Interactive Quantum Mechanics

Interactive Quantum Mechanics

S. Brandt H.D. Dahmen T. Stroh Interactive Quantum Mechanics Quantum Experiments on the Computer Second Edition With CD-ROM, 128 Figures, and 344 Exercises

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 Additional material to this book can be downloaded from http://extras.springer.com ISBN 978-1-4419-7423-5 e-isbn 978-1-4419-7424-2 DOI 10.1007/978-1-4419-7424-2 Springer New York Dordrecht Heidelberg London Springer Science+Business Media, LLC 2011 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface to the Second Edition For the present edition the concept of the book and of INTERQUANTA, the accompanying Interactive Program of Quantum Mechanics, (IQ, for short), was left unchanged. However, the physics scope of the text and the capabilities of the program were widened appreciably. The most conspicuous addition to IQ is the capability to produce and display movies of quantum-mechanical phenomena. So far, IQ presented time dependence 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 under study and can be examined quantitatively at leisure, the new movies give a more direct impression of what happens as time passes. For such movies, as for the conventional simulations, the parameters defining the physical phenomenon and the graphical appearance can be changed interactively. Movies can be produced and also stored for many quantum-mechanical problems such as 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, identical bosons). Concerning the physics scope, these are the main additions: One-dimensional bound and scattering states are now discussed and computed also for piecewise linear potentials. These, as opposed to the usual step potentials (which are piecewise constant), allow for much better approximations of arbitrary smooth potentials. Another interesting addition to one-dimensional quantum mechanics is the juxtaposition of quantum-mechanical wave packets with classical phase-space distributions. The treatment of quantum mechanics in three dimensions is extended by the hybridization of bound states and by 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 as movie is more than tripled. In this way users have a much richer choice of ready-made examples from which to start their exploits. Moreover, solution descriptors are now provided for the exercises. Siegen, Germany May 2010 Siegmund Brandt Hans Dieter Dahmen Tilo Stroh v

Preface to the First Edition This book can be regarded as a concise introduction to basic quantum mechanics: free particle, bound states, and scattering in one and in three dimensions, two-particle systems, special functions of mathematical physics. But the book can also be seen as an extensive user s guide for INTERQUANTA, the Interactive Program of Quantum Mechanics, which we will abbreviate henceforth as IQ. The book also contains a large number of exercises. The program can be used in two ways. By working through (at least a part of) these exercises, the user of IQ explores a computer laboratory in quantum mechanics by performing computer experiments. A simpler way to use IQ is to study one or several of the ready-made demonstrations. In each demonstration the user is taken through one chapter of quantum mechanics. Graphics illustrating 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 essentially self-explanatory. In addition, extensive help functions are provided not only on technical questions but also on quantum-mechanical concepts. All in all using INTERQUANTA is not more difficult than surfing the Internet. The modern user interface is the main improvement over older versions of IQ. 1 Moreover, new physics topics are added and there are also new graphical features. The present version of INTERQUANTA is easily installed and run on personal computers (running under Windows or Linux) or Macintosh (running under Mac OS X). We do hope that by using INTERQUANTA on their own computer many students will gain experience with different quantum phenomena without having to do tedious calculations. From this experience an intuition for this important but abstract field of modern science can be developed. Siegen, Germany February 2003 Siegmund Brandt Hans Dieter Dahmen Tilo Stroh 1 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 York 1991 and 1995; Pasocon de manebu ryoushi nikigacu, Springer, Tokyo 1992; Quantenmechanik auf dem Personalcomputer, Springer, Berlin 1993 vi

