Thaller b visual quantum mechanics selected topics with computer generated animations of quantum mechanical phenomena ( 1999)(ISBN 0387989293)(314s)

Visual Quantum Mechanics: Selected Topics with Computer-Generated Animations of Quantum-Mechanical Phenomena Bernd Thaller Springer Visual Quantum Mechanics This page intentionally left blank Bernd Thaller Visual Quantum Mechanics Selected Topics with Computer-Generated Animations of Quantum-Mechanical Phenomena CD-ROM INCLUDED Bernd Thaller Institute for Mathematics University of Graz A-8010 Graz Austria bernd.thaller@kfunigraz.ac.at Library of Congress Cataloging-in-Publication Data Visual quantum mechanics : selected topics with computer-generated animations of quantum-mechanical phenomena / Bernd Thaller p cm Includes bibliographical references and index ISBN 0-387-98929-3 (hc : alk paper) Quantum theory Quantum theory—Computer simulation I Title QC174.12.T45 2000 99-42455 530.12 0113—dc21 Printed on acid-free paper Mathematica is a registered trademark of Wolfram Research, Inc QuickTimeTM is a registered trademark of Apple Computer, Inc., registered in the United States and other countries Used by license Macromedia and Macromedia R DirectorTM are registered trademarks of Macromedia, Inc., in the United States and other countries C 2000 Springer-Verlag New York, Inc TELOS R , The Electronic Library of Science, is an imprint of Springer-Verlag New York, Inc This Work consists of a printed book and a CD-ROM packaged with the book, both of which are protected by federal copyright law and international treaty The book may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis For copyright information regarding the CD-ROM, please consult the printed information packaged with the CD-ROM in the back of this publication, and which is also stored as a “readme” file on the CD-ROM Use of the printed version of this Work 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, other than those uses expressly granted in the CD-ROM copyright notice and disclaimer information, is forbidden The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone Where those designations appear in the book and Springer-Verlag was aware of a trademark claim, the designations follow the capitalization style used by the manufacturer Production managed by Steven Pisano; manufacturing supervised by Jacqui Ashri Photocomposed pages prepared from the author’s LATEX files Printed and bound by Hamilton Printing Co., Rensselaer, NY Printed in the United States of America ISBN 0-387-98929-3 Springer-Verlag New York Berlin Heidelberg SPIN 10743163 Preface In the strange world of quantum mechanics the application of visualization techniques is particularly rewarding, for it allows us to depict phenomena that cannot be seen by any other means Visual Quantum Mechanics relies heavily on visualization as a tool for mediating knowledge The book comes with a CD-ROM containing about 320 digital movies in QuickTimeTM format, which can be watched on every multimedia-capable computer These computer-generated animations are used to introduce, motivate, and illustrate the concepts of quantum mechanics that are explained in the book If a picture is worth a thousand words, then my hope is that each short animation (consisting of about a hundred frames) will be worth a hundred thousand words The collection of films on the CD-ROM is presented in an interactive environment that has been developed with the help of Macromedia DirectorTM This multimedia presentation can be used like an adventure game without special computer skills I hope that this presentation format will attract the interest of a wider audience to the beautiful theory of quantum mechanics Usually, in my own courses, I first show a movie that clearly depicts some phenomenon and then I explain step-by-step what can be learned from the animation The theory is further impressed on the students’ memory by watching and discussing several related movies Concepts presented in a visually appealing way are easier to remember Moreover, the visualization should trigger the students’ interest and provide some motivation for the effort to understand the theory behind it By “watching” the solutions of the Schră odinger equation the student will hopefully develop a feeling for the behavior of quantum-mechanical systems that cannot be gained by conventional means The book itself is self-contained and can be read without using the software This, however, is not recommended, because the phenomenological background for the theory is provided mainly by the movies, rather than the more traditional approach to motivating the theory using experimental results The text is on an introductory level and requires little previous knowledge, but it is not elementary When I considered how to provide the v vi PREFACE theoretical background for the animations, I found that only a more mathematical approach would lead the reader