W Greiner

QUANTUM MECHANICS SPECIAL CHAPTERS

Springer-V erlag Berlin Heidelberg GmbH

Greiner Quantum Mechanics An Introduction 3rd Edition

Greiner Quantum Mechanics Special Chapters

Greiner· Milller Quantum Mechanics Symmetries 2nd Edition

Greiner Relativistic Quantum Mechanics Wave Equations 2nd Edition

Greiner· Reinhardt Field Quantization

Greiner· Reinhardt Quantum Electrodynamics 2nd Edition

Greiner· Schafer Quantum Chromodynamics

Greiner· Maruhn Nuclear Models

Greiner· Milller Gauge Theory of Weak Interactions 2nd Edition

Greiner Mechanics I (in preparation)

Greiner Mechanics II (in preparation)

Greiner Electrodynamics (in preparation)

Greiner· Neise . StOcker Thermodynamics and Statistical Mechanics

Walter Greiner

QUANTUM MECHANICS

SPECIAL CHAPTERS

With a Foreword by D. A. Bromley

With 120 Figures, 75 Worked Examples and Problems

Springer

Professor Dr. Walter Greiner Institut fiir Theoretische Physik der Johann Wolfgang Goethe-Universităt Frankfurt Postfach Il 19 32 D-60054 Frankfurt am Main Germany

Street address:

Robert-Mayer-Strasse 8-10 D-60325 Frankfurt am Main Germany

email: greiner@th.physik.uni-frankfurt.de

Title of the original German edition: Theoretische Physik, Ein Lehr- und Obungsbuch,

Band 4a: Quantentheorie, Spezielle Kapitel, 3. Aufl., © Verlag Ham Deutsch, Thun 1989

1st Edition 1998, 2nd Printing 2001

ISBN 978-3-540-60073-2

Library of Congress Cataloging-in-Publication Data.

Greiner, Walter, 1935 - [Quantenmechanik, English] Quantum mechanics. Special ehapters / Walter Greiner; with a foreword by D. A. Bromley, p. cm. Includes bibliographical referenees and index ISBN 978-3-540-60073-2 ISBN 978-3-642-58847-1 (eBook) DOI 10.1007/978-3-642-58847-1

1. Quantum theory, 2. Electrodynamies, 3. Quantum field theory, 4. Mathematical physics. 1. Greiner, Walter, 1935 -Theoretische Physik, English, Band 4a. Il. Title. QCI74.12.G74513 1998 530.12-dc21 97-24126

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© Springer-Verlag Berlin Heidelberg 1998 Originally published by Springer-Verlag Berlin Heidelberg New York in 1998

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Foreword to Ear Her Series Editions

More than a generation of German-speaking students around the world have worked their way to an understanding and appreciation of the power and beauty of modern theoretical physics - with mathematics, the most funda­mental of sciences - using Walter Greiner's textbooks as their guide.

The idea of developing a coherent, complete presentation of an entire field of science in a series of closely related textbooks is not a new one. Many older physicists remember with real pleasure their sense of adventure and discovery as they worked their ways through the classic series by Sommerfeld, by Planck and by Landau and Lifshitz. From the students' viewpoint, there are a great many obvious advantages to be gained through use of consistent notation, logical ordering of topics and coherence of presentation; beyond this, the complete coverage of the science provides a unique opportunity for the author to convey his personal enthusiasm and love for his subject.

The present five-volume set, Theoretical Physics, is in fact only that part of the complete set of textbooks developed by Greiner and his students that presents the quantum theory. I have long urged him to make the remaining vol­umes on classical mechanics and dynamics, on electromagnetism, on nuclear and particle physics, and on special topics available to an English-speaking audience as well, and we can hope for these companion volumes covering all of theoretical physics some time in the future.

What makes Greiner's volumes of particular value to the student and professor alike is their completeness. Greiner avoids the all too common "it follows that ... " which conceals several pages of mathematical manipulation and confounds the student. He does not hesitate to include experimental data to illuminate or illustrate a theoretical point and these data, like the theoret­ical content, have been kept up to date and topical through frequent revision and expansion of the lecture notes upon which these volumes are based.