Contents Preface to the Second Edition.................. v Preface to the First Edition................... vi 1 Introduction........................... 1 1.1 Interquanta......................... 1 1.2 The Structure of This Book................. 2 1.3 The Demonstrations..................... 3 1.4 The Computer Laboratory.................. 3 1.5 Literature.......................... 4 2 Free Particle Motion in One Dimension............. 5 2.1 Physical Concepts...................... 5 2.1.1 Planck s Constant. Schrödinger s Equation for a Free Particle.................. 5 2.1.2 The Wave Packet. Group Velocity. Normalization. 6 2.1.3 Probability-Current Density. Continuity Equation. 7 2.1.4 Quantile Position. Quantile Trajectory....... 8 2.1.5 Relation to Bohm s Equation of Motion....... 9 2.1.6 Analogies in Optics................. 10 2.1.7 Analogies in Classical Mechanics: The Phase-Space Probability Density........ 11 2.2 A First Session with the Computer............. 15 2.2.1 Starting IQ..................... 15 2.2.2 An Automatic Demonstration............ 16 2.2.3 A First Dialog.................... 16 2.3 The Free Quantum-Mechanical Gaussian Wave Packet... 17 2.3.1 The Subpanel Physics Comp. Coord......... 18 2.3.2 The Subpanel Physics Wave Packet......... 19 2.3.3 The Subpanel Physics Quantile........... 19 2.3.4 The Subpanel Movie................. 20 2.4 The Free Optical Gaussian Wave Packet........... 20 2.5 Quantile Trajectories.................... 21 vii

viii Contents 2.6 The Spectral Function of a Gaussian Wave Packet...... 22 2.7 The Wave Packet as a Sum of Harmonic Waves....... 23 2.8 The Phase-Space Distribution of Classical Mechanics.... 25 2.9 Classical Phase-Space Distribution: Covariance Ellipse... 26 2.10 Exercises.......................... 28 3 Bound States in One Dimension................. 32 3.1 Physical Concepts...................... 32 3.1.1 Schrödinger s Equation with a Potential. Eigenfunctions. Eigenvalues............ 32 3.1.2 Normalization. Discrete Spectra. Orthonormality.. 33 3.1.3 The Infinitely Deep Square-Well Potential..... 33 3.1.4 The Harmonic Oscillator.............. 34 3.1.5 The Step Potential.................. 34 3.1.6 The Piecewise Linear Potential........... 36 3.1.7 Time-Dependent Solutions............. 37 3.1.8 Harmonic Particle Motion. Coherent States. Squeezed States................... 37 3.1.9 Quantile Motion in the Harmonic-Oscillator Potential 38 3.1.10 Harmonic Motion of a Classical Phase-Space Distribution.............. 38 3.1.11 Particle Motion in a Deep Square Well....... 41 3.2 Eigenstates in the Infinitely Deep Square-Well Potential and in the Harmonic-Oscillator Potential.......... 43 3.3 Eigenstates in the Step Potential............... 45 3.4 Eigenstates in the Step Potential Quasiperiodic...... 46 3.5 Eigenstates in the Piecewise Linear Potential........ 47 3.6 Eigenstates in the Piecewise Linear Potential Quasiperiodic 48 3.7 Harmonic Particle Motion.................. 49 3.8 Harmonic Oscillator: Quantile Trajectories......... 51 3.9 Classical Phase-Space Distribution: Harmonic Motion... 52 3.10 Harmonic Motion of Classical Phase-Space Distribution: Covariance Ellipse..................... 53 3.11 Particle Motion in the Infinitely Deep Square-Well Potential 54 3.12 Exercises.......................... 56 4 Scattering in One Dimension.................. 63 4.1 Physical Concepts...................... 63 4.1.1 Stationary Scattering States. Continuum Eigenstates and Eigenvalues. Continuous Spectra................. 63 4.1.2 Time-Dependent Solutions of the Schrödinger Equation............. 64

