Visual Quantum Mechanics

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Preface for Book One

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. This quantum mechanics course relies heavily on visualization as a tool for mediating knowledge. Included in the course is a CD-ROM containing about 320 digital movies in QuickTime 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 which has been developed with the help of Macromedia Director. This multimedia presentation can be used like an adventure game without special computer skills. I hope that this presentation format will help to 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 student's 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 student's interest and provide some motivation for the effort to understand the theory behind it. By "watching" the solutions of the Schrödinger 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 rather conventional -- it 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. Due to a lack of time and space the more traditional approach to motivating the theory using experimental results falls a little short. The text is on an introductory level and requires little previous knowledge, but it is not elementary. When I considered how to provide the 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 me 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 which are usually considered to belong to a basic course in quantum mechanics are postponed until Book Two. A preliminary table of contents for Book Two encompasses 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 1. 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 {\sl Mathematica}.

Chapter 2 introduces some mathematical concepts needed for quantum mechanics. Fourier analysis is an essential tool for solving the Schrödinger 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 Schrödinger equation are already available when it is introduced in Chapter 3 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 the corresponding sections (marked as "special topics") may be skipped at first reading.

Quantum Mechanics starts with Chapter 3. We describe the free motion of approximately localized wave packets and put some emphasis on the statistical interpretation and the measurement process. The Schrödinger equation for particles in external fields is given in Chapter 4. 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 5. 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 6 which focuses on linear operators. Among the mathematically more sophisticated topics that usually do 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 4).

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 9. 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 which can be reduced to two dimensions by exploiting their symmetry. For example, Chapter 8 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 do occur in nature. A good example are the surface state of electrons which can be depicted by a scanning tunneling microscope.

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 that 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., Landau/Lifshitz, Merzbacher, Messiah, Schiff) and the mathematically advanced presentations (e.g., Berezin/Shubin, Prugovecki, Thirring). For those who like a more intuitive approach I recommend that they first read a book which tries to avoid technicalities as long as possible (e.g., Feynman or Lévy-Leblond/Balibar).

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 by Feagin and by Horbatsch, 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 trains 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 the 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 own computer experiments.

In many cases it is not possible to obtain explicit solutions of the Schrödinger 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 and 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.

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 also read the entire manuscript carefully correcting many errors and misprints. My son Wolfgang deserves special thanks. Despite his own numerous projects he helped me a lot with his unparalleled computer skills. Finally, a project preparation grant from Springer-Verlag is gratefully acknowledged.

Bernd Thaller

Graz, June 1999




< Previous \| Index \| Next > Preface for Book One 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. This quantum mechanics course relies heavily on visualization as a tool for mediating knowledge. Included in the course is a CD-ROM containing about 320 digital movies in QuickTime 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 which has been developed with the help of Macromedia Director. This multimedia presentation can be used like an adventure game without special computer skills. I hope that this presentation format will help to 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 student's 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 student's interest and provide some motivation for the effort to understand the theory behind it. By "watching" the solutions of the Schrödinger 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 rather conventional -- it 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. Due to a lack of time and space the more traditional approach to motivating the theory using experimental results falls a little short. The text is on an introductory level and requires little previous knowledge, but it is not elementary. When I considered how to provide the 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 me 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 which are usually considered to belong to a basic course in quantum mechanics are postponed until Book Two. A preliminary table of contents for Book Two encompasses 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 1. 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 {\sl Mathematica}. Chapter 2 introduces some mathematical concepts needed for quantum mechanics. Fourier analysis is an essential tool for solving the Schrödinger 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 Schrödinger equation are already available when it is introduced in Chapter 3 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 the corresponding sections (marked as "special topics") may be skipped at first reading. Quantum Mechanics starts with Chapter 3. We describe the free motion of approximately localized wave packets and put some emphasis on the statistical interpretation and the measurement process. The Schrödinger equation for particles in external fields is given in Chapter 4. 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 5. 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 6 which focuses on linear operators. Among the mathematically more sophisticated topics that usually do 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 4). 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 9. 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 which can be reduced to two dimensions by exploiting their symmetry. For example, Chapter 8 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 do occur in nature. A good example are the surface state of electrons which can be depicted by a scanning tunneling microscope. 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 that 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., Landau/Lifshitz, Merzbacher, Messiah, Schiff) and the mathematically advanced presentations (e.g., Berezin/Shubin, Prugovecki, Thirring). For those who like a more intuitive approach I recommend that they first read a book which tries to avoid technicalities as long as possible (e.g., Feynman or Lévy-Leblond/Balibar). 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 by Feagin and by Horbatsch, 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 trains 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 the 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 own computer experiments. In many cases it is not possible to obtain explicit solutions of the Schrödinger 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 and 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. 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 also read the entire manuscript carefully correcting many errors and misprints. My son Wolfgang deserves special thanks. Despite his own numerous projects he helped me a lot with his unparalleled computer skills. Finally, a project preparation grant from Springer-Verlag is gratefully acknowledged. Bernd Thaller Graz, June 1999