Moreover, Greiner greatly increases the value of his presentation by includ­ing something like one hundred completely worked examples in each volume. Nothing is of greater importance to the student than seeing, in detail, how the theoretical concepts and tools under study are applied to actual problems of interest to a working physicist. And, finally, Greiner adds brief biographical sketches to each chapter covering the people responsible for the development of the theoretical ideas and/or the experimental data presented. It was Auguste Comte (1798-1857) in his Positive Philosophy who noted, "To understand a science it is necessary to know its history". This is all too often forgotten in

VI Foreword to Earlier Series Editions

modern physics teaching and the bridges that Greiner builds to the pioneering figures of our science upon whose work we build are welcome ones.

Greiner's lectures, which underlie these volumes, are internationally noted for their clarity, their completeness and for the effort that he has devoted to making physics an integral whole; his enthusiasm for his science is contagious and shines through almost every page.

These volumes represent only a part of a unique and Herculean effort to make all of theoretical physics accessible to the interested student. Beyond that, they are of enormous value to the professional physicist and to all others working with quantum phenomena. Again and again the reader will find that, after dipping into a particular volume to review a specific topic, he will end up browsing, caught up by often fascinating new insights and developments with which he had not previously been familiar.

Having used a number of Greiner's volumes in their original German in my teaching and research at Yale, I welcome these new and revised English translations and would recommend them enthusiastically to anyone searching for a coherent overview of physics.

Yale University New Haven, CT, USA 1989

D. Allan Bromley Henry Ford II Professor of Physics

Preface

Theoretical physics has become a many-faceted science. For the young stu­dent it is difficult enough to cope with the overwhelming amount of new scientific material that has to be learned, let alone obtain an overview of the entire field, which ranges from mechanics through electrodynamics, quantum mechanics, field theory, nuclear and heavy-ion science, statistical mechanics, thermodynamics, and solid-state theory to elementary-particle physics. And this knowledge should be acquired in just 8-10 semesters, during which, in addition, a Diploma (Masters) thesis has to be worked on and examinations prepared for. All this can be achieved only if the academic teachers help to introduce the student to the new disciplines as early on as possible, in order to create interest and excitement that in turn set free essential new energy.

At the Johann Wolfgang Goethe University in Frankfurt am Main we therefore confront the student with theoretical physics immediately, in the first semester. Theoretical Mechanics I and II, Electrodynamics, and Quantum Mechanics I - An Introduction are the basic courses during the first two years. These lectures are supplemented with many mathematical explanations and much support material. After the fourth semester of studies, graduate work begins, and Quantum Mechanics II - Symmetries, Statistical Mechanics and Thermodynamics, Relativistic Quantum Mechanics, Quantum Electrodynam­ics, the Gauge Theory of Weak Interactions, and Quantum Chromo dynamics are obligatory. Apart from these, a number of supplementary courses on spe­cial topics are offered, such as Hydrodynamics, Classical Field Theory, Special and General Relativity, Many-Body Theories, Nuclear Models, Models of El­ementary Particles, and Solid-State Theory.

This volume of lectures provides an important supplement on the subject of Quantum Mechanics. These Special Chapters are in the form of overviews on various subjects in modern theoretical physics. The book is devised for students in their fifth semester who are still trying to decide on an area of research to follow, whether they would like to focus on experiments or on theory later on.

The observation by Planck and Einstein that a classical field theory -electrodynamics - had to be augmented by corpuscular and nondeterministic aspects stood at the cradle of quantum theory. At around 1930 it was recog­nized that not only the radiation field with photons but also matter fields, e.g. electrons, can be described by the same procedure of second quantization.

VIII Preface

Within this formalism, matter is represented by operator-valued fields that are subject to certain (anti-)commutation relations. In this way one arrives at a theory describing systems of several particles (field quanta) which in partic­ular provides a very natural way to formulate the creation and annihilation of particles. Quantum field theory has become the language of modern the­oretical physics. It is used in particle and high-energy physics, but also the description of many-body systems encountered in solid-state, plasma, nuclear, and atomic physics make use of the methods of quantum field theory.

We use second quantization (creation and annihilation operators for par­ticles and modes) extensively. The lectures begin with the quantization of the electromagnetic fields. As well as the state vectors with a well-defined (sharp) number of photons, the coherent (Glauber) states are discussed, followed by absorption and emission processes, the lifetime of exited states, the width of spectral lines, the self-energy problem, photon scattering, and Cherenkov radiation. In between it seemed fit to elucidate on the Aharanov-Bohm and Casimir effects. Many applications are hidden in Exercises and Examples (e.g. two-photon decay, the Compton effect, photon spectra of black bodies).

