics Department Siegen University 57068 Siegen Germany dahmen@physik.uni-siegen.de

Tilo Stroh Physics Department Siegen University 57068 Siegen Germany stroh@sirs02.physik.uni-siegen.de

Preface to the Second Edition

For the present edition the concept of the book and of INTERQUANTA, theaccompanying Interactive Program of Quantum Mechanics, (IQ, for short),was left unchanged. However, the physics scope of the text and the capabili-ties of the program were widened appreciably.

The most conspicuous addition to IQ is the capability to produce and dis-play movies of quantum-mechanical phenomena. So far, IQ presented timedependence as a series of graphs in one frame. While such plots (which can,of course, still be shown) lead to a good understanding of the phenomenon un-der study and can be examined quantitatively at leisure, the new movies givea more direct impression of what happens as time passes. For such movies,as for the conventional simulations, the parameters defining the physical phe-nomenon and the graphical appearance can be changed interactively. Moviescan be produced and also stored for many quantum-mechanical problems suchas bound states and scattering states in various one-dimensional potentials,wave packets in three dimensions (free or in a harmonic-oscillator potential),and two-particle systems (distinguishable particles, identical fermions, iden-tical bosons).

Concerning the physics scope, these are the main additions: One-dimen-sional bound and scattering states are now discussed and computed also forpiecewise linear potentials. These, as opposed to the usual step potentials(which are piecewise constant), allow for much better approximations of ar-bitrary smooth potentials. Another interesting addition to one-dimensionalquantum mechanics is the juxtaposition of quantum-mechanical wave packetswith classical phase-space distributions. The treatment of quantum mechan-ics in three dimensions is extended by the hybridization of bound states andby the simulation of magnetic resonance.

In the present edition the number of data sets (we call them descriptors),defining a complete simulation – either presented as conventional plot or asmovie – is more than tripled. In this way users have a much richer choice ofready-made examples from which to start their exploits. Moreover, solutiondescriptors are now provided for the exercises.

Siegen, Germany Siegmund BrandtMay 2010 Hans Dieter Dahmen

Tilo Stroh

v

Preface to the First Edition

This book can be regarded as a concise introduction to basic quantum mechan-ics: free particle, bound states, and scattering in one and in three dimensions,two-particle systems, special functions of mathematical physics. But the bookcan also be seen as an extensive user’s guide for INTERQUANTA, the Inter-

active Program of Quantum Mechanics, which we will abbreviate henceforthas IQ. The book also contains a large number of exercises. The programcan 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

by performing computer experiments. A simpler way to use IQ is to studyone or several of the ready-made demonstrations. In each demonstration theuser is taken through one chapter of quantum mechanics. Graphics illustrat-ing quantum-mechanical problems that are solved by the program are shown,while short explanatory texts are either also displayed or can be listened to.

INTERQUANTA has a user interface based on tools provided by the Java

programming language. With this interface using the program is essentiallyself-explanatory. In addition, extensive help functions are provided not onlyon technical questions but also on quantum-mechanical concepts. All in allusing INTERQUANTA is not more difficult than surfing the Internet.

The modern user interface is the main improvement over older versions ofIQ.1 Moreover, new physics topics are added and there are also new graphicalfeatures.

The present version of INTERQUANTA is easily installed and run onpersonal computers (running under Windows or Linux) or Macintosh (runningunder Mac OS X).

We do hope that by using INTERQUANTA on their own computer manystudents will gain experience with different quantum phenomena without hav-ing to do tedious calculations. From this experience an intuition for this im-portant but abstract field of modern science can be developed.

