Springer-V erlag Berlin Heidelberg GmbH

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1 W Greiner QUANTUM MECHANICS SPECIAL CHAPTERS Springer-V erlag Berlin Heidelberg GmbH

2 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 Mechanics I (in preparation) Greiner Mechanics II (in preparation) Greiner Electrodynamics (in preparation) Greiner Neise. StOcker Thermodynamics and Statistical Mechanics 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

3 Walter Greiner QUANTUM MECHANICS SPECIAL CHAPTERS With a Foreword by D. A. Bromley With 120 Figures, 75 Worked Examples and Problems Springer

4 Professor Dr. Walter Greiner Institut fiir Theoretische Physik der Johann Wolfgang Goethe-Universităt Frankfurt Postfach Il D Frankfurt am Main Germany Street address: Robert-Mayer-Strasse 8-10 D Frankfurt am Main Germany Title of the original German edition: Theoretische Physik, Ein Lehr- und Obungsbuch, Band 4a: Quantentheorie, Spezielle Kapitel, 3. Aufl., Verlag Ham Deutsch, Thun st Edition 1998, 2nd Printing 2001 ISBN Library of Congress Cataloging-in-Publication Data. Greiner, Walter, [Quantenmechanik, English] Quantum mechanics. Special ehapters / Walter Greiner; with a foreword by D. A. Bromley, p. cm. Includes bibliographical referenees and index ISBN ISBN (ebook) DOI / Quantum theory, 2. Electrodynamies, 3. Quantum field theory, 4. Mathematical physics. 1. Greiner, Walter, Theoretische Physik, English, Band 4a. Il. Title. QCI74.12.G dc This work is subject ta copyright. AII rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction an microfilm Of in any other way, and storage in data banks. Duplication of this publicatian or parts thereof is pennitted only underthe provisions of the German Copyright Law of September 9, 1965, in its current vers ion, and permission for use must always be obtained from Springer-Verlag. Viol.tions are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg 1998 Originally published by Springer-Verlag Berlin Heidelberg New York in 1998 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Data conversion by A. Leinz, Karlsruhe Cover design: Design Concept, Emil Smejkal, Heidelberg Copy Editor: V. Wicks Productian Editor: P. Treiber SPIN / O - Printed on acid-free paper

5 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 ( ) in his Positive Philosophy who noted, "To understand a science it is necessary to know its history". This is all too often forgotten in

6 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

7 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.

8 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

9 Preface IX 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

10 Contents 1. Quantum Theory of Free Electromagnetic Fields Maxwell's Equations Electromagnetic Plane Waves Quantization of Free Electromagnetic Fields Eigenstates of Electromagnetic Fields Coherent States (Glauber States) of Electromagnetic Fields Biographical Notes Interaction of Electromagnetic Fields with Matter Emission of Radiation from an Excited Atom Lifetime of an Excited State Absorption of Photons Photon Scattering from Free Electrons Calculation of the Total Photon Scattering Cross Section Cherenkov Radiation of a Schrodinger Electron Natural Linewidth and Self-energy Noninteracting Fields Spin-Statistics Theorem Relationship Between Second Quantization and Elementary Quantum Mechanics Quantum Fields with Interaction Infinities in Quantum Electrodynamics: Renormalization Problems Attraction of Parallel, Conducting Plates Due to Field Quantum Fluctuations (Casimir Effect) Renormalization of the Electron Mass The Splitting of the Hydrogen States 2S 1/ 2-2p3/2: The Lamb Shift Is There an Inconsistency in Bethe's Approach? Nonrelativistic Quantum Field Theory of Interacting Particles and Its Applications Quantum Gases Nearly Ideal, Degenerate Bose-Einstein Gases

11 XII Contents 7. Superfluidity Basics of a Microscopic Theory of Superfluidity Landau's Theory of Superfluidity Pair Correlations Among Fermions and Bosons Pair-Correlation Function for Fermions Pair-Correlation Function for Bosons The Hanbury-Brown and Twiss Effect Cooper Pairs Quasiparticles in Plasmas and Metals: Selected Topics Plasmons and Phonons Basics of Quantum Statistics Concept of Quantum Statistics and the Notion of Entropy Density Operator of a Many-Particle State Dynamics of a Quantum-Statistical Ensemble Ordered and Disordered Systems: The Density Operator and Entropy Stationary Ensembles Structure of Atoms Atoms with Two Electrons The Hartree Method Thomas-Fermi Method The Hartree--Fock Method On the Periodic System of the Elements Splitting of Orbital Multiplets Spin-Orbit Interaction Treatment of the Spin-Orbit Splitting in the Hartree-Fock Approach The Zeeman Effect Biographical Notes Elementary Structure of Molecules Born-Oppenheimer Approximation The Ht Ion as an Example The Hydrogen Molecule Electron Pairing Spatially Oriented Orbits Hybridization Hydrocarbons Biographical Notes.,