Contents ix 4.1.3 Right-Moving and Left-Moving Stationary Waves of a Free Particle......... 64 4.1.4 Orthogonality and Continuum Normalization of Stationary Waves of a Free Particle. Completeness 65 4.1.5 Boundary Conditions for Stationary Scattering Solutions in Step Potentials 66 4.1.6 Stationary Scattering Solutions in Step Potentials.. 67 4.1.7 Constituent Waves................. 68 4.1.8 Normalization of Continuum Eigenstates...... 68 4.1.9 Harmonic Waves in a Step Potential........ 68 4.1.10 Time-Dependent Scattering Solutions in a Step Potential.................. 69 4.1.11 Generalization to Piecewise Linear Potentials.... 69 4.1.12 Transmission and Reflection. Unitarity. The Argand Diagram................ 70 4.1.13 The Tunnel Effect.................. 71 4.1.14 Resonances..................... 72 4.1.15 Phase Shifts upon Reflection at a Steep Rise or Deep Fall of the Potential............. 72 4.1.16 Transmission Resonances upon Reflection at More- and Less-Dense Media.......... 74 4.1.17 The Quantum-Well Device and the Quantum-Effect Device........... 75 4.1.18 Stationary States in a Linear Potential........ 77 4.1.19 Wave Packet in a Linear Potential.......... 77 4.1.20 Quantile Motion in a Linear Potential........ 78 4.1.21 Classical Phase-Space Density in a Linear Potential 78 4.1.22 Classical Phase-Space Density Reflected by a High Potential Wall......... 79 4.2 Stationary Scattering States in the Step Potential and in the Piecewise Linear Potential............ 81 4.3 Time-Dependent Scattering by the Step Potential and by the Piecewise Linear Potential............ 83 4.4 Transmission and Reflection. The Argand Diagram..... 87 4.5 Stationary Wave in a Linear Potential............ 90 4.6 Gaussian Wave Packet in a Linear Potential......... 91 4.7 Quantile Trajectories in a Linear Potential.......... 92 4.8 Classical Phase-Space Density in a Linear Potential..... 92 4.9 Classical Phase-Space Distribution: Covariance Ellipse... 94 4.10 Classical Phase-Space Density Reflected by a High Potential Wall............. 95 4.11 Exercises.......................... 97

x Contents 4.12 Analogies in Optics..................... 109 4.13 Reflection and Refraction of Stationary Electromagnetic Waves............ 113 4.14 Time-Dependent Scattering of Light............. 114 4.15 Transmission, Reflection, and Argand Diagram for a Light Wave...................... 117 4.16 Exercises.......................... 119 5 A Two-Particle System: Coupled Harmonic Oscillators.... 122 5.1 Physical Concepts...................... 122 5.1.1 The Two-Particle System.............. 122 5.1.2 Initial Condition for Distinguishable Particles.... 124 5.1.3 Time-Dependent Wave Functions and Probability Distributions for Distinguishable Particles 125 5.1.4 Marginal Distributions for Distinguishable Particles 125 5.1.5 Wave Functions for Indistinguishable Particles. Symmetrization for Bosons. Antisymmetrization for Fermions.......... 126 5.1.6 Marginal Distributions of the Probability Densities of Bosons and Fermions 127 5.1.7 Normal Oscillations................. 127 5.2 Stationary States....................... 128 5.3 Time Dependence of Global Variables............ 129 5.4 Joint Probability Densities.................. 130 5.5 Marginal Distributions.................... 131 5.6 Exercises.......................... 133 6 Free Particle Motion in Three Dimensions........... 138 6.1 Physical Concepts...................... 138 6.1.1 The Schrödinger Equation of a Free Particle in 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..... 142 6.1.4 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... 145 6.1.6 Partial-Wave Decomposition of the Gaussian Wave Packet............ 145 6.2 The 3D Harmonic Plane Wave................ 148 6.3 The Plane Wave Decomposed into Spherical Waves..... 150 6.4 The 3D Gaussian Wave Packet............... 151