Siegen, Germany Siegmund BrandtFebruary 2003 Hans Dieter Dahmen

Tilo Stroh1 S. Brandt and H. D. Dahmen, Quantum Mechanics on the Personal Computer, Springer,

Berlin 1989, 1992, and 1994; Quantum Mechanics on the Macintosh, Springer, New York1991 and 1995; Pasocon de manebu ryoushi nikigacu, Springer, Tokyo 1992; Quanten-

mechanik auf dem Personalcomputer, Springer, Berlin 1993

vi

Contents

Preface to the Second Edition . . . . . . . . . . . . . . . . . . v

Preface to the First Edition . . . . . . . . . . . . . . . . . . . vi

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Interquanta . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 The Structure of This Book . . . . . . . . . . . . . . . . . 21.3 The Demonstrations . . . . . . . . . . . . . . . . . . . . . 31.4 The Computer Laboratory . . . . . . . . . . . . . . . . . . 31.5 Literature . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Free Particle Motion in One Dimension . . . . . . . . . . . . . 52.1 Physical Concepts . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Planck’s Constant. Schrödinger’s Equationfor a Free Particle . . . . . . . . . . . . . . . . . . 5

2.1.2 The Wave Packet. Group Velocity. Normalization . 62.1.3 Probability-Current Density. Continuity Equation . 72.1.4 Quantile Position. Quantile Trajectory . . . . . . . 82.1.5 Relation to Bohm’s Equation of Motion . . . . . . . 92.1.6 Analogies in Optics . . . . . . . . . . . . . . . . . 102.1.7 Analogies in Classical Mechanics:

The Phase-Space Probability Density . . . . . . . . 112.2 A First Session with the Computer . . . . . . . . . . . . . 15

2.2.1 Starting IQ . . . . . . . . . . . . . . . . . . . . . 152.2.2 An Automatic Demonstration . . . . . . . . . . . . 162.2.3 A First Dialog . . . . . . . . . . . . . . . . . . . . 16

2.3 The Free Quantum-Mechanical Gaussian Wave Packet . . . 172.3.1 The Subpanel Physics—Comp. Coord. . . . . . . . . 182.3.2 The Subpanel Physics—Wave Packet . . . . . . . . . 192.3.3 The Subpanel Physics—Quantile . . . . . . . . . . . 192.3.4 The Subpanel Movie . . . . . . . . . . . . . . . . . 20

2.4 The Free Optical Gaussian Wave Packet . . . . . . . . . . . 202.5 Quantile Trajectories . . . . . . . . . . . . . . . . . . . . 21

vii

viii Contents

2.6 The Spectral Function of a Gaussian Wave Packet . . . . . . 222.7 The Wave Packet as a Sum of Harmonic Waves . . . . . . . 232.8 The Phase-Space Distribution of Classical Mechanics . . . . 252.9 Classical Phase-Space Distribution: Covariance Ellipse . . . 262.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3 Bound States in One Dimension . . . . . . . . . . . . . . . . . 323.1 Physical Concepts . . . . . . . . . . . . . . . . . . . . . . 32

3.1.1 Schrödinger’s Equation with a Potential.Eigenfunctions. Eigenvalues . . . . . . . . . . . . 32

3.1.2 Normalization. Discrete Spectra. Orthonormality . . 333.1.3 The Infinitely Deep Square-Well Potential . . . . . 333.1.4 The Harmonic Oscillator . . . . . . . . . . . . . . 343.1.5 The Step Potential . . . . . . . . . . . . . . . . . . 343.1.6 The Piecewise Linear Potential . . . . . . . . . . . 363.1.7 Time-Dependent Solutions . . . . . . . . . . . . . 373.1.8 Harmonic Particle Motion. Coherent States.

