Text and Exercise Books

Volume 1 Quantum Mechanics -An Introduction

Volume2 Quantum Mechanics -Symmetries

Volume3 Relativistic Quantum Mechanics -Wave Equations

Volume4 Quantum Electrodynamics

Volume5 Gauge Theory of Weak Interactions

Walter Greiner

Relativistic Quantum Mechanies Wave Equations

With a Foreword by D. A. Bromley

With 62 Figures

Springer-Verlag Berlin Heidelberg GmbH

Professor Dr. Walter Greiner Institut für Theoretische Physik, Universität Frankfurt, Robert-Mayer-Str. 8-10 D-6000 Frankfurt, Fed. Rep. of Germany

Title of the original German edition: Theoretische Physik, Band 6, Relativistische Quantenmechanik, Wellengleichungen 2 .. überarbeitete und erweiterte Auflage 1987 © Verlag Harri Deutsch, Thun 1981

ISBN 978-3-540-50986-8 ISBN 978-3-662-02634-2 (eBook)

DOI 10.1007/978-3-662-02634-2

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifi­cally the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permiUed under the pro­visions of the German Copyright Law of September 9, 1965, in its current version, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law.

© Springer-Verlag Berlin Heidelberg 1990

Originally published by Springer-Verlag Berlin Heidelberg New York in 1990

The use of 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.

2156/3150-543210 - Printing on acid-free paper

Foreword

More than a generation of Gennan-speaking students around the world have worked their way to an understanding and appreciation of the power and beauty of modem 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 aseries of elosely 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 elassic 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 bis 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. 1 have long urged hirn to make the remaining volumes on elassical mechanics and dynamics, on electromagnetism, on nuelear and partiele physics, and on special topics available to an English-speaking audience as weIl, 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 ineluding 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 modem physics teaching and the bridges that Greiner builds to the pioneering figures of our science upon whose work we build are welcome ones.

v

Greiner's lectures, which underlie these volumes, are intemationally 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 apart 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.

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D. Al/an Bromley Henry Ford 11 Professor of Physics Yale University New Haven, er USA

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 or Master's thesis has to be worked on or examinations prepared for. All this can be achieved only if the university 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. Naturally, all inessential material must simply be eliminated

At the Johann Wolfgang Goethe University in Frankfurt we therefore confront the stu­dent with theoretical physics immediately, in the first semester. Theoretical Mechanics I and 11, 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 11 - Symmetries, Statistical Mechanics and Thermody­namics, Relativistic Quantum Mechanics, Quantum Electrodynamics, the Gauge Theory of Weak Interactions, and Quantum Chromodynamics 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. Some of them, for ex­ample the two-semester courses Theoretical Nuclear Physics and Theoretical Solid-State Physics, are also obligatory.

The form of the lectures that comprise Relativistic Quantum Mechanics - Wave Equa­tions follows that of all the others: together with a broad presentation of the necessary mathematical tools, many examples and exercises are worked through. We try to offer science in as interesting a way as possible. With relativistic quantum mechanics we are dealing with a broad, yet beautiful, theme. Therefore we have had to restrict ourselves to relativistic wave equations. The selected material is perhaps unconventional, but corre­sponds, in our opinion, to the importance of this field in modern physics:

The Klein-Gordon equation (for spin-O particles) and the Dirac equation (for spin-~ particles) and their applications constitute the backbone of these lectures. Wave equations for partieles with higher spin (the Rarita-Schwinger, spin- ~, Kemmer and Proca, spin-I, and general Bargman~Wigner equations) are confined to the last chapters.

After introducing the Klein-Gordon equation we discuss its properties and difficulties (especially with respect to the single-particle interpretation); the Feshbach-Villars repre-

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sentation is given. In many worked-out exercises and examples its practical applications can be found: pionic atoms as a modern field of research and the particularly challeng­ing examples on the effective pion-nucleus potential (the Kisslinger potential) and its improvement by Ericson and Ericson stand in the foreground.

