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The Raman Effect A Unified Treatment of the Theory of Raman Scattering by Molecules Derek A. Long free download

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Contents
Preface xix
Acknowledgements xxiii
Part One Theory 1
1 Survey of Light-scattering Phenomena 3
1.1 Introduction 3
1.2 Some Basic Definitions 4
1.3 Rayleigh and Raman Scattering 5
1.3.1 Description 5
1.3.2 Energy transfer model 7
1.4 Hyper-Rayleigh and Hyper-Raman Scattering 10
1.4.1 Description 10
1.4.2 Energy transfer model 10
1.5 Second Hyper-Rayleigh and Second Hyper-Raman Scattering 11
1.5.1 Description 11
1.5.2 Energy transfer model 11
1.6 Coherent anti-Stokes and Coherent Stokes Raman Scattering 11
1.7 Stimulated Raman Gain and Loss Spectroscopy 13
1.8 Typical Spectra 14
1.9 Bases for the Theoretical Treatment of Rayleigh and
Raman Scattering 16
1.10 Historical Perspective 16
viii Contents
1.11 Caveat 17
References 17
2 Introduction to Theoretical Treatments of Incoherent Light Scattering 19
2.1 General Considerations 19
2.2 Induced Oscillating Electric Dipoles as Sources of
Scattered Radiation 21
2.3 Basis of the Classical Theory of Light Scattering 22
2.4 Basis of the Quantum Mechanical Treatment
of Incoherent Light-Scattering Phenomena: Electric Dipole Case 24
2.5 Extension of Quantum Mechanical Treatment of Incoherent
Light Scattering to Include Magnetic Dipole and Electric
Quadrupole Cases 27
2.6 Comparison of the Classical and Quantum Mechanical
Treatments of Light Scattering 28
2.7 The Way Ahead 29
3 Classical Theory of Rayleigh and Raman Scattering 31
3.1 Introduction 31
3.2 First-order Induced Electric Dipole 31
3.3 Frequency Dependence of the First-order Induced Electric
Dipole 34
3.4 Classical Scattering Tensors aRay and aRam
k 35
3.5 Selection Rules for Fundamental Vibrations 36
3.5.1 General considerations 36
3.5.2 Diatomic molecules 36
3.5.3 Polyatomic molecules 38
3.6 Selection Rules for Overtones and Combinations 43
3.7 Coherence Properties of Rayleigh and Raman Scattering 44
3.8 Limitations of the Classical Theory 45
3.9 Example of Rayleigh and Raman Scattering 45
3.10 Presentation of Raman Spectra 47
References 48
4 Quantum Mechanical Theory of Rayleigh and Raman Scattering 49
4.1 Introduction 49
4.2 Time-dependent Perturbation Theory and a fi 50
4.3 Qualitative Discussion of ˛ fi 54
4.3.1 Frequency denominators 55
4.3.2 Transition electric dipole numerators 56
4.3.3 Selection rules 58
4.4 Tensorial Nature of the Transition Polarizability and its Symmetry 58
4.5 Born–Oppenheimer Approximation and the Transition
Polarizability Tensor 61
Contents ix
4.6 Simplification of ˛ efvf:egvi : General Considerations 64
4.7 Simplification by Radical Approximation: the Placzek
Transition Polarizability 65
4.8 Simplification of ˛ efvf:eivi by Stages 68
4.8.1 Introduction of Herzberg–Teller vibronic coupling 68
4.8.2 Identification of non-resonance and resonance
situations 75
4.9 Normal Electronic (and Vibronic) Raman Scattering 77
4.10 Normal Pure Vibrational Raman Scattering 78
4.11 Electronic (and Vibronic) Resonance Raman Scattering 81
4.12 Vibrational Resonance Raman Scattering 83
4.13 Units and Orders of Magnitude 83
References 84
5 Vibrational Raman Scattering 85
5.1 Introduction 85
5.2 The Placzek Vibrational Transition Polarizability:
Recapitulation 86
5.2.1 Cartesian basis 86
5.2.2 The spherical basis 88
5.3 Definition of Illumination–Observation Geometry 89
5.4 Intensity of Scattered Radiation: Some General Considerations 94
5.4.1 Development of a symbol for scattered intensity 94
5.4.2 Scattering cross-section 95
5.5 Intensity Formulae and Polarization Characteristics for a General