quickly to the level necessary to understand the more intricate details of the movies So I took the opportunity to combine a vivid discussion of the basic principles with a more advanced presentation of some mathematical aspects of the formalism Therefore, the book will certainly serve best as a companion in a theoretical physics course, while the material on the CD-ROM will be useful for a more general audience of science students The choice of topics and the organization of the text is in part due to purely practical considerations The development of software parallel to writing a text is a time-consuming process In order to speed up the publication I decided to split the text into two parts (hereafter called Book One and Book Two), with this first book containing selected topics This enables me to adapt to the technological evolution that has taken place since this project started, and helps provide the individual volumes at an affordable price The arrangement of the topics allows us to proceed from simple to more and more complicated animations Book One mainly deals with spinless particles in one and two dimensions, with a special emphasis on exactly solvable problems Several topics that are usually considered to belong to a basic course in quantum mechanics are postponed until Book Two Book Two will include chapters about spherical symmetry in three dimensions, the hydrogen atom, scattering theory and resonances, periodic potentials, particles with spin, and relativistic problems (the Dirac equation) Let me add a few remarks concerning the contents of Book One The first two chapters serve as a preparation for different aspects of the course The ideas behind the methods of visualizing wave functions are fully explained in Chapter We describe a special color map of the complex plane that is implemented by Mathematica packages for plotting complex-valued functions These packages have been created especially for this book They are included on the CD-ROM and will, hopefully, be useful for the reader who is interested in advanced graphics programming using Mathematica Chapter introduces some mathematical concepts needed for quantum mechanics Fourier analysis is an essential tool for solving the Schră odinger equation and for extracting physical information from the wave functions This chapter also presents concepts such as Hilbert spaces, linear operators, and distributions, which are all basic to the mathematical apparatus of quantum mechanics In this way, the methods for solving the Schră odinger equation are already available when it is introduced in Chapter and the student is better prepared to concentrate on conceptual problems Certain more abstract topics have been included mainly for the sake of completeness Initially, a beginner does not need to know all this “abstract nonsense,” and PREFACE vii the corresponding sections (marked as “special topics”) may be skipped at first reading Moreover, the symbol Ψ has been used to designate some paragraphs intended for the mathematically interested reader Quantum mechanics starts with Chapter We describe the free motion of approximately localized wave packets and put some emphasis on the statistical interpretation and the measurement process The Schrăodinger equation for particles in external fields is given in Chapter This chapter on states and observables describes the heuristic rules for obtaining the correct quantum observables when performing the transition from classical to quantum mechanics We proceed with the motion under the influence of boundary conditions (impenetrable walls) in Chapter The particle in a box serves to illustrate the importance of eigenfunctions of the Hamiltonian and of the eigenfunction expansion Once again we come back to interpretational difficulties in our discussion of the double-slit experiment Further mathematical results about unitary groups, canonical commutation relations, and symmetry transformations are provided in Chapter which focuses on linear operators Among the mathematically more sophisticated topics that usually not appear in textbooks are the questions related to the domains of linear operators I included these topics for several reasons For example, solutions that are not in the domain of the Hamiltonian have strange temporal behavior and produce interesting effects when visualized in a movie Some of these often surprising phenomena are perhaps not widely known even among professional scientists Among these I would like to mention the strange behavior of the unit function in a Dirichlet box shown in the movie CD 4.11 (Chapter 5) The remaining chapters deal with subjects of immediate physical importance: the harmonic oscillator in Chapter 7, constant electric and magnetic fields in Chapter 8, and some elements of scattering theory in Chapter The exactly solvable quantum systems serve to underpin the theory by examples for which all results can be obtained