Fermi and Bose statistics and their relationship with the way of quantiza­tion (commutators, anticommutators) are discussed in the third chapter. Here also, tripple commutators leading to para-Bose and para-Fermi statistics are reflected upon. After describing quantum fields with interaction (Chap. 4) we address renormalization problems, not in full (as done in the lectures on quan­tum electrodynamics and on field quantization), but in a rather elementary way such that the student gets a feeling for the problems, their difficulties, and their solution.

In Chaps.6 to 9 the methods of quantum field theory are applied to topics in solid-state and plasma physics: quantum gases, superfluidity, pair correlations (Hanbury-Brown-Twiss effect and Cooper pairs), plasmons and phonons, and the quasiparticle concept give an impression of the flavor of these fields. The following chapters are devoted to the structure of atoms and molecules, containing many fascinating subjects (Hartree, Hartree-Fock, Thomas-Fermi methods, the periodic system of elements, the Born-Oppen­heimer approach, various types of elementary molecules, oriented orbitals, hybridization, etc.).

Finally we present an elementary exhibition of Feynman path integrals. The method of quantization using path integrals, which essentially is equiv­alent to the canonical formalism, has gained increasing popularity over the years. Apart from their elegance and formal appeal, path-integral quantiza­tion and the related functional techniques are particulary well suited to the implementation of conditions of constraint, which is necessary for the treat­ment of gauge fields. Nowadays any student of physics should at least know where and how the canonical and the path-integral formalisms are connected.

Like all other lectures, these special chapters are presented together with the necessary mathematical tools. Many detailed examples and worked-out problems are included in order to further illuminate the material.

It is clear from what we have said so far that these lectures are meant to give an elementary (but not naive) overview of special subjects a student may

Preface

hear about in colloquia and seminars. The lectures may help to furnish better orientation in the vast field of interesting modern physics.

We have profitted a lot from excellent text books, such as

E.G. Harris: A Pedestrian Approach to Quantum Field Theory (Wiley, New York 1972), G. Baym: Lectures on Quantum Mechanics (W.A. Benjamin, Reading, MA 1974), L.D. Landau, E.M. Lifshitz: Quantum Mechanics (Pergamon, Oxford 1977),

which have guided us to some extend in devising certain chapters, examples, and exercises. We recommend them for additional reading. The biographi­cal notes on outstanding physicists and mathematicians were taken from the Brockhaus Lexikon.

This book is not intended to provide an exhaustive introduction to all aspects of quantum mechanics. Our main goal has been to present an elemen­tary introduction to the methods of field quantization and their applications in many-body physics as well as to special aspects of atomic and nuclear physics. We hope to attain this goal by presenting the subjects in considerable detail, explaining the mathematical tools in a rather informal way, and by including a large number of examples and worked exercises.

We would like to express our gratitude to Drs. J. Reinhardt, G. Plunien, and S. Schramm for their help in preparing some exercises and examples and in proofreading the German edition of the text. For the preparation of the English edition we enjoyed the help of Priv. Doz. Dr. Martin Greiner. Once again we are pleased to acknowledge the agreeable collaboration with Dr. H.J. K6lsch and his team at Springer-Verlag, Heidelberg. The English manuscript was copy edited by Dr. Victoria Wicks.

Frankfurt am Main, August 1997

Walter Greiner

IX

Contents

1. Quantum Theory of Free Electromagnetic Fields . . . . . . . 1 1.1 Maxwell's Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Electromagnetic Plane Waves. . . . . . . . . . . . . . . . . . . . . . 3 1.3 Quantization of Free Electromagnetic Fields . . . . . . . . . . . 5 1.4 Eigenstates of Electromagnetic Fields. . . . . . . . . . . . . . .. 12 1.5 Coherent States (Glauber States) of Electromagnetic Fields 16 1.6 Biographical Notes .... . . . . . . . . . . . . . . . . . . . . . . . .. 29