Squeezed States . . . . . . . . . . . . . . . . . . . 373.1.9 Quantile Motion in the Harmonic-Oscillator Potential 383.1.10 Harmonic Motion of a Classical

Phase-Space Distribution . . . . . . . . . . . . . . 383.1.11 Particle Motion in a Deep Square Well . . . . . . . 41

3.2 Eigenstates in the Infinitely Deep Square-Well Potentialand in the Harmonic-Oscillator Potential . . . . . . . . . . 43

3.3 Eigenstates in the Step Potential . . . . . . . . . . . . . . . 453.4 Eigenstates in the Step Potential – Quasiperiodic . . . . . . 463.5 Eigenstates in the Piecewise Linear Potential . . . . . . . . 473.6 Eigenstates in the Piecewise Linear Potential – Quasiperiodic 483.7 Harmonic Particle Motion . . . . . . . . . . . . . . . . . . 493.8 Harmonic Oscillator: Quantile Trajectories . . . . . . . . . 513.9 Classical Phase-Space Distribution: Harmonic Motion . . . 523.10 Harmonic Motion of Classical Phase-Space Distribution:

Covariance Ellipse . . . . . . . . . . . . . . . . . . . . . 533.11 Particle Motion in the Infinitely Deep Square-Well Potential 543.12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4 Scattering in One Dimension . . . . . . . . . . . . . . . . . . 634.1 Physical Concepts . . . . . . . . . . . . . . . . . . . . . . 63

4.1.1 Stationary Scattering States.Continuum Eigenstates and Eigenvalues.Continuous Spectra . . . . . . . . . . . . . . . . . 63

4.1.2 Time-Dependent Solutionsof the Schrödinger Equation . . . . . . . . . . . . . 64

Contents ix

4.1.3 Right-Moving and Left-MovingStationary Waves of a Free Particle . . . . . . . . . 64

4.1.4 Orthogonality and Continuum Normalizationof Stationary Waves of a Free Particle. Completeness 65

4.1.5 Boundary Conditionsfor Stationary Scattering Solutions in Step Potentials 66

4.1.6 Stationary Scattering Solutions in Step Potentials . . 674.1.7 Constituent Waves . . . . . . . . . . . . . . . . . 684.1.8 Normalization of Continuum Eigenstates . . . . . . 684.1.9 Harmonic Waves in a Step Potential . . . . . . . . 684.1.10 Time-Dependent Scattering Solutions

in a Step Potential . . . . . . . . . . . . . . . . . . 694.1.11 Generalization to Piecewise Linear Potentials . . . . 694.1.12 Transmission and Reflection. Unitarity.

The Argand Diagram . . . . . . . . . . . . . . . . 704.1.13 The Tunnel Effect . . . . . . . . . . . . . . . . . . 714.1.14 Resonances . . . . . . . . . . . . . . . . . . . . . 724.1.15 Phase Shifts upon Reflection at a Steep Rise

or Deep Fall of the Potential . . . . . . . . . . . . . 724.1.16 Transmission Resonances upon Reflection

at ‘More- and Less-Dense Media’ . . . . . . . . . . 744.1.17 The Quantum-Well Device

and the Quantum-Effect Device . . . . . . . . . . . 754.1.18 Stationary States in a Linear Potential . . . . . . . . 774.1.19 Wave Packet in a Linear Potential . . . . . . . . . . 774.1.20 Quantile Motion in a Linear Potential . . . . . . . . 784.1.21 Classical Phase-Space Density in a Linear Potential 784.1.22 Classical Phase-Space Density

Reflected by a High Potential Wall . . . . . . . . . 794.2 Stationary Scattering States in the Step Potential

and in the Piecewise Linear Potential . . . . . . . . . . . . 814.3 Time-Dependent Scattering by the Step Potential

and by the Piecewise Linear Potential . . . . . . . . . . . . 834.4 Transmission and Reflection. The Argand Diagram . . . . . 874.5 Stationary Wave in a Linear Potential . . . . . . . . . . . . 904.6 Gaussian Wave Packet in a Linear Potential . . . . . . . . . 914.7 Quantile Trajectories in a Linear Potential . . . . . . . . . . 924.8 Classical Phase-Space Density in a Linear Potential . . . . . 924.9 Classical Phase-Space Distribution: Covariance Ellipse . . . 944.10 Classical Phase-Space Density