Vibrational Transition in Various Illumination–Observation
Geometries 97
5.5.1 General considerations 97
5.5.2 Linearly polarized incident radiation 98
5.5.3 Natural incident radiation 102
5.5.4 Angular dependence of scattered intensity 103
5.5.5 Circularly polarized incident radiation 106
5.5.6 Symmetry and depolarization ratios, reversal
coefficients and degrees of circularity 109
5.6 Stokes Parameters for Scattered Radiation 113
5.7 Specific Vibrational Transitions 116
5.8 Vibrational Selection Rules 120
5.9 Patterns of Vibrational Spectra 123
5.10 Orders of Magnitude 126
5.11 Epilogue 127
References 131
Reference Tables for Chapter 5 132
Reference Table 5.1: Definitions for I ; ps, pi 132
x Contents
Reference Table 5.2(a) to 5.2(g): Intensities, Polarization
Properties and Stokes Parameters for Vibrational Raman
(and Rayleigh) Scattering 132
Reference Table 5.3: Symmetry classes for x, y, z, the rotations
Rx, Ry and Rz, and the components of the cartesian basis
tensor ca. 145
6 Rotational and Vibration–Rotation Raman Scattering 153
6.1 Introduction 153
6.2 Irreducible Transition Polarizability Components 154
6.3 Symmetric Top 156
6.3.1 Selection rules 156
6.3.2 Placzek invariants G j fi 157
6.3.3 Intensities 167
6.3.4 Subsequent development 169
6.4 Rotational and Vibrational Terms 169
6.5 Statistical Distribution of Molecular Population 171
6.6 Diatomic Molecule 173
6.6.1 Introduction 173
6.6.2 Heteronuclear diatomic molecule: pure rotation 174
6.6.3 Heteronuclear diatomic molecule: vibration–rotation 175
6.6.4 Homonuclear diatomic molecule: nuclear spin degeneracy 179
6.6.5 Intensity distribution 180
6.7 Symmetric Top Molecule 186
6.7.1 Introduction 186
6.7.2 Symmetric top: pure rotation 187
6.7.3 Symmetric top: vibration–rotation 191
6.7.4 Intensities 203
6.8 Linear Molecules 204
6.8.1 Rotation and vibration-rotation Raman spectra 204
6.8.2 Intensities 207
6.9 Contributions from Electronic Orbital and Spin Angular Momenta 208
6.10 Spherical Top Molecules 210
6.11 Asymmetric Top Molecules 211
6.12 Epilogue 211
References 213
Reference Tables for Chapter 6 214
Introduction 214
Reference Tables 6.1 to 6.4 216
7 Vibrational Resonance Raman Scattering 221
7.1 Introduction 221
7.2 Vibrational Transition Polarizability Tensor Components in the
Resonance Case, Based on Perturbation Theory 222
Contents xi
7.3 Comparison of the AVI, BVI,CVI and DVI Terms 224
7.3.1 The AVI term 224
7.3.2 The BVI term 227
7.3.3 The CVI term 229
7.3.4 The DVI term 229
7.3.5 Subsequent developments 230
7.4 AVI Term Raman Scattering from Molecules with Totally
Symmetric Modes 231
7.4.1 AVI term Raman scattering from molecules with one
totally symmetric mode 231
7.4.2 AVI term Raman scattering from molecules
with more than one totally symmetric mode: general
considerations 237
7.4.3 AVI term Raman scattering from totally symmetric modes
when k is very small 238
7.5 AVI Term Raman Scattering Involving Non-Totally Symmetric
Modes 239
7.5.1 General considerations 239
7.5.2 AVI term scattering involving a change of molecular
symmetry of the resonant excited state 239
7.5.3 AVI term scattering involving excited state Jahn–Teller
coupling 240
7.5.4 Summary of excited state Jahn–Teller effects in
resonance Raman scattering 240
7.6 BVI Term Scattering Involving Vibronic Coupling of the Resonant
Excited State to a Second Excited State 241
7.6.1 Introduction 241
7.6.2 BVI term scattering from molecules with non-totally
symmetric modes 241
7.6.3 BVI term scattering from molecules with totally
symmetric modes 244
7.7 Symmetry, Raman Activity and Depolarization Ratios 246
7.7.1 General symmetry considerations 246
7.7.2 The AVI term 247