2. Interaction of Electromagnetic Fields with Matter. . . . .. 31 2.1 Emission of Radiation from an Excited Atom .......... 33 2.2 Lifetime of an Excited State. . . . . . . . . . . . . . . . . . . . . .. 35 2.3 Absorption of Photons. . . . . . . . . . . . . . . . . . . . . . . . . .. 48 2.4 Photon Scattering from Free Electrons . . . . . . . . . . . . . .. 55 2.5 Calculation of the Total Photon Scattering Cross Section.. 57 2.6 Cherenkov Radiation of a Schrodinger Electron. . . . . . . .. 63 2.7 Natural Linewidth and Self-energy. . . . . . . . . . . . . . . . .. 74

3. Noninteracting Fields ............................. 81 3.1 Spin-Statistics Theorem. . . . . . . . . . . . . . . . . . . . . . . . .. 98 3.2 Relationship Between Second Quantization

and Elementary Quantum Mechanics .. . . . . . . . . . . . . .. 99

4. Quantum Fields with Interaction . . . . . . . . . . . . . . . . . . .. 109

5. Infinities in Quantum Electrodynamics: Renormalization Problems . . . . . . . . . . . . . . . . . . . . . . . .. 133 5.1 Attraction of Parallel, Conducting Plates Due

to Field Quantum Fluctuations (Casimir Effect) ........ 133 5.2 Renormalization of the Electron Mass. . . . . . . . . . . . . . .. 143 5.3 The Splitting of the Hydrogen States 2S1/ 2-2p3/2:

The Lamb Shift. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 149 5.4 Is There an Inconsistency in Bethe's Approach? . . . . . . . .. 156

6. Nonrelativistic Quantum Field Theory of Interacting Particles and Its Applications. . . . . . . . . .. 161 6.1 Quantum Gases ............................... 165 6.2 Nearly Ideal, Degenerate Bose-Einstein Gases. . . . . . . . .. 174

XII Contents

7. Superfluidity..................................... 193 7.1 Basics of a Microscopic Theory of Superfluidity . . . . . . . .. 194 7.2 Landau's Theory of Superfluidity . . . . . . . . . . . . . . . . . .. 205

8. Pair Correlations Among Fermions and Bosons ........ 213 8.1 Pair-Correlation Function for Fermions . . . . . . . . . . . . . .. 213 8.2 Pair-Correlation Function for Bosons ................ 218 8.3 The Hanbury-Brown and Twiss Effect. . . . . . . . . . . . . . .. 223 8.4 Cooper Pairs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 226

9. Quasiparticles in Plasmas and Metals: Selected Topics. .. 241 9.1 Plasmons and Phonons .... . . . . . . . . . . . . . . . . . . . . .. 246

10. Basics of Quantum Statistics ....................... 255 10.1 Concept of Quantum Statistics and the Notion of Entropy. 255 10.2 Density Operator of a Many-Particle State . . . . . . . . . . .. 256 10.3 Dynamics of a Quantum-Statistical Ensemble . . . . . . . . .. 272 10.4 Ordered and Disordered Systems:

The Density Operator and Entropy. . . . . . . . . . . . . . . . .. 276 10.5 Stationary Ensembles ........................... 278

11. Structure of Atoms ... . . . . . . . . . . . . . . . . . . . . . . . . . . .. 285 11.1 Atoms with Two Electrons. . . . . . . . . . . . . . . . . . . . . . .. 285 11.2 The Hartree Method . . . . . . . . . . . . . . . . . . . . . . . . . . .. 292 11.3 Thomas-Fermi Method . . . . . . . . . . . . . . . . . . . . . . . . .. 293 11.4 The Hartree--Fock Method . . . . . . . . . . . . . . . . . . . . . . .. 297 11.5 On the Periodic System of the Elements . . . . . . . . . . . . .. 305 11.6 Splitting of Orbital Multiplets . . . . . . . . . . . . . . . . . . . .. 306 11.7 Spin-Orbit Interaction. . . . . . . . . . . . . . . . . . . . . . . . . .. 312 11.8 Treatment of the Spin-Orbit Splitting

in the Hartree-Fock Approach . . . . . . . . . . . . . . . . . . . .. 324 11.9 The Zeeman Effect ... . . . . . . . . . . . . . . . . . . . . . . . . .. 327 11.10 Biographical Notes ....... . . . . . . . . . . . . . . . . . . . . .. 332