explicitly Therefore, these systems play a special role in this course although they are an exception in nature Many of the animations on the CD-ROM show wave packets in two dimensions Hence the text pays more attention than usual to two-dimensional problems, and problems that can be reduced to two dimensions by exploiting their symmetry For example, Chapter presents the angular-momentum decomposition in two dimensions The investigation of two-dimensional systems is not merely an exercise Very good approximations to such systems occur in nature A good example is the surface states of electrons which can be depicted by a scanning tunneling microscope viii PREFACE The experienced reader will notice that the emphasis in the treatment of exactly solvable systems has been shifted from a mere calculation of eigenvalues to an investigation of the dynamics of the system The treatment of the harmonic oscillator or the constant magnetic field makes it very clear that in order to understand the motion of wave packets, much more is needed than just a derivation of the energy spectrum Our presentation includes advanced topics such as coherent states, completeness of eigenfunctions, and Mehler’s integral kernel of the time evolution Some of these results certainly go beyond the scope of a basic course, but in view of the overwhelming number of elementary books on quantum mechanics the inclusion of these subjects is warranted Indeed, a new book must also contain interesting topics which cannot easily be found elsewhere Despite the presentation of advanced results, an effort has been made to keep the explanations on a level that can be understood by anyone with a little background in elementary calculus Therefore I hope that the text will fill a gap between the classical texts (e.g., [39], [48], [49], [68]) and the mathematically advanced presentations (e.g., [4], [17], [62], [76]) For those who like a more intuitive approach it is recommended that first a book be read that tries to avoid technicalities as long as possible (e.g., [19] or [40]) Most of the films on the CD-ROM were generated with the help of the computer algebra system Mathematica While Mathematica has played an important role in the creation of this book, the reader is not required to have any knowledge of a computer algebra system Alternate approaches which use symbolic mathematics packages on a computer to teach quantum mechanics can be found, for example, in the books [18] and [36], which are warmly recommended to readers familiar with both quantum mechanics and Mathematica or Maple However, no interactive computer session can replace an hour of thinking just with the help of a pencil and a sheet of paper Therefore, this text describes the mathematical and physical ideas of quantum mechanics in the conventional form It puts no special emphasis on symbolic computation or computational physics The computer is mainly used to provide quick and easy access to a large collection of animated illustrations, interactive pictures, and lots of supplementary material The book teaches the concepts, and the CD-ROM engages the imagination It is hoped that this combination will foster a deeper understanding of quantum mechanics than is usually achieved with more conventional methods While knowledge of Mathematica is not necessary to learn quantum mechanics with this text, there is a lot to find here for readers with some experience in Mathematica The supplementary material on the CD-ROM includes many Mathematica notebooks which may be used for the reader’s own computer experiments PREFACE ix In many cases it is not possible to obtain explicit solutions of the Schrăodinger equation For the numerical treatment we used external C++ routines linked to Mathematica using the MathLink interface This has been done to enhance computation speed The simulations are very large and need a lot of computational power, but all of them can be managed on a modern personal computer On the CD-ROM will be found all the necessary information as well as the software needed for the student to produce similar films on his/her own The exploration of quantum-mechanical systems usually requires more than just a variation of initial conditions and/or potentials (although this is sometimes very instructive) The student will soon notice that a very detailed understanding of the system is needed in order to produce a useful film illustrating its typical behavior This book has a home page on the internet with URL http://www.kfunigraz.ac.at/imawww/vqm/ As this site evolves, the reader will find more supplementary material, exercises and solutions, additional animations, links to other sites with quantummechanical visualizations, etc Acknowledgments During the preparation of both the book and the software I have profited from many suggestions offered by students and colleagues My thanks to M Liebmann for his contributions to the software, and to K