Most of these lectures deal with Dirac's theory. The covariance properties of the Dirac equation are discussed in detail. So, for example, its free solutions are on the one hand determined directly and on the other hand through Lorentz transformations from the simple solutions in the rest frame. Here the methodical issue is emphasized: the same physical phenomenon is illuminated from different angles. We proceed in a similar manner in the discussion of single-particle operators (the odd and even parts of an operator) and the so-called Zitterbewegung, wbich is also derived from the consideration of wave packets of plane Dirac waves. In many worked-out problems and examples the new tools are exercised. Thus the whole of Chap. 9 is dedicated to the motion of Dirac particles in external potentials. It contains simple potential problems, extensively the case of the electron in a Coulomb potential (the fine-structure formula), and muonic atoms. In Chap. 10 we present the two-centre Dirac equation, which is of importance in the modem field of heavy-ion atomic physics. The fundamental problem of overcritical fields and the decay of the electron-positron vacuum is only touched upon. A full treatment is reserved for Quantwn Electrodynamics (Vol. 4 of this series). However, we give an extended discussion of hole theory and also of Klein's paradox. The Weyl equation for the neutrino (Chap.14) and relativistic wave equations for particles with arbitrary spin (Chap. 15) follow. Starting with the Bargmann-Wigner equations the general frame for these equations is set, and in numerous worked-out examples and exercises special cases (spin-l particles with and without mass, and spin-~ particles according to Rarita and Schwinger) are considered in greater detail. In the last chapter we give an overview of relativistic symmetry principles, which we enjoy from a superior point of view, since by now we have studied Quantwn Mechanies - Symmetries (Vol. 2 of tbis series).

We hope that in this way the lectures will become ever more complete and may lead to new insights.

Biographical notes help to obtain an impression, however short, of the life and work of outstanding physicists and mathematicians. We gratefully acknowledge the publishers Harri Deutsch and F.A. Brockhaus (Brockhaus Enzyklopädie, F.A. Brockhaus - Wiesbaden indicated by BR) for giving permission to use relevant information from their publications.

Special thanks go to Prof. Dr. Gerhard Soff, Dr. Joachim Reinhardt, and Dr. David Vasak for their critical reading of the original draft of these lectures. Many students and collaborators have helped during the years to work out examples and exercises. For this first English edition we enjoyed the help of Maria Berenguer, Christian Borchert, Snjezana Butorac, Christian Derreth, Carsten Greiner, Kordt Griepenkerl, Christian Hofmann, Raf­faele Mattiello, Dieter Neubauer, Jochen Rau, Wolfgang Renner, Dirk Rischke, Alexander Scherdin, Thomas Schönfeld, and Dr. Stefan Schramm. Miss Astrid Steidl drew the graphs and prepared the figures. To all of them we express our sincere thanks.

We would especially like to thank Mr. Bela Waldhauser, Dipl.-Phys., for his overall assistance. His organizational talent and his advice in technical matters are very much appreciated.

Finally, we wish to thank Springer-Verlag; in particular, Dr. H.-U. Daniel, for his encouragement and patience, Mr. Michael Edmeades for expertly copy-editing the English edition, and Mr. R. Michels and his team for the excellent layout.

Frankfurt am Main, May 1990 Walter Greiner

Contents

1. Relativistic Wave Equation for Spin-O Particles The Klein-Gordon Equation and Its Applications . . . . . . . . . . . . . . . . . . . 1 1.1 The Notation ............................................... 1 1.2 The Klein-Gordon Equation .................................. 4 1.3 The Nonrelativistie Limit ..................................... 7 1.4 Free Spin-Zero Particles .......... . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.5 Energy-Momentum Tensor of the Klein-Gordon Field ............. 11 1.6 The Klein-Gordon Equation in Sehrödinger Form ................. 18 1.7 Charge Conjugation ......................................... 23 1.8 Free Spin-O Partieles in the Feshbaeh-Villars Representation ........ 26 1.9 The Interaetion of a Spin-O Particle with an Electromagnetie Field .. 34 1.10 Gauge Invariance of the Coupling ............................. 39 1.11 The Nonrelativistie Limit With Fields . . . . . . . . . . . . . . . . . . . . . . . . . . 41 1.12 Interpretation of One-Partiele Operators

in Relativistie Quantum Mechanies ............................. 52 Biographieal Notes .......................................... 72