Reflected by a High Potential Wall . . . . . . . . . . . . . 954.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 97

x Contents

4.12 Analogies in Optics . . . . . . . . . . . . . . . . . . . . . 1094.13 Reflection and Refraction

of Stationary Electromagnetic Waves . . . . . . . . . . . . 1134.14 Time-Dependent Scattering of Light . . . . . . . . . . . . . 1144.15 Transmission, Reflection, and Argand Diagram

for a Light Wave . . . . . . . . . . . . . . . . . . . . . . 1174.16 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 119

5 A Two-Particle System: Coupled Harmonic Oscillators . . . . 1225.1 Physical Concepts . . . . . . . . . . . . . . . . . . . . . . 122

5.1.1 The Two-Particle System . . . . . . . . . . . . . . 1225.1.2 Initial Condition for Distinguishable Particles . . . . 1245.1.3 Time-Dependent Wave Functions and

Probability Distributions for Distinguishable Particles 1255.1.4 Marginal Distributions for Distinguishable Particles 1255.1.5 Wave Functions for Indistinguishable Particles.

Symmetrization for Bosons.Antisymmetrization for Fermions . . . . . . . . . . 126

5.1.6 Marginal Distributionsof the Probability Densities of Bosons and Fermions 127

5.1.7 Normal Oscillations . . . . . . . . . . . . . . . . . 1275.2 Stationary States . . . . . . . . . . . . . . . . . . . . . . . 1285.3 Time Dependence of Global Variables . . . . . . . . . . . . 1295.4 Joint Probability Densities . . . . . . . . . . . . . . . . . . 1305.5 Marginal Distributions . . . . . . . . . . . . . . . . . . . . 1315.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 133

6 Free Particle Motion in Three Dimensions . . . . . . . . . . . 1386.1 Physical Concepts . . . . . . . . . . . . . . . . . . . . . . 138

6.1.1 The Schrödinger Equation of a Free Particlein Three Dimensions. The Momentum Operator . . 138

6.1.2 The Wave Packet. Group Velocity.Normalization. The Probability Ellipsoid . . . . . . 140

6.1.3 Angular Momentum. Spherical Harmonics . . . . . 1426.1.4 The Stationary Schrödinger Equation

in Polar Coordinates. Separation of Variables.Spherical Bessel Functions.Continuum Normalization. Completeness . . . . . . 144

6.1.5 Partial-Wave Decomposition of the Plane Wave . . . 1456.1.6 Partial-Wave Decomposition

of the Gaussian Wave Packet . . . . . . . . . . . . 1456.2 The 3D Harmonic Plane Wave . . . . . . . . . . . . . . . . 1486.3 The Plane Wave Decomposed into Spherical Waves . . . . . 1506.4 The 3D Gaussian Wave Packet . . . . . . . . . . . . . . . 151

Contents xi

6.5 The Probability Ellipsoid . . . . . . . . . . . . . . . . . . 1526.6 Angular-Momentum Decomposition of a Wave Packet . . . 1536.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 155

7 Bound States in Three Dimensions . . . . . . . . . . . . . . . 1587.1 Physical Concepts . . . . . . . . . . . . . . . . . . . . . . 158

7.1.1 The Schrödinger Equation for a Particleunder the Action of a Force.The Centrifugal Barrier. The Effective Potential . . . 158

7.1.2 Bound States. Scattering States.Discrete and Continuous Spectra . . . . . . . . . . 160

7.1.3 The Infinitely Deep Square-Well Potential . . . . . 1617.1.4 The Spherical Step Potential . . . . . . . . . . . . 1627.1.5 The Harmonic Oscillator . . . . . . . . . . . . . . 1657.1.6 The Coulomb Potential. The Hydrogen Atom . . . . 1667.1.7 Harmonic Particle Motion . . . . . . . . . . . . . . 167