12. Elementary Structure of Molecules .................. 335 12.1 Born-Oppenheimer Approximation. . . . . . . . . . . . . . . . .. 337 12.2 The Ht Ion as an Example . . . . . . . . . . . . . . . . . . . . . .. 339 12.3 The Hydrogen Molecule. . . . . . . . . . . . . . . . . . . . . . . . .. 346 12.4 Electron Pairing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 349 12.5 Spatially Oriented Orbits. . . . . . . . . . . . . . . . . . . . . . . .. 351 12.6 Hybridization................................. 353 12.7 Hydrocarbons................................. 356 12.8 Biographical Notes .,. . . . . . . . . . . . . . . . . . . . . . . . . .. 358

Contents

13. Feynman's Path Integral Formulation of Schrodinger's Wave Mechanics. . . . . . . . . . . . . . . . . . .. 361 13.1 Action Functional in Classical Mechanics

and Schrodinger's Wave Mechanics . . . . . . . . . . . . . . . . .. 362 13.2 Transition Amplitude as a Path Integral. . . . . . . . . . . . .. 365 13.3 Path Integral Representation

of the Schrodinger Propagator . . . . . . . . . . . . . . . . . . . .. 370 13.4 Alternative Derivation of the Schrodinger Equation. . . . .. 374 13.5 Biographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 376

Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 377

XIII

Contents of Examples and Exercises

1.1 The Coulomb Gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Computation of the Magnetic Contributions to the Energy

of an Electromagnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3 Momentum Operator of Electromagnetic Fields. . . . . . . . . . . . . 13 1.4 Matrix Elements with Coherent States . . . . . . . . . . . . . . . . . .. 18 1.5 The Mean Quadratic Deviation of the Electric Field

Within the Coherent State. . . . . . . . . . . . . . . . . . . . . . . . . . .. 20 1.6 The Aharonov-Bohm Effect. . . . . . . . . . . . . . . . . . . . . . . . . .. 21 2.1 Selection Rules for Electric Dipole Transitions. . . . . . . . . . . . .. 38 2.2 Lifetime of the 2p State with m = 0 in the Hydrogen Atom

with Respect to Decay Into the Is State . . . . . . . . . . . . . . . . .. 40 2.3 Impossibility of the Decay of the 2s State of the Hydrogen Atom

via the p . A Interaction .... . . . . . . . . . . . . . . . . . . . . . . . .. 41 2.4 The Hamiltonian for Interaction Between the Electron Spin

and the Electromagnetic Field . . . . . . . . . . . . . . . . . . . . . . . .. 42 2.5 Lifetime of the Ground State of the Hydrogen Atom

with Hyperfine Splitting ............................. 43 2.6 One-Photon Decay of the 2s State in the Hydrogen Atom. . . . .. 46 2.7 Differential Cross Section dO'/drl for Photoelectric Emission

of an Electron in the Hydrogen Atom (Dipole Approximation) .. 50 2.8 Spectrum of Black-Body Radiation. . . . . . . . . . . . . . . . . . . . .. 53 2.9 The Compton Effect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 59 2.10 Two-Photon Decay of the 2s State of the Hydrogen Atom ..... 60 2.11 The Field Energy in Media with Dispersion. . . . . . . . . . . . . . .. 64 2.12 The Cherenkov Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 73 2:13 Plemlj's Formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 78 3.1 Do the Commutators and Anticommutators

Fulfill the Poisson Bracket Algebra? . . . . . . . . . . . . . . . . . . . .. 85 3.2 Threefold Commutators from an Expansion of Paraoperators . .. 87 3.3 More on Paraoperators: Introduction of the Operator Gjk ..... 89 3.4 Occupation Numbers of Para-Fermi States ................ 91 3.5 On the Boson Commutation Relations ..... . . . . . . . . . . . . .. 95 3.6 Consistency of the Phase Choice for Fermi States

with the Fermion Commutation Relations . . . . . . . . . . . . . . . .. 97 3.7 Constancy of the Total Particle-Number Operator. . . . . . . . . .. 102 4.1 Nonrelativistic Bremsstrahlung. . . . . . . . . . . . . . . . . . . . . . . .. 112 4.2 Rutherford Scattering Cross Section. . . . . . . . . . . . . . . . . . . .. 121