Contents xi 6.5 The Probability Ellipsoid.................. 152 6.6 Angular-Momentum Decomposition of a Wave Packet... 153 6.7 Exercises.......................... 155 7 Bound States in Three Dimensions............... 158 7.1 Physical Concepts...................... 158 7.1.1 The Schrödinger Equation for a Particle under 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..... 161 7.1.4 The Spherical Step Potential............ 162 7.1.5 The Harmonic Oscillator.............. 165 7.1.6 The Coulomb Potential. The Hydrogen Atom.... 166 7.1.7 Harmonic Particle Motion.............. 167 7.2 Radial Wave Functions in Simple Potentials......... 168 7.3 Radial Wave Functions in the Step Potential......... 172 7.4 Probability Densities.................... 173 7.5 Contour Lines of the Probability Density.......... 176 7.6 Contour Surface of the Probability Density......... 177 7.7 Harmonic Particle Motion.................. 179 7.8 Exercises.......................... 180 8 Scattering in Three Dimensions................. 185 8.1 Physical Concepts...................... 185 8.1.1 Radial Scattering Wave Functions.......... 185 8.1.2 Boundary and Continuity Conditions. Solution of the System of Inhomogeneous Linear Equations for the Coefficients........ 187 8.1.3 Scattering of a Plane Harmonic Wave........ 188 8.1.4 Scattering Amplitude and Phase. Unitarity. The Argand Diagram................ 192 8.1.5 Coulomb Scattering................. 193 8.2 Radial Wave Functions................... 195 8.3 Stationary Wave Functions and Scattered Waves...... 197 8.4 Differential Cross Sections................. 199 8.5 Scattering Amplitude. Phase Shift. Partial and Total Cross Sections............... 201 8.6 Coulomb Scattering: Radial Wave Function......... 204 8.7 Coulomb Scattering: 3D Wave Function........... 206 8.8 Exercises.......................... 207

xii Contents 9 Spin and Magnetic Resonance.................. 213 9.1 Physical Concepts...................... 213 9.1.1 Spin Operators. Eigenvectors and Eigenvalues... 213 9.1.2 Magnetic Moment and Its Motion in a Magnetic Field. Pauli Equation......... 215 9.1.3 Magnetic Resonance................ 216 9.1.4 Rabi Formula.................... 219 9.2 The Spin-Expectation Vector near and at Resonance.... 220 9.3 Resonance Form of the Rabi Amplitude........... 222 9.4 Exercises.......................... 223 10 Hybridization........................... 224 10.1 Physical Concepts...................... 224 10.1.1 Hybrid States in the Coulomb Potential....... 224 10.1.2 Some Qualitative Details of Hybridization..... 226 10.1.3 Hybridization Parameters and Orientations of Highly Symmetric Hybrid States......... 227 10.2 Hybrid Wave Functions and Probability Densities...... 231 10.3 Contour Lines of Hybrid Wave Functions and Probability Densities.................. 232 10.4 Contour Surfaces of Hybrid Probability Densities...... 234 10.5 Exercises.......................... 236 11 Special Functions of Mathematical Physics........... 238 11.1 Basic Formulae....................... 238 11.1.1 Hermite Polynomials................ 238 11.1.2 Harmonic-Oscillator Eigenfunctions........ 239 11.1.3 Legendre Polynomials and Legendre Functions... 239 11.1.4 Spherical Harmonics................ 240 11.1.5 Bessel Functions.................. 241 11.1.6 Spherical Bessel Functions............. 242 11.1.7 Airy Functions................... 243 11.1.8 Laguerre Polynomials................ 244 11.1.9 Radial Eigenfunctions of the Harmonic Oscillator. 245 11.1.10 Radial Eigenfunctions of the Hydrogen Atom.... 245 11.1.11 Gaussian Distribution and Error Function...... 246 11.1.12 Binomial Distribution and Poisson Distribution... 247 11.2 Hermite Polynomials and Related Functions........ 248 11.3 Legendre Polynomials and Related Functions........ 249 11.4 Spherical Harmonics: Surface over Cartesian Grid..... 250 11.5 Spherical Harmonics: 2D Polar Diagram.......... 251 11.6 Spherical Harmonics: Polar Diagram in 3D......... 252 11.7 Bessel Functions and Related Functions........... 253