7.2 Radial Wave Functions in Simple Potentials . . . . . . . . . 1687.3 Radial Wave Functions in the Step Potential . . . . . . . . . 1727.4 Probability Densities . . . . . . . . . . . . . . . . . . . . 1737.5 Contour Lines of the Probability Density . . . . . . . . . . 1767.6 Contour Surface of the Probability Density . . . . . . . . . 1777.7 Harmonic Particle Motion . . . . . . . . . . . . . . . . . . 1797.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 180

8 Scattering in Three Dimensions . . . . . . . . . . . . . . . . . 1858.1 Physical Concepts . . . . . . . . . . . . . . . . . . . . . . 185

8.1.1 Radial Scattering Wave Functions . . . . . . . . . . 1858.1.2 Boundary and Continuity Conditions.

Solution of the System of InhomogeneousLinear Equations for the Coefficients . . . . . . . . 187

8.1.3 Scattering of a Plane Harmonic Wave . . . . . . . . 1888.1.4 Scattering Amplitude and Phase. Unitarity.

The Argand Diagram . . . . . . . . . . . . . . . . 1928.1.5 Coulomb Scattering . . . . . . . . . . . . . . . . . 193

8.2 Radial Wave Functions . . . . . . . . . . . . . . . . . . . 1958.3 Stationary Wave Functions and Scattered Waves . . . . . . 1978.4 Differential Cross Sections . . . . . . . . . . . . . . . . . 1998.5 Scattering Amplitude. Phase Shift.

Partial and Total Cross Sections . . . . . . . . . . . . . . . 2018.6 Coulomb Scattering: Radial Wave Function . . . . . . . . . 2048.7 Coulomb Scattering: 3D Wave Function . . . . . . . . . . . 2068.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 207

xii Contents

9 Spin and Magnetic Resonance . . . . . . . . . . . . . . . . . . 2139.1 Physical Concepts . . . . . . . . . . . . . . . . . . . . . . 213

9.1.1 Spin Operators. Eigenvectors and Eigenvalues . . . 2139.1.2 Magnetic Moment and Its Motion

in a Magnetic Field. Pauli Equation . . . . . . . . . 2159.1.3 Magnetic Resonance . . . . . . . . . . . . . . . . 2169.1.4 Rabi Formula . . . . . . . . . . . . . . . . . . . . 219

9.2 The Spin-Expectation Vector near and at Resonance . . . . 2209.3 Resonance Form of the Rabi Amplitude . . . . . . . . . . . 2229.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 223

10 Hybridization . . . . . . . . . . . . . . . . . . . . . . . . . . . 22410.1 Physical Concepts . . . . . . . . . . . . . . . . . . . . . . 224

10.1.1 Hybrid States in the Coulomb Potential . . . . . . . 22410.1.2 Some Qualitative Details of Hybridization . . . . . 22610.1.3 Hybridization Parameters and Orientations

of Highly Symmetric Hybrid States . . . . . . . . . 22710.2 Hybrid Wave Functions and Probability Densities . . . . . . 23110.3 Contour Lines of Hybrid Wave Functions

and Probability Densities . . . . . . . . . . . . . . . . . . 23210.4 Contour Surfaces of Hybrid Probability Densities . . . . . . 23410.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 236

11 Special Functions of Mathematical Physics . . . . . . . . . . . 23811.1 Basic Formulae . . . . . . . . . . . . . . . . . . . . . . . 238

11.1.1 Hermite Polynomials . . . . . . . . . . . . . . . . 23811.1.2 Harmonic-Oscillator Eigenfunctions . . . . . . . . 23911.1.3 Legendre Polynomials and Legendre Functions . . . 23911.1.4 Spherical Harmonics . . . . . . . . . . . . . . . . 24011.1.5 Bessel Functions . . . . . . . . . . . . . . . . . . 24111.1.6 Spherical Bessel Functions . . . . . . . . . . . . . 24211.1.7 Airy Functions . . . . . . . . . . . . . . . . . . . 24311.1.8 Laguerre Polynomials . . . . . . . . . . . . . . . . 24411.1.9 Radial Eigenfunctions of the Harmonic Oscillator . 24511.1.10 Radial Eigenfunctions of the Hydrogen Atom . . . . 24511.1.11 Gaussian Distribution and Error Function . . . . . . 24611.1.12 Binomial Distribution and Poisson Distribution . . . 247