XVI Contents of Examples and Exercises

4.3 Lifetime of the Hydrogen 2s State with Respect to Two-Photon Decay (in Second Quantization) ............ 123

4.4 Second-Order Corrections to Rutherford's Scattering Cross Section. . . . . . . . . . . . . . . . .. 128

5.1 Attraction of Parallel, Conducting Plates Due to the Casimir Effect .. . . . . . . . . . . . . . . . . . . . . . . . . .. 137

5.2 Measurement of the Casimir Effect. . . . . . . . . . . . . . . . . . . . .. 140 5.3 Casimir's Approach Towards a Model for the Electron. . . . . . .. 142 5.4 Supplement: Historical Remark on the Electron Mass. . . . . . . .. 144 5.5 Lamb and Retherford's Experiment . . . . . . . . . . . . . . . . . . . .. 150 5.6 The Lamb Shift ................................... 157 6.1 The Field-Theoretical Many-Particle Problem. . . . . . . . . . . . .. 162 6.2 Equilibrium Solution

of the Quantum-Mechanical Boltzmann Equation ........... 167 6.3 Equilibrium Solution of the Classical Boltzmann Equation . . . .. 172 6.4 From the Entropy Formula for the Bose (Fermi) Gas

to the Classical Entropy Formula . . . . . . . . . . . . . . . . . . . . . .. 173 6.5 Proof of the H Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 174 6.6 Entropy of a Quantum Gas . . . . . . . . . . . . . . . . . . . . . . . . . .. 181 6.7 Distribution of N Particles over G States

(Number of Combinations) ........................... 186 6.8 Stirling's Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 186 6.9 Entropy and Information. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 187 6.10 Maxwell's Demon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 191 7.1 Choice of Coefficients for the Bogoliubov Transformation. . . . .. 199 7.2 An Analogy to Superftuidity in Hydrodynamics. . . . . . . . . . . .. 209 8.1 Pair-Correlation Function for a Beam of Bosons ............ 220 8.2 Boson Pair-Correlation Function as a Function

of the Quantization Volume . . . . . . . . . . . . . . . . . . . . . . . . . .. 222 8.3 The Debye Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 233 8.4 Correlation Length of a Cooper Pair. . . . . . . . . . . . . . . . . . . .. 236 8.5 Determination of the Coupling Strength of a Bound Cooper Pair 238 9.1 Electrostatic Potential of a Charge in a Plasma. . . . . . . . . . . .. 249 9.2 Classical Dielectric Function .......................... 250 9.3 Details of Calculating the Dielectric Function E(q,W) . . . . . . . .. 252 10.1 Density Operators in Second Quantization ................ 264 10.2 Transformation Equations for Field Operators. . . . . . . . . . . . .. 268 10.3 Commutation Relations for Fermion Field Operators. . . . . . . .. 270 10.4 Density Operator of a Mixture. . . . . . . . . . . . . . . . . . . . . . . .. 274 10.5 Construction of the Density Operator

for a System of U npolarized Electrons. . . . . . . . . . . . . . . . . . .. 275 10.6 Systems of Noninteracting Fermions and Bosons . . . . . . . . . . .. 281 11.1 Calculation of Some Frequently used Integrals. . . . . . . . . . . . .. 288 11.2 Proof of (11.49) ................................... 298 11.3 The Hartree-Fock Equation as a Nonlocal Schrodinger Equation. 301 11.4 An Approximation for the Hartree-Fock Exchange Term .. . . .. 304 11.5 Application of Hund's Rules. . . . . . . . . . . . . . . . . . . . . . . . . .. 311 11.6 The Wigner-Eckart Theorem. . . . . . . . . . . . . . . . . . . . . . . . .. 314

Contents of Examples and Exercises

11.7 Derivation of the Spin-Orbit Interaction. . . . . . . . . . . . . . . . .. 317 11.8 Transformation of the Spin-Orbit Interaction ..... . . . . . . . .. 323 11.9 The Stark Effect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 329 12.1 Calculation of an Overlap Integral and Some Matrix Elements

for the Ht Ion .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 343 13.1 Momentum and Energy at the End Point

of a Classical Trajectory ................. . . . . . . . . . . .. 364 13.2 The Transition Amplitude for a Free Particle .............. 369 13.3 Trotter's Product Rule .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 372

XVII