7.7.3 The BVI term 250
7.8 Time-Dependent Formulation of Resonance Raman Scattering 262
7.8.1 Introduction 262
7.8.2 Transformation of the AVI term to a time-dependent
expression 263
7.8.3 The time-dependent interpretation of resonance Raman
scattering 264
7.9 Continuum Resonance Raman Scattering 266
References 270
xii Contents
8 Rotational and Vibration–Rotation Resonance Raman Scattering 271
8.1 Introduction 271
8.2 General Expression for ˛ fi for a Symmetric Top Molecule 272
8.3 General Expression for ˛ j
m fi 274
8.4 Contraction of General Expression for ˛ j
m fi 274
8.5 The Quadratic Term 275
8.6 Selection Rules 276
8.7 Evaluation of j ˛ j
m fij2 277
8.8 Intensities and Depolarization Ratios 279
8.9 An Illustrative Example 283
8.10 Concluding Remarks 287
Reference 287
9 Normal and Resonance Electronic and Vibronic Raman Scattering 289
9.1 Introduction 289
9.2 Normal Electronic and Vibronic Raman Scattering 289
9.2.1 General considerations 289
9.2.2 AIII-term scattering 290
9.2.3 BIII C CIII -term scattering 291
9.2.4 DIII-term scattering 292
9.2.5 Transition tensor symmetry 292
9.3 Resonant Electronic and Vibronic Raman Scattering 292
9.3.1 General considerations 292
9.3.2 AV-term scattering 293
9.3.3 BV-term scattering 296
9.3.4 CV-term scattering 297
9.3.5 DV-term scattering 297
9.4 Selection Rules in Electronic Raman Spectra 297
9.4.1 General symmetry considerations 297
9.5 Intensities and Polarization Properties of Electronic Raman
Scattering 301
9.5.1 Intensities: general considerations 301
9.5.2 Excitation profiles 301
9.5.3 Depolarization ratios 302
10 Rayleigh and Raman Scattering by Chiral Systems 303
10.1 Introduction 303
10.2 Outline of the Theoretical Treatment 305
10.3 Intensities of Optically Active Rayleigh Scattering 310
10.3.1 General considerations 310
10.3.2 Intensity formulae 314
10.3.3 Stokes parameters 317
Contents xiii
10.4 Intensities of Optically Active Raman Scattering 321
10.4.1 General considerations 321
10.4.2 Discussion of intensities and isotropic invariants 323
10.4.3 Placzek polarizability theory and optically
active scattering 324
10.5 Symmetry Considerations 326
10.6 Concluding Remarks 327
Reference 327
Reference Tables for Chapter 10 328
Part Two Appendices 337
Introduction 339
A1 The Right-handed Cartesian Axis System and Related Coordinate
Systems 341
A1.1 Introduction 341
A1.2 The Right-handed Cartesian Axis System 342
A1.3 Cartesian Coordinate System 344
A1.4 Cylindrical Coordinate System 344
A1.5 Polar Coordinate System 345
A1.6 Complex Coordinate Systems 347
A2 The Summation Convention 349
A2.1 General Definitions 349
A3 Direction Cosines 351
A3.1 Introduction 351
A3.2 Definitions and Properties of Direction Cosines 351
A3.3 Definitions of Direction Cosines in Other Coordinate Systems 354
A4 Isotropic Averages of Products of Direction Cosines 355
A4.1 Introduction 355
A4.2 Specific Isotropic Averages of Products of Direction Cosines 356
A4.3 General Formulae 358
References 358
A5 The Euler Angles and the Rotation Operator 359
A5.1 Introduction 359
A5.2 Definitions of the Euler Angles and the Rotation Operator 359
A5.3 The Relationship of the Euler Angles to the Polar Coordinates 362
A5.4 Direction Cosines and Euler Angles 362
References 363
xiv Contents
A6 Complex Numbers and Quantities 365
A6.1 Introduction 365
A6.2 Definitions and Operations 365
A6.3 Graphical Representation of Complex Numbers 367
A6.4 Complex Numbers and Polar Coordinates 369
A6.5 Complex Quantities and Physical Phenomena 370
A6.6 Spherical Coordinates 371
Reference 372
A7 Some Properties of Matrices 373
A7.1 Introduction 373
A7.2 Nomenclature 373
A7.3 Some Special Matrices 374
A7.4 Matrix Representation of Simultaneous Linear Equations 377