2. A Wave Equation for Spin-~ Particles - The Dirac Equation . . . . . . . . . 75 2.1 Free Motion of a Dirae Particle ..... . . . . . . . . . . . . . . . . . . . . . . . . . . 82 2.2 Single-Particle Interpretation of the Plane (Free) Dirae Waves ....... 86 2.3 Nonrelativistie Limit of the Dirae Equation ...................... 93

Biographical Notes .......................................... 98

3. Lorentz-Covariance of the Dirac Equation . . . . . . . . . . . . . . . . . . . . . . . . . 99 3.1 Formulation of Covarianee (Form Invariance) .................... 102 3.2 Construetion of the S Operator for Infinitesimal

Lorentz Transformations ..................................... 110 3.3 Finite Proper Lorentz Transformations .......................... 112

3.4 The S Operator for Proper Lorentz Transformations ............... 113 3.5 The Four-Current Density .................................... 116

Biographical Notes .......................................... 117

4. Spinors under Spatial Reftection .................................. 119

5. Bilinear Covariants of the Dirac Spinors ........................... 121 Biographical Notes .......................................... 125

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6. Another Way of Constructing Solutions of the Free Dirac Equation: Construction by Lorentz Transformations .......................... 127 6.1 Plane Waves in Arbitrary Directions ............................ 131 6.2 The General Fonn of the Free Solutions and Their Properties ....... 134 6.3 Polarized Electrons in Relativistic Theory ....................... 141

7. Projection Operators for Energy and Spin ... .. .. . . . . . . . .. . . . . . .. . . 145 7.1 Simultaneous Projections of Energy and Spin .................... 149

8. Wave Packets of Plane Dirac Waves 151

9. Dirac Particles in External Fields: Examples and Problems 161

10. The Two-Centre Dirac Equation .................................. 201

11. The Foldy-Wouthuysen Representation for Free Particles . .. . .. . . . . .. 217 11.1 The Foldy-Wouthuysen Representation in the Presence

of External Fields ........................................... 225

12. The Hole Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 12.1 Charge Conjugation ......................................... 239 12.2 Charge Conjugation of Eigenstates with Arbitrary Spin and Momentum 246 12.3 Charge Conjugation of Bound States ........................... 248 12.4 TIme Reversal and PCT Symmetry ............................. 249

Biographical Notes .......................................... 259

13. Klein's Paradox ................................................ 261

14. The Weyl Equation - The Neutrino 269

15. Wave Equations for ParticIes with Arbitrary Spins .................. 281 15.1 Partieles with Finite Mass .................................... 281 15.2 Massless Particles ........................................... 287 15.3 Spin-l Fields for Particles with Finite Mass; Proca Equations ....... 290 15.4 Kemmer Equation .......................................... 292 15.5 The Maxwell Equations ...................................... 295 15.6 Spin-! Fields .............................................. 307

Biographical Notes .......................................... 312

16. Lorentz Invariance and Relativistic Symmetry Principles . . . . . . . . . . .. 313 16.1 Orthogonal Transfonnations in Four Dimensions .................. 313 16.2 Infinitesimal Transfonnations and the Proper Subgroup of 0(4) ...... 314 16.3 Classification of the Subgroups of 0(4) ......................... 320 16.4 The Inhomogeneous Lorentz Group ............................ 322 16.5 The Confonnal Group ....................................... 323 16.6 Representations of the Four-Dimensional

Orthogonal Group and Its Subgroups ........................... 326

x

16.7 Representation of SL(2, C) ................................... 329 16.8 Representations of SO(3, R) .................................. 330 16.9 Representations of the Lorentz Group r." ........................ 331 16.10 Spin and the Rotation Group .................................. 333

Biographical Notes .......................................... 338

Subject Index ......... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 339

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Contents of Examples and Exercises