12 Contents XIII 13. Feynman's Path Integral Formulation of Schrodinger's Wave Mechanics Action Functional in Classical Mechanics and Schrodinger's Wave Mechanics Transition Amplitude as a Path Integral Path Integral Representation of the Schrodinger Propagator Alternative Derivation of the Schrodinger Equation Biographical Notes Subject Index

13 Contents of Examples and Exercises 1.1 The Coulomb Gauge Computation of the Magnetic Contributions to the Energy of an Electromagnetic Field Momentum Operator of Electromagnetic Fields Matrix Elements with Coherent States The Mean Quadratic Deviation of the Electric Field Within the Coherent State The Aharonov-Bohm Effect Selection Rules for Electric Dipole Transitions Lifetime of the 2p State with m = 0 in the Hydrogen Atom with Respect to Decay Into the Is State Impossibility of the Decay of the 2s State of the Hydrogen Atom via the p. A Interaction The Hamiltonian for Interaction Between the Electron Spin and the Electromagnetic Field Lifetime of the Ground State of the Hydrogen Atom with Hyperfine Splitting One-Photon Decay of the 2s State in the Hydrogen Atom Differential Cross Section do'/drl for Photoelectric Emission of an Electron in the Hydrogen Atom (Dipole Approximation) Spectrum of Black-Body Radiation The Compton Effect Two-Photon Decay of the 2s State of the Hydrogen Atom The Field Energy in Media with Dispersion The Cherenkov Angle :13 Plemlj's Formula Do the Commutators and Anticommutators Fulfill the Poisson Bracket Algebra? Threefold Commutators from an Expansion of Paraoperators More on Paraoperators: Introduction of the Operator Gjk Occupation Numbers of Para-Fermi States On the Boson Commutation Relations Consistency of the Phase Choice for Fermi States with the Fermion Commutation Relations Constancy of the Total Particle-Number Operator Nonrelativistic Bremsstrahlung Rutherford Scattering Cross Section

14 XVI Contents of Examples and Exercises 4.3 Lifetime of the Hydrogen 2s State with Respect to Two-Photon Decay (in Second Quantization) Second-Order Corrections to Rutherford's Scattering Cross Section Attraction of Parallel, Conducting Plates Due to the Casimir Effect Measurement of the Casimir Effect Casimir's Approach Towards a Model for the Electron Supplement: Historical Remark on the Electron Mass Lamb and Retherford's Experiment The Lamb Shift The Field-Theoretical Many-Particle Problem Equilibrium Solution of the Quantum-Mechanical Boltzmann Equation Equilibrium Solution of the Classical Boltzmann Equation From the Entropy Formula for the Bose (Fermi) Gas to the Classical Entropy Formula Proof of the H Theorem Entropy of a Quantum Gas Distribution of N Particles over G States (Number of Combinations) Stirling's Formula Entropy and Information Maxwell's Demon Choice of Coefficients for the Bogoliubov Transformation An Analogy to Superftuidity in Hydrodynamics Pair-Correlation Function for a Beam of Bosons Boson Pair-Correlation Function as a Function of the Quantization Volume The Debye Frequency Correlation Length of a Cooper Pair Determination of the Coupling Strength of a Bound Cooper Pair Electrostatic Potential of a Charge in a Plasma Classical Dielectric Function Details of Calculating the Dielectric Function E(q,W) Density Operators in Second Quantization Transformation Equations for Field Operators Commutation Relations for Fermion Field Operators Density Operator of a Mixture Construction of the Density Operator for a System of U npolarized Electrons Systems of Noninteracting Fermions and Bosons Calculation of Some Frequently used Integrals Proof of (11.49) The Hartree-Fock Equation as a Nonlocal Schrodinger Equation An Approximation for the Hartree-Fock Exchange Term Application of Hund's Rules The Wigner-Eckart Theorem

15 Contents of Examples and Exercises XVII 11.7 Derivation of the Spin-Orbit Interaction Transformation of the Spin-Orbit Interaction The Stark Effect Calculation of an Overlap Integral and Some Matrix Elements for the Ht Ion Momentum and Energy at the End Point of a Classical Trajectory The Transition Amplitude for a Free Particle Trotter's Product Rule