Contents xiii 11.8 Bessel Function and Modified Bessel Function with Real Index....................... 255 11.9 Airy Functions........................ 256 11.10 Laguerre Polynomials.................... 257 11.11 Laguerre Polynomials as Function of x and the Upper Index α 258 11.12 Gaussian Distribution.................... 258 11.13 Error Function and Complementary Error Function..... 259 11.14 Bivariate Gaussian Distribution............... 260 11.15 Bivariate Gaussian: Covariance Ellipse........... 260 11.16 Binomial Distribution.................... 262 11.17 Poisson Distribution..................... 263 11.18 Simple Functions of a Complex Variable.......... 264 11.19 Exercises.......................... 266 12 Additional Material and Hints for the Solution of Exercises.. 269 12.1 Units and Orders of Magnitude............... 269 12.1.1 Definitions..................... 269 12.1.2 SI Units....................... 269 12.1.3 Scaled Units..................... 270 12.1.4 Atomic and Subatomic Units............ 271 12.1.5 Data-Table Units.................. 272 12.1.6 Special Scales.................... 274 12.2 Argand Diagrams and Unitarity for One-Dimensional Problems............... 275 12.2.1 Probability Conservation and the Unitarity of the Scattering Matrix............... 275 12.2.2 Time Reversal and the Scattering Matrix...... 277 12.2.3 Diagonalization of the Scattering Matrix...... 278 12.2.4 Argand Diagrams.................. 280 12.2.5 Resonances..................... 281 12.3 Hints and Answers to the Exercises............. 284 A A Systematic Guide to IQ.................... 315 A.1 Overview.......................... 315 A.1.1 Starting IQ..................... 315 A.1.2 Introductory Demonstration............. 315 A.1.3 Selecting a Descriptor File............. 316 A.1.4 Selecting a Descriptor and Producing a Plot..... 316 A.1.5 Creating and Running a Movie........... 318 A.1.6 Printing a Plot.................... 320 A.1.7 Changing Colors and Line Widths......... 320 A.1.8 Changing Parameters................ 322 A.1.9 Saving a Changed Descriptor............ 322

xiv Contents A.1.10 Creating a Mother Descriptor............ 324 A.1.11 Editing Descriptor Files............... 324 A.1.12 Printing a Set of Plots................ 326 A.1.13 Running a Demonstration.............. 326 A.1.14 Customizing.................... 327 A.1.15 Help and Context-Sensitive Help.......... 330 A.2 Coordinate Systems and Transformations.......... 330 A.2.1 The Different Coordinate Systems......... 330 A.2.2 Defining the Transformations............ 332 A.3 The Different Types of Plot................. 335 A.3.1 Surface over Cartesian Grid in 3D......... 335 A.3.2 Surface over Polar Grid in 3D............ 337 A.3.3 2D Function Graph................. 337 A.3.4 Contour-Line Plot in 2D.............. 340 A.3.5 Contour-Surface Plot in 3D............. 341 A.3.6 Polar Diagram in 3D................ 342 A.3.7 Probability-Ellipsoid Plot.............. 344 A.3.8 3D Column Plot................... 345 A.4 Parameters The Subpanel Movie.............. 346 A.5 Parameters The Subpanel Physics............. 346 A.5.1 The Subpanel Physics Comp. Coord......... 347 A.5.2 The Subpanel Multiple Plot.............. 347 A.6 Parameters The Subpanel Graphics............. 347 A.6.1 The Subpanel Graphics Geometry......... 347 A.6.2 The Subpanel Graphics Accuracy.......... 348 A.6.3 The Subpanel Graphics Hidden Lines........ 349 A.7 Parameters The Subpanel Background........... 350 A.7.1 The Subpanel Background Box........... 350 A.7.2 The Subpanel Background Scales.......... 352 A.7.3 The Subpanel Background Arrows......... 353 A.7.4 The Subpanel Background Texts.......... 355 A.8 Parameters The Subpanel Format.............. 356 A.9 Coding Mathematical Symbols and Formulae........ 356 A.10 A Combined Plot and Its Mother Descriptor......... 358 A.10.1 The Subpanel Type and Format............ 358 A.10.2 The Subpanel Table of Descriptors.......... 359 A.10.3 Special Cases.................... 360 A.11 Details of Printing...................... 360 A.11.1 Preview. Colors and Line Widths.......... 360 A.11.2 Using a System Printer............... 361 A.11.3 Creating PostScript Files: IQ Export........ 362 A.12 Preparing a Demonstration................. 363

Contents xv B How to Install IQ......................... 365 B.1 Contents of the CD-ROM.................. 365 B.2 Computer Systems on which INTERQUANTA Can Be Used 365 B.3 Installation with Options. The File ReadMe.txt....... 366 B.4 Quick Installation for the Impatient User.......... 366 Index............................... 369