Unterkofler for his critical remarks and for his hospitality in Millstatt, where part of this work was completed This book would not have been written without my wife Sigrid, who not only showed patience and understanding when I spent 150% of my time with the book and only -50% with my family, but who also read the entire manuscript carefully, correcting many errors and misprints My son Wolfgang deserves special thanks Despite numerous projects of his own, he helped me a lot with his unparalleled computer skills I am grateful to the people at Springer-Verlag, in particular to Steven Pisano for his professional guidance through the production process Finally, a project preparation grant from Springer-Verlag is gratefully acknowledged Bernd Thaller Im z i -1 Re z -i Color Plate Color map of the complex plane Each complex number has a hue proportional to its phase, the lightness corresponds to its absolute value This color map of the plane is obtained by a stereographic projection from the colored sphere in Color Plate (Section 1.2.3.) 1 x -1 -5 (a) Im x (b) Re -5 (c) Color Plate Various visualizations of the complexvalued function x → exp(ix) See CD 1.8 for other examples (a) Separate plots of the real part (red) and the imaginary part (yellow-green), (b) representation as a space curve, (c) plot of the absolute value with a color code for the phase (Section 1.3.1.) x Color Plate Visualizations of a wave function in two dimensions The left graphic shows the function as a “density plot” with additional contour lines for the absolute value In the three-dimensional surface plot the height of the surface gives the absolute value of the wave function In both cases, the color describes the complex values according to Color Plate (Section 1.3.2.) Color Plate Visualization of the function ψ(x, y) = (x + iy)3 − using the color map of Color Plate The left graphic shows the function ψ and the right graphic shows its square ψ The zeros of ψ are of second order This can be easily recognized because all colors appear twice on a small circle around each zero (Section 1.3.2.) Color Plate Visualization of a wave function in three dimensions The picture shows the wave function of a highly excited state of the hydrogen atom (with quantum numbers n = 10, l = 5, m = 3) A certain level of the absolute value of the wave function is indicated by an isosurface The hue of the color is given by the phase of the wave function (Section 1.3.2.) 0.6 0.6 0.4 0.4 0.2 0.2 -6 -4 -2 -4 -2 -10 -5 10 0.5 0.8 0.4 0.6 0.3 0.4 0.2 0.2 0 0.1 -4 -2 0.5 0.3 0.4 0.3 0.2 0.2 0.1 0.1 -6 -4 -2 0.3 0.3 0.2 0.2 0.1 0.1 -4 -2 -5 -10 -9 -6 -3 10 Color Plate Various functions and the spectrum of the ˆ n(L) ) defined in Eq (2.13), with kn(L) = Fourier amplitudes ψ(k nπ/L The lines describe the absolute values (length) and the phases (color) of the Fourier amplitudes (Section 2.1.2.) L = 20 0.8 0.6 0.4 0.2 -4 -2 Color Plate The Fourier spectrum of the Gaussian in Fig 8, but with respect to a much larger interval [−L, L] This illustrates the transition from the Fourier spectrum to ˆ (Section 2.3.1.) the Fourier transformed function ψ 1 x -4 -2 k -4 -2 Color Plate 10 The Fourier transform of a Gaussian function exp(−x2 /2) is again a Gaussian function The picture shows the function exp(2ix − x2 /2) (left) and its Fourier transform exp(−(k − 2)2 /2) (right) The translation by in momentum space corresponds to a phase shift by 2ix in position space (Section 2.6.1.) t = 0.00 0.8 t = 1.00 0.8 0.4 0.4 x -4 -4 t = 2.00 0.8 x t = 3.00 0.8 0.4 0.4 x -4 x -4 8 Color Plate 11 Time evolution of a Gaussian wave packet with average momentum It can be seen that the maximum of the wave packet moves according to classical physics with velocity During the time evolution the wave packet spreads and contributions of higher momenta accumulate in front of the maximum (Section 3.3.2.) 0.8 t = 0.00 t = 0.12 0.6 0.4 0.2 x -4 -2 -4 -2 Color Plate 12 There is always a flow in the direction of increasing phase Hence if a yellow region of a wave packet is surrounded by red, then the wave function will increase in the yellow region (Section 3.5.) 0.8 0.8 0.4 0.4 x -6 -3 k -6 -3 Color Plate 13 Here the function sin(4x) exp(−x2 /2) is shown together with its Fourier transform, which has two well-separated peaks in momentum space (Section 3.6.) t = 0.05 t = 0.10 x t = 0.20 x t = 0.40 x t = 0.60 x t = 1.00 x x Color Plate 14 Some snapshots from the time evolution of the function in Color Plate 13 For sufficiently large times the localization in position space can be understood from the distribution of momenta in the Fourier transform of the initial wave packet (Section 3.6 See also CD 3.14.) 