1.1 The Charged Klein-Gordon Field ................................. 11 1.2 Derivation of the Field Equations for Wavefields ..... . . . . . . . . . . . . . . . 12 1.3 Determination of the Energy-Momentum Tensor

for a General Lagrange Density C('l/Ju, 8'I/Ju/8xIJ) ................... 15 1.4 Lagrange Density and Energy-Momentum Tensor

of the Schrödinger Equation ..................................... 16 1.5 Lorentz Invariance of the Klein-Gordon Equation .................... 17 1.6 C Parity ..................................................... 24 1.7 Lagrange Density and Energy-Momentum Tensor of the

Free Klein-Gordon Equation in the Feshbach-ViHars Representation ..... 28 1.8 The Hamiltonian in the Feshbach-Villars Representation .............. 31 1.9 Solution of the Free Klein-Gordon Equation in the

Feshbach-ViHars Representation .................................. 32 1.10 Separation of Angular and Radial Parts of the Wave Function

for the Stationary Klein-Gordon Equation with a Coulomb Potential ..... 36 1.11 Pionic Atom with Point-Like Nucleus ............................. 37 1.12 Lagrange Density and Energy-Momentum Tensor

for a Klein-Gordon Particle in an Electromagnetic Field .............. 42 1.13 Solution of the Klein-Gordon Equation for the Potential

of an Homogeneously Charged Sphere ............................. 43 1.14 The Solution of tbe Klein-Gordon Equation for

a Square WeH Potential ......................................... 44 1.15 Solution of the Klein-Gordon Equation for an Exponential Potential ..... 46 1.16 Solution of the Klein-Gordon Equation for a Scalar l/r Potential ....... 47 1.17 Basics of Pionic Atoms ......................................... 49 1.18 Calculation of the Position Operator in the ~ Representation ........... 57 1.19 Calculation of the Current Density in the ~ Representation

for Particles and Antiparticles .................................... 59 1.20 Calculation of the Position Eigenfunctions in the Coordinate

Representation ................................................ 60 1.21 Mathematical Supplement: Modified Bessel Functions

of the Second Type, Kv(z) ...................................... 62 1.22 The Kisslinger Potential ......................................... 63 1.23 Evaluation of the Kisslinger Potential in Coordinate Space ............ 68 1.24 Lorentz-Lorenz Effect in Electrodynamics and Its Analogy

in Pion-Nucleon Scattering (the Ericson-Ericson Correction) ........... 69

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2.1

2.2

2.3

2.4

3.1 3.2

3.3

5.1

5.2

6.1

6.2

6.3 6.4 6.5

6.6

8.1 8.2 8.3 8.4 8.5

9.1

9.2

9.3

9.4

9.5 9.6

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Representation of the Maxwell Equations in the Fonn of the Dirae Equation .......................................... . Lagrange Density and Energy-Momentum Tensor of the Free Dirae Equation ..................................... . Caleulation of the Energies of the Solutions to the Free Dirae Equation in the Canonieal Fonnalism .............. . Time Derivative of the Position and Momentum Operators for Dirae Particles in an Electromagnetie Field ..................... .

Pauli's Fundamental Theorem: The 16 Complete 4 x 4 Matriees FA Proof of the Coeffieients aOtß for the Infinitesimal Lorentz Transfonnation ....................... . Caleulation of the Inverse Spinor Transfonnation Operators

Transfonnation Properties of Some Bilinear Expressions of Dirae Spinors ................................... . Majorana Representation of the Dirac Equation ..................... .

Caleulation of the Spinor Transformation Operator in Matrix Fonn ............................................... . Caleulation of the Spinor u(p + q) by Means of a Lorentz Transfonnation from the Spinor u(P) of a Free Partiele ......................................... . Normalization of the Spinor wr(P) ............................... .

Proof of the Relation Wrt(crP)Wr' (cr'P) = örr,(E /moc2) ••••••••••••••

Independenee of the Closure Relation from the Representation of the Dirae Spinors .............................. . Proof of Another Closure Relation for Dirae Spinors ................ .

The Gordon Decomposition ..................................... . Caleulation of the Expectation Value of a Velocity Operator .......... . Caleulation of the Nonn of a Wave Packet ........................ . Caleulation of the Current for a Wave Packet ...................... . Temporal Development of a Wave Packet with Gaussian Density Distribution ................................... .