A7.5 Eigenvalues and Eigenvectors 377
A7.6 Example of Diagonalization of a Matrix 379
A8 Vectors, I 381
A8.1 Introduction: Scalars, Vectors and Tensors 381
A8.2 Basic Definition of a Vector 382
A8.3 Unit Vectors 383
A8.4 Vector Addition, Subtraction and Multiplication by a Scalar 385
A8.5 Multiplication of Two Vectors 385
A8.6 Triple Products of Vectors 390
A8.6.1 A B Ð C 391
A8.6.2 A Ð B ð C 391
A8.6.3 A ð B ð C 392
A8.6.4 ABC 392
A8.7 Formal Definition of a Vector in Terms of its Transformation
upon Rotation of Axes 393
A8.8 Polar and Axial Vectors: Time Even and Time Odd Vectors 394
A8.9 Vector Differentiation 395
A8.9.1 The operator ∇ 395
A8.9.2 The gradient 396
A8.9.3 The divergence 397
A8.9.4 The curl, ∇ð V 397
A8.9.5 The divergence and the curl illustrated and compared 398
A8.9.6 Composite functions involving ∇ 398
A8.9.7 Successive applications of ∇ 400
A8.9.8 Time derivative of a vector 401
A8.9.9 Caveat 401
A8.10 Change of Basis Vectors and Effect Upon Coordinates
of a Fixed Vector 401
A8.11 The Effect of a Symmetry Operation on Vectors and Basis Vectors 404
Contents xv
A9 Vectors, II 407
A9.1 Introduction 407
A9.2 Cylindrical Coordinates and Basis Vectors 407
A9.3 Polar Coordinates and Polar Basis Vectors 408
A9.4 Spherical Components and Spherical Basis Vectors and Direction
Cosines 409
A9.5 Rotation of Vectors using Spherical Coordinates 413
A9.6 Vectors in n-Dimensional Space 415
References 416
A10 Tensors 417
A10.1 General Definitions 417
A10.2 Representation or Specification of a Tensor 419
A10.3 Transformation of Tensors upon Rotation of Axes 422
A10.4 Some Properties of Tensors 425
A10.4.1 General 425
A10.4.2 Tensors of rank two 425
A10.4.3 Tensors of rank three: the alternating or
Levi–Civit`a tensor 426
A10.4.4 Isotropic tensors of various ranks 427
A10.4.5 Tensor contraction 427
A10.5 Irreducible Tensorial Sets 428
A11 Electrostatics 433
A11.1 Introduction 433
A11.2 Force Between Charges 433
A11.3 Electric Field Strength 435
A11.4 Electrostatic Potential 436
A11.5 Gauss’s Law 437
A11.6 The Equations of Poisson and Laplace 438
A12 Magnetostatics 439
A12.1 Introduction 439
A12.2 Magnetic Forces 441
A12.3 The Magnetic Induction 442
A12.4 The Lorentz Force on a Point Charge Moving in a
Magnetic Field 445
A12.5 The Divergence of the Magnetic Induction B 446
A12.6 The Vector Potential A 446
A13 The Interaction of a System of Electric Charges with Electric
and Magnetic Fields 449
A13.1 Introduction 449
A13.2 Point Charges in Molecular Systems 450
xvi Contents
A13.3 Electric Dipole in a Molecular System 450
A13.4 Basic Treatment of the Energy of a Distribution of Point Charges
at Rest in a Uniform Electric Field 451
A13.5 Basic Treatment of Permanent and Induced Molecular Dipoles
in a Uniform Static Electric Field 454
A13.6 Basic Treatment of Macroscopic Polarization and Electric
Susceptibilities 457
A13.7 Basic Treatment of the Electric Displacement for a Uniform
Static Electric Field 460
A13.8 The Implications of Using Dynamic Electric Fields 461
A13.9 More General Treatment of Energy of Interaction of Point Charges
at Rest in a Static Electric Field 462
A13.10 Interaction of Charges in Motion with a Static Magnetic Field 466
Reference 470
A14 The Polarizability Tensor 471
A14.1 Introduction 471
A14.2 The Polarizability Tensor in the Cartesian Basis 472
A14.2.1 General considerations 472
A14.2.2 Reduction of the tensor 473
A14.2.3 The polarizability ellipsoid 475
A14.2.4 Transformation of ca under rotation of axes 477
A14.3 The Polarizability Tensor in the Spherical Basis 477
A14.3.1 General definitions 477
A14.3.2 Reduction of the tensor sa 479