11.2 Hermite Polynomials and Related Functions . . . . . . . . 24811.3 Legendre Polynomials and Related Functions . . . . . . . . 24911.4 Spherical Harmonics: Surface over Cartesian Grid . . . . . 25011.5 Spherical Harmonics: 2D Polar Diagram . . . . . . . . . . 25111.6 Spherical Harmonics: Polar Diagram in 3D . . . . . . . . . 25211.7 Bessel Functions and Related Functions . . . . . . . . . . . 253

Contents xiii

11.8 Bessel Function and Modified Bessel Functionwith Real Index . . . . . . . . . . . . . . . . . . . . . . . 255

11.9 Airy Functions . . . . . . . . . . . . . . . . . . . . . . . . 25611.10 Laguerre Polynomials . . . . . . . . . . . . . . . . . . . . 25711.11 Laguerre Polynomials as Function of x and the Upper Index α 25811.12 Gaussian Distribution . . . . . . . . . . . . . . . . . . . . 25811.13 Error Function and Complementary Error Function . . . . . 25911.14 Bivariate Gaussian Distribution . . . . . . . . . . . . . . . 26011.15 Bivariate Gaussian: Covariance Ellipse . . . . . . . . . . . 26011.16 Binomial Distribution . . . . . . . . . . . . . . . . . . . . 26211.17 Poisson Distribution . . . . . . . . . . . . . . . . . . . . . 26311.18 Simple Functions of a Complex Variable . . . . . . . . . . 26411.19 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 266

12 Additional Material and Hints for the Solution of Exercises . . 26912.1 Units and Orders of Magnitude . . . . . . . . . . . . . . . 269

12.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . 26912.1.2 SI Units . . . . . . . . . . . . . . . . . . . . . . . 26912.1.3 Scaled Units . . . . . . . . . . . . . . . . . . . . . 27012.1.4 Atomic and Subatomic Units . . . . . . . . . . . . 27112.1.5 Data-Table Units . . . . . . . . . . . . . . . . . . 27212.1.6 Special Scales . . . . . . . . . . . . . . . . . . . . 274

12.2 Argand Diagrams and Unitarityfor One-Dimensional Problems . . . . . . . . . . . . . . . 27512.2.1 Probability Conservation and the Unitarity

of the Scattering Matrix . . . . . . . . . . . . . . . 27512.2.2 Time Reversal and the Scattering Matrix . . . . . . 27712.2.3 Diagonalization of the Scattering Matrix . . . . . . 27812.2.4 Argand Diagrams . . . . . . . . . . . . . . . . . . 28012.2.5 Resonances . . . . . . . . . . . . . . . . . . . . . 281

12.3 Hints and Answers to the Exercises . . . . . . . . . . . . . 284

A A Systematic Guide to IQ . . . . . . . . . . . . . . . . . . . . 315A.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . 315

A.1.1 Starting IQ . . . . . . . . . . . . . . . . . . . . . 315A.1.2 Introductory Demonstration . . . . . . . . . . . . . 315A.1.3 Selecting a Descriptor File . . . . . . . . . . . . . 316A.1.4 Selecting a Descriptor and Producing a Plot . . . . . 316A.1.5 Creating and Running a Movie . . . . . . . . . . . 318A.1.6 Printing a Plot . . . . . . . . . . . . . . . . . . . . 320A.1.7 Changing Colors and Line Widths . . . . . . . . . 320A.1.8 Changing Parameters . . . . . . . . . . . . . . . . 322A.1.9 Saving a Changed Descriptor . . . . . . . . . . . . 322

xiv Contents

A.1.10 Creating a Mother Descriptor . . . . . . . . . . . . 324A.1.11 Editing Descriptor Files . . . . . . . . . . . . . . . 324A.1.12 Printing a Set of Plots . . . . . . . . . . . . . . . . 326A.1.13 Running a Demonstration . . . . . . . . . . . . . . 326A.1.14 Customizing . . . . . . . . . . . . . . . . . . . . 327A.1.15 Help and Context-Sensitive Help . . . . . . . . . . 330