Eigenvalue Spectrum of a Dirae Particle in a One-Dimensional Square-WeIl Potential ........................... . Eigenvalues of the Dirae Equation in a One-Dimensional Square Potential Well with Sealar Coupling ........................ . Separation of the Variables for the Dirae Equation with Central Potential (Minimally Coupled) ........................ . Commutation of the Total Angular Momentum Operator with the Hamiltonian in a Spherieally Symmetrie Potential ........................................... . A Dirac Particle in a Spherieal Potential Box ...................... . Solution of the Radial Equations for a Dirae Particle in a Coulomb Potential ........................................ .

80

88

90

95

104

111 116

124 124

132

133 135 136

139 139

152 153 155 156

156

161

166

169

172 174

178

9.7 Discussion of the Sommerfeld Fine Strueture Fonnula and the Classifieation of the Electron Levels in the Dirae Theory .............. 182

9.8 Solution of the Dirac Equation for a Coulomb and a Sealar Potential '" . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

9.9 Stationary Continuum States of a Dirae Particle in a Coulomb Field ............................................ 187

9.10 Muonic Atoms ................................................ 193 9.11 Dirac Equation for the Interaction between a Nuclear

and an Extemal Field Taking Account of the Anomalous Magnetic Moment .............................. 197

9.12 The Impossibility of Additional Solutions for the Dirac-Coulomb Problem Beyond Z = 118 ....................... 198

12.1 Radiative Transition Probability from the Hydrogen Ground State to the States of Negative Energy .......... 236

12.2 Expectation Values of Some Operators in Charge-Conjugate States ...................................... 242

12.3 Proof of (H(-e»)c = -(H(e») ................................... 244 12.4 Effect of Charge Conjugation on an Electron with Negative Energy ..... 244 12.5 Representation of Operators for Charge Conjugation

and Time Reversal ............................................. 245 12.6 Behaviour of the Current with Time Reversal

and Charge Conjugation ....................................... .

13.1 Klein's Paradox and the Hole Theory ............................. .

14.1 Dirac Equation for Neutrinos ................................... . 14.2 CP as a Symmetry for the Dirac Neutrino ......................... . 14.3 Solutions of the Weyl Equation with Good Angular Momentum ....... .

15.1 Eigenvalue Equation for Multispinors ............................. .

15 2 M 1·· (+) E' f ~2 . u tlspmor w as 1genvector 0 ~ •••••.•••••••••••••••••••••••

15.3 Multispinor of Negative Energy ................................. .

15.4 Construetion of the Spinor w~~ ... r(P' i) ........................... .

15.5 The Bargmann-Wigner Equations ................................ . 15.6 ')'-Matriees in Weyl-Representation ............................... . 15.7 Commutation Relations of Kemmer Matrices ...................... . 15.8 Properties of the Kemmer Equation under Lorentz Transfonnations .... . 15.9 Verification of the Kemmer Algebra .............................. . 15.10 Verification of the Proca Equations from the Lagrange Density ........ . 15.11 Conserved Current of Vector Fields .............................. . 15.12 Lorentz Covariance of Veetor Field Theory ........................ . 15.13 Maxwell-Similar Form of the Vector Fields ........................ . 15.14 Plane Waves for the Proca Equation .............................. . 15.15 Transfonnation from the Kemmer to the Proca Representation ........ . 15.16 Lagrange Density for Kemmer Theory ............................ .

255

265

277 277 279

282

283 284

285

286 287 296 298 300 301 301 301 302 303 303 304

xv

15.17 The Weinberg-Shay-Good Equations .............................. 304 15.18 Lagrangian Density for the Weinberg-Shay-Good Theory .............. 306 15.19 Coupling of Charged Vector Mesons to the Electromagnetic Field ...... 306 15.20 A Useful Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

16.1 Transformation Relations of the Rest Mass under Dilatations ........... 325 16.2 D1 Representation of SU(2) ..................................... 328 16.3 Vector Representation and Spin ................................... 337

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