A14.3.3 Transformation of sa under rotation of axes 481
A14.4 The Relation Between the ˛ and the ˛ 481
A14.5 Irreducible Polarizability Tensors and their Components 482
A14.6 Transformation Properties of the ˛ j
m under Rotations 486
A14.7 Isotropic Averages and Tensor Invariants 486
A14.7.1 General considerations 486
A14.7.2 Isotropic averages and rotational invariants G j in terms
of the ˛ j
m 487
A14.7.3 Isotropic averages and rotational invariants G j in terms
of the ˛ 488
A14.7.4 Isotropic averages and rotational invariants G j in terms
of the ˛ 490
A14.7.5 Isotropic averages and the rotational invariants G j ,
a, υ and
for the cartesian basis 490
A14.7.6 Isotropic averages and rotational invariants, G j , a,
υ and
for the spherical basis 494
References 495
Contents xvii
A15 The Optical Activity Tensors, G, G , A and A 497
A15.1 Introduction 497
A15.2 Isotropic Averages of the Type haG0i 497
A15.3 Isotropic Averages of the Type haAi 499
A15.4 Other Tensor Invariants 500
Reference 501
A16 Maxwell Equations in Vacuum and in Media 503
A16.1 The Fundamental Equations 503
A16.2 Case I: the Maxwell Equations in Vacuum 507
A16.3 Case II: the Maxwell Equations in a Linear Medium
with D 0 and D 0 508
A16.4 Case III: the Maxwell Equations in a Non-Linear Medium
with D 0 and D 0 509
A16.5 Case IV: the Maxwell Equations in a Linear Medium
with 6D 0 and 6D 0 511
A17 Monochromatic Plane Harmonic Waves in Vacuum and in
a Non-absorbing Linear Medium 513
A17.1 General Wave Equation in Vacuum 513
A17.2 Monochromatic Plane Harmonic Electromagnetic Wave in
Vacuum 514
A17.2.1 Solution of the wave equation for the vector E 514
A17.2.2 Solution of the wave equation for the vector B 518
A17.2.3 Energy considerations for a plane harmonic
electromagnetic wave in vacuum 522
A17.3 The Exponential Representation of Harmonic Waves 526
A17.4 Monochromatic Plane Harmonic Wave in a Homogeneous,
Isotropic and Linear Medium 530
A18 The Transition Polarizability Tensor a fi 533
A18.1 Introduction 533
A18.2 The Restrictions r 6D i, f 533
A18.3 The Relative Signs of ir 535
References 536
A19 Clebsch–Gordan Coefficients and Wigner 3-j and 6-j Symbols 537
A19.1 Introduction 537
A19.2 Clebsch–Gordan Coefficients 538
A19.3 Wigner 3-j Symbols 544
A19.4 Wigner 6-j Symbols 546
Reference 550
Reference Table A19.1 551
xviii Contents
A20 Sources of Electromagnetic Radiation 555
A20.1 Introduction 555
A20.2 The Oscillating Electric Dipole as a Source 555
A20.3 The Oscillating Magnetic Dipole as a Source 561
A20.4 The Oscillating Electric Quadrupole as a Source 562
A20.5 Scattering from Chiral Molecules 564
A21 Polarization of Electromagnetic Radiation 565
A21.1 Introduction 565
A21.2 States of Polarization: Monochromatic Radiation 565
A21.2.1 Linear polarization 565
A21.2.2 Elliptical and circular polarization 566
A21.2.3 Stokes parameters 570
A21.2.4 Stokes parameters for scattered radiation 572
A21.3 States of Polarization: Quasi-Monochromatic Radiation 573
A21.4 Change of Polarization: Depolarization Ratios, Reversal
Coefficients and Degrees of Circularity 575
Further Reading 579
Index 585
xix
Preface
Many ingenious practizes in all trades, by a connexion
and transferring of the observations of one Arte, to the
use of another, when the experiences of severall misteries
shall fall under the consideration of one man’s mind.
Francis Bacon
Raman spectroscopy is now finding wide-ranging application in pure and applied science
and the number of original papers devoted to this area of spectroscopy continues to grow.
This is largely the result of significant advances in the equipment available, particularly
laser excitation sources, spectrometers, detectors, signal processors and computers.