0.5 t = 0.00 0.5 t = 0.30 0.5 t = 0.60 0.5 t = 0.90 0.5 t = 1.20 0.5 t = 1.50 0.5 t = 1.80 0.5 t = 2.10 x x x x x x x -6 12 Color Plate 15 Particle in a box by the method of mirrors The initial function is a centered Gaussian function The motion between the walls can be understood as a superposition of mirror wave packets Initially, only the mirror waves that move toward the box are shown The part of the wave function in the physical region (inside the box) is drawn with higher saturation (Section 5.3.2.) x 2 t = 0.0006 1 0 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 t = 0.0516 0 t = 0.0777 0 t = 0.0746 t = 0.0361 0 t = 0.0118 t = 0.0199 t = 0.0050 0.2 0.4 0.6 0.8 t = 0.0796 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 Color Plate 16 Particle in a box This graphic shows some frames from the time evolution of a state in a Dirichlet box The initial state is the unit function ψ0 (x) = which is not in the domain of the generator of the time evolution The motion is periodic with period T = 4/π The function ψ(x, t) is continuous with respect to t in the L2 -topology, but it is not differentiable At times t for which t/T is a rational number, ψ(x, t) is a step function (Section 5.5.) Color Plate 17 Aharonov-Bohm effect Scattering from the left at an obstacle with a magnetic field inside The node line behind the obstacle is due to the influence of the magnetic vector potential (Section 8.3.) Color Plate 18 Potential barrier with two holes (doubleslit experiment) The initial state has an uncertainty of position which is larger than the distance between the two holes So one cannot predict through which of the holes the particle will actually go (Section 5.7.2.) t = 0.80 4 2 0 -2 -2 -4 -4 -6 -6 -4 -2 t = 1.60 6 -6 -6 -4 -2 Color Plate 19 Time evolution of the initial function shown in Color Plate 18 A part of the wave function penetrates through the holes in the screen Behind the screen emerges an interference pattern which shows that in certain directions there is only a small probability of observing the particle In this example, the probability of being scattered to angles about ±15◦ off the forward direction has a minimum (Section 5.7.4.) t = 0.80 4 2 0 -2 -2 -4 -4 -6 -6 -4 -2 t = 1.60 6 -6 -6 -4 -2 Color Plate 20 The double slit experiment with one hole closed The wave emerging from the hole behind the screen is an approximately spherical wave without visible interference pattern, as one might expect from Huygens’ principle (Section 5.7.4.) t = 0.00 t = 1.57 Oscillating function Oscillating function 0.8 0.8 0.4 0.4 0 -4 -2 and its Fourier transform -4 0.8 0.8 0.4 0.4 0 -4 -2 x -2 and its Fourier transform k -4 4 -2 Color Plate 21 Two snapshots of an oscillating state in a harmonic-oscillator potential The wave function and its Fourier transform are shown at times t = and t = π/2 The initial state is a superposition of φ0 and φ1 (Section 7.3.) t = 0.00 t = 0.79 t = 1.57 t = 2.36 t = 3.14 t = 3.93 t = 4.71 t = 5.50 t = 6.28 0.8 0.4 0.8 0.4 0.8 0.4 -6 -4 -2 -6 -4 -2 -6 -4 -2 Color Plate 22 The time evolution of a coherent state of the harmonic oscillator (Section 7.4.3.) 0.8 0.6 0.4 0.2 15 10 20 V0 0.8 0.6 0.4 0.2 V0 Color Plate 23 Reflection and transmission coefficients for the scattering at a potential barrier The barrier has a width R = and the energy is E = 18 (Section 9.6.3.) 0.6 0.4 V0 =16.8 E 0.2 x -6 -4 -2 V0 =15 E 0.2 x Color Plate 24 Scattering of a plane waves with energy E = 18 at a potential barrier The image above shows the solution at V0 = 16.8 For this height the reflection coefficient is zero, as one can see from Color Plate 23 In the image below, at V0 = 15, the part of the solution on the left-hand side shows the interference between an incoming and a reflected plane wave (Section 9.6.3.) 1 1 g(E ) 16 18 20 E 18 20 E 18 20 16 E 18 20 16 18 20 18 20 16 (b) Color Plate 25 Energy representation g(E) of an incoming wave packet, and the scattering coefficients at a barrier (a) V0 = 16.8 (b) V0 = 15 (Section 9.6.3.) t = 2.0 0.5 0.4 0.3 V0 =16.8 0.2 E 0.1 -20 0.3 -10 t =2.0 10 20 V0 =15 0.2 E 0.1 -20 -10 E T(E ) (a) 0.6 E R(E ) T(E ) 16 R(E ) 16 g(E ) 10 20 Color Plate 26 Scattering with energies strictly higher than the potential barrier for the two situations of Color Plate 25 (Section 9.6.3.) E t = 0.30 t = 0.90 6 4 2 0 -2 -2 -4 -4 -6 -6 -6 -4 -2 -6 -4 -2 t = 1.50 6 t = 2.10 6 4 2 0 -2 -2 -4 -4 -6 -6 -6 -4 -2 -6 -4 -2 Color Plate 27 Tunneling through a thin barrier in two dimensions Here the energies in the wave packet are strictly below the height of the potential barrier (Section 9.7.) ... distance between any two colors C (1 ) = (R(1) , G(1) , B (1 ) ) and C (2 ) = (R(2) , G(2) , B (2 ) ) in the color cube is given by the maximum metric d(C (1 ) , C (2 ) ) = max{|R(1) − R(2) |, |G(1) − G(2).. .Visual Quantum Mechanics This page intentionally left blank Bernd Thaller Visual Quantum Mechanics Selected Topics with Computer- Generated Animations of Quantum- Mechanical Phenomena. .. |, |B (1 ) − B (2 ) |} (1 .6) The distance of a color C = (R, G, B) from the black origin O = (0 , 0, 0) is called the brightness b of C, b( C) = d(C, O) = max{R, G, B} (1 .7) The saturation s(C) is