A.2 Coordinate Systems and Transformations . . . . . . . . . . 330A.2.1 The Different Coordinate Systems . . . . . . . . . 330A.2.2 Defining the Transformations . . . . . . . . . . . . 332

A.3 The Different Types of Plot . . . . . . . . . . . . . . . . . 335A.3.1 Surface over Cartesian Grid in 3D . . . . . . . . . 335A.3.2 Surface over Polar Grid in 3D . . . . . . . . . . . . 337A.3.3 2D Function Graph . . . . . . . . . . . . . . . . . 337A.3.4 Contour-Line Plot in 2D . . . . . . . . . . . . . . 340A.3.5 Contour-Surface Plot in 3D . . . . . . . . . . . . . 341A.3.6 Polar Diagram in 3D . . . . . . . . . . . . . . . . 342A.3.7 Probability-Ellipsoid Plot . . . . . . . . . . . . . . 344A.3.8 3D Column Plot . . . . . . . . . . . . . . . . . . . 345

A.4 Parameters – The Subpanel Movie . . . . . . . . . . . . . . 346A.5 Parameters – The Subpanel Physics . . . . . . . . . . . . . 346

A.5.1 The Subpanel Physics—Comp. Coord. . . . . . . . . 347A.5.2 The Subpanel Multiple Plot . . . . . . . . . . . . . . 347

A.6 Parameters – The Subpanel Graphics . . . . . . . . . . . . . 347A.6.1 The Subpanel Graphics—Geometry . . . . . . . . . 347A.6.2 The Subpanel Graphics—Accuracy . . . . . . . . . . 348A.6.3 The Subpanel Graphics—Hidden Lines . . . . . . . . 349

A.7 Parameters – The Subpanel Background . . . . . . . . . . . 350A.7.1 The Subpanel Background—Box . . . . . . . . . . . 350A.7.2 The Subpanel Background—Scales . . . . . . . . . . 352A.7.3 The Subpanel Background—Arrows . . . . . . . . . 353A.7.4 The Subpanel Background—Texts . . . . . . . . . . 355

A.8 Parameters – The Subpanel Format . . . . . . . . . . . . . . 356A.9 Coding Mathematical Symbols and Formulae . . . . . . . . 356A.10 A Combined Plot and Its Mother Descriptor . . . . . . . . . 358

A.10.1 The Subpanel Type and Format . . . . . . . . . . . . 358A.10.2 The Subpanel Table of Descriptors . . . . . . . . . . 359A.10.3 Special Cases . . . . . . . . . . . . . . . . . . . . 360

A.11 Details of Printing . . . . . . . . . . . . . . . . . . . . . . 360A.11.1 Preview. Colors and Line Widths . . . . . . . . . . 360A.11.2 Using a System Printer . . . . . . . . . . . . . . . 361A.11.3 Creating PostScript Files: IQ Export . . . . . . . . 362

A.12 Preparing a Demonstration . . . . . . . . . . . . . . . . . 363

Contents xv

B How to Install IQ . . . . . . . . . . . . . . . . . . . . . . . . . 365B.1 Contents of the CD-ROM . . . . . . . . . . . . . . . . . . 365B.2 Computer Systems on which INTERQUANTA Can Be Used 365B.3 Installation with Options. The File ReadMe.txt . . . . . . . 366B.4 Quick Installation for the Impatient User . . . . . . . . . . 366

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369