It seems timely, therefore, to provide an integrated treatment of the theory underlying
Raman spectroscopy. Of course there are already a number of edited books and reviews
dealing with various aspects of the subject, but this book is the result of the phenomenon of
Raman spectroscopy falling ‘under the consideration of one man’s mind’ as Francis Bacon
put it. My objective has been to present a unified theoretical treatment which is reasonably
complete and adequately rigorous but nonetheless readable. My hope is that this will
provide a sound basis for the effective use of more highly specialized review articles.
As to completeness, I have had to put some restrictions on the coverage, partly because
the subject is so vast and partly because of my own limitations. Therefore the treatments
developed here relate mainly to scattering by a system of freely orienting, non-interacting
molecules or by systems which approximate to this. As to rigour, I have endeavoured to
explain in words, as far as possible, the inwardness of the mathematics and physics which
are necessarily involved. I have particularly tried to avoid taking refuge behind that often
overworked phrase ‘as is well known’.
An effective theoretical treatment demands a variety of carefully honed mathematical
and physical tools. To keep the treatment in the main text uncluttered, these tools are
xx Preface
developed in comprehensive Appendices to which cross-references are made in the main
text. These Appendices should also ensure that the main text is useful to readers with a
wide variety of scientific backgrounds and experience.
As far as possible the symbols used to represent physical quantities are based on the
IUPAC recommendations but to avoid excessive embroidery with subscripts and superscripts
it has been necessary to introduce a few new symbols, all of which are clearly
defined. The SI system of units is used throughout, except in those few instances where
spectroscopists commonly adhere to historical units, as for example the unit cm 1 for
wavenumber and related quantities.
In the main text I have limited citations of the literature. This has enabled me to use the
Harvard system and quote names of authors and the dates of publication directly in the
text. I find this preferable to the anonymity and lack of historical sequence which result
from the use merely of reference numbers in the text. More complete lists of publications
are provided in the section entitled Further Reading, located at the end of the book.
The writing of this book has been a somewhat lengthy process and I have had to learn a
great deal along the way despite more than 50 years of work in this field. Happily I have
been able to find the time and the energy required. The University of Bradford enabled me
to free myself of administrative responsibilities by allowing me to take early retirement
and then reappointed me in an honorary capacity and provided excellent facilities for
research and scholarship. I am very grateful for these arrangements.
I am also much indebted to a number of other universities and institutions which
invited me to spend short periods with them so that I could use their libraries and benefit
from discussions with colleagues. In France I would mention the National Laboratory for
Aerospace Research (ONERA), and the Universities of Bordeaux I, Lille, Paris VI and
Reims. In Italy I would mention the University of Bologna and the European Laboratory
for Nonlinear Spectroscopy (LENS), attached to the University of Florence. In this country
I have made frequent use of the excellent facilities of the Radcliffe Science Library,
Oxford University. My periods in Oxford were made all the more pleasurable by the
kindness of my old Oxford College, Jesus, in making me a supernumerary member of its
Senior Common Room for a period. Nearer home, the J. B. Priestley Library of Bradford
University has also been much consulted and I am particularly indebted to Mr John
Horton, deputy librarian, for his unstinting help.
The following friends in the community of Raman spectroscopists have kindly read
and commented fruitfully upon sections of this book: A. Albrecht, D. L. Andrews,
L. D. Barron, J. Bendtsen, H. Berger, A. D. Buckingham, R. J. H. Clark, T. J. Dines,
H. Hamaguchi, S. Hassing, L. Hecht, M. Hollas, W. J. Jones, W. Kiefer, I. M. Mills,
O. Mortensen, H. W. Schr¨otter, G. Turrell, A. Weber, R. Zare and L. D. Ziegler. I
acknowledge with gratitude their efforts which have eliminated many errors and
ambiguities. I am also grateful to Claude Coupry and Marie-Th´er`ese Gousset for providing
Plate 5.1 and to H. G. M. Edwards for providing Plate 5.2.
I would also like to record my appreciation of the patience and continued support
of John Wiley and Sons. I would mention particularly Dr Helen McPherson, Chemistry
Publisher, who has been most considerate throughout; and Mr Martin Tribe who has been
an efficient and imperturbable Production Coordinator.
Preface xxi
Very special thanks are due to my wife Moira. She has very ably undertaken most
of the word-processing of the text, helped considerably with the style and presentation,
and provided loving encouragement. Rupert, another member of the family, has also
earned honourable mention. His insistence upon regular walks has helped to offset the
sedentary effects of authorship and his relaxed presence in my study has been calming
and companionable.
It would be unrealistic to expect that this wide-ranging book will be entirely free of
errors. In the words of Evan Lloyd, an eighteenth century Welshman, also a graduate of
Jesus College, Oxford, I can only plead that
‘Earnest is each Research, and deep;
And where it is its Fate to err,
Honest its Error, and Sincere.’
December 2001 Derek A. Long
Acknowledgements
Comme quelqu’un pourrait dire de moi
que j’ai seulement fait ici un amas de fleurs ´etrang`eres,
n’ayant fourni du mien que le filet `a les lier.
M. E. Montaigne
I am grateful for permission to reproduce the material listed below.
Figure 5.11, Plate 5.2 – Edwards, H. G. M., Russell, N. C. and Wynn-Williams, D. D.
(1997) Fourier-Transform Raman Spectroscopic and Scanning Electron Microscopic Study
of Cryptoendolithic Lichens from Antarctica. J. Raman Spectrosc., 28, 685, John Wiley
& Sons, Ltd, Chichester. Reproduced with permission from John Wiley & Sons, Ltd,
Chichester.
Figure 6.13 – from Bendtsen, J. (1974). J. Raman Spectrosc., 2, 133. Reproduced with
permission from John Wiley & Sons, Ltd, Chichester.
Figure 6.14 – from Bendtsen, J. and Rasmussen, F. (2000). J. Raman Spectrosc., 31, 433.
Reproduced with permission from John Wiley & Sons, Ltd, Chichester.
Table 7.1 – from Mortensen, O. S. and Hassing, S. (1980). Advances in Infra-red and
Raman Spectroscopy, volume 6, 1, eds. R. J. Clark and R. E. Hester, Wiley-Heyden,
London. Reproduced with permission from John Wiley & Sons, Ltd, Chichester.
Figures 7.6–7.12 inclusive – from Clark, R. J. H. and Dines, T. J. (1986). Resonance
Raman Spectroscopy and its Application to Inorganic Chemistry. Angew. Chem. Int. Ed.
Engl., 25, 131. Reproduced with permission from Wiley-VCH Verlag GmbH, Weinheim.
xxiv The Raman Effect
Figures 7.14–7.18 inclusive – from Kiefer, W. (1995). Resonance Raman Spectroscopy
in Infrared and Raman Spectroscopy, ed. B. Schrader. VCH Verlag GmbH, Weinheim.
Reproduced with permission from Wiley-VCH Verlag GmbH, Weinheim.
Figures 8.2, 8.3, 8.5, 8.6 – from Ziegler, L. D. (1986). J. Chem. Phys., 84, 6013. Reproduced
with permission of the American Institute of Physics.
Table 8.2 – from Ziegler, L. D. (1986). J. Chem. Phys., 84, 6013. Reproduced with permission
of the American Institute of Physics.
Figure 10.1 – from Nafie, L. A. and Che, D. (1994). Theory and Measurement of Raman
Optical Activity, in Modern Non-linear Optics, part 3, eds. M. Evans and S. Kielich.
Reproduced with permission from John Wiley & Sons, Inc., New York.
Figures A8.6, A8.7, A8.8 – from Barron, L. D. (1983). Molecular Light Scattering and
Optical Activity. Cambridge University Press, Cambridge. Reproduced with permission
from Cambridge University Press.
Figure A8.10a, b, c – from Atkins, P. W. (1983). Molecular Quantum Mechanics, Oxford
University Press, Oxford. Reproduced by permission of Oxford University Press.
Reference Table A19.1 – from Zare, R. N. (1988). Angular Momentum, John Wiley &
Sons, Inc.: New York. Reproduced with permission from John Wiley & Sons, Inc., New
York.
I also acknowledge my indebtedness to the literature of Raman spectroscopy in general
and in particular to the following authors and their publications upon which I have
drawn for the topics stated. For scattering by chiral systems, reviews and books by
L. D. Barron, L. Hecht and L. A. Nafie; for rotational and vibration-rotation Raman scattering,
reviews by S. Brodersen, by W. J. Jones and by A. Weber; for vibrational and
electronic resonance Raman scattering, reviews by R. J. H. Clark and T. J. Dines, by
H. Hamaguchi and by W. Kiefer; for rotational and vibration-rotation resonance Raman
scattering, reviews by L. Ziegler; and for irreducible transition polarizability tensors, a
review by O. S. Mortensen and S. Hassing. Full references will be found in the Section
entitled Further Reading.
The treatments of the classical theory of Rayleigh and Raman scattering, vibrational
Raman scattering and the properties of electromagnetic radiation and many of the Reference
Tables are based upon my earlier book, the copyright of which has been assigned
to me by the original publishers.
D. A. Long.

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PREFACE
It was already an important question to the philosophers of antiquity: of
what does matter fundamentally consist? When repeatedly dividing some
piece of matter such as wood or metal or a diamond, does one reach a
limit? If so, how does this limit manifest itself? Are there indivisible
objects; is there a smallest possible object? Or does the limit exist only
in the sense that any further division would seem to make no sense or be
experimentally impossible?
Every careful observer of natural phenomena is bound to be impressed
by the colorful and stunning diversity of the material world. One soon
notes, however, that there does not exist total chaos in these phenomena.
Things repeat themselves. A diamond here and another in some other
place are as alike as peas in a pod. The leaves of an oak in Boston are
indistinguishable from those of an oak in Denver.
So, within the uncountable diversity of phenomena, there are
also constants, things that repeat themselves. It was this duality of
vi THE FUNDAMENTAL CONSTANTS
multiplicity and constancy that inspired the Greek philosophers, most
notably Leucippus of Milet and his student Democritus of Abdera in the
5th century BC, to formulate the hypothesis that the universe is made
up of many small, indivisible building blocks called atoms (derived
from the Greek word atomos, which means essentially the same thing as
indivisible). A small number of different atoms and their unending new
combinations should suffice, so they claimed, to make up the diversity
of things. “Nothing exists,” spoke Democritus, “except atoms and empty
space.”
A slightly different concept was brought into play by Anaxagoras
around 500 BC. He spoke of an infinity of basic elements which, by
mixing, would produce the diversity of objects in the world. These basic
elements were furthermore said to be indestructible, and the changes
observable in physical objects were considered the result of motion
causing new combinations of the elements. Empedocles, who was ten
years younger than Anaxagoras, postulated that the basic materials of
the world were the four elements earth, water, air, and fire.
It is interesting to note that it is in the conceptualization of the atom
that empty space plays a role for the first time. Up to this point, space
had been seen as filled with matter, and the idea of empty space was
therefore unthinkable. In the context of the theory of the atom, empty
space took on an important function. It became the bearer of geometry,
the structure in which the atoms moved. So now matter and geometry
are two different things.
Atoms move in space and they have geometrical qualities.
Democritus said: “Just as tragedy and comedy can be written down with
the same letters, different phenomena can be realized in our world by
the same atoms, provided they assume different positions and different
movements. Some given matter may have the appearance of a given
color; may appear to us as tasting sweet or bitter — but in reality there
are only atoms and empty space.”
PREFACE vii >>>
Later the Greek philosophy adopted the elements of the theory of
the atom and developed the idea further. Plato, in his dialogue Timaeus,
discusses possible connections between atoms and the Pythagorean
theory of the harmony of numbers. For instance, he identified the
atoms of the elements earth, water, air, and fire with the regular solids,
the cube, octahedron, icosahedron, and tetrahedron. In speaking of the
motion of the atoms, special reference is made to natural causality.
Atoms are not moved by forces like love and hate, rather their motion is
the consequence of true natural laws.
What started two and a half millennia ago on what is nowadays the
west coast of Turkey was nothing less than the beginning of a revolution
that continues today. For thousands of years before then, mankind had
seen what occurred in the world as coming from some primarily mystic
source. Magic and superstition ruled the world.
That changed 2500 years ago on the Ionian coast. The time and place
were no accident. In the city states of the Ionian coast democratic values
had taken hold. New ideas were accepted easily and could spread quickly.
This was due in part to the switch around that time from hieroglyphic
symbols to an alphabet. Religion played no or only a subordinate role.
Thus the idea that our world is somehow in the end knowable, and that
natural processes can be analyzed with a rational mind, gained ground.
“Atomism” was the very beginning of this development. The thread that
winds its way through history from the Ionian coast 2500 years ago to
this highly scientific and technological present day essentially concerns
our knowledge of the building blocks of matter.
Many details about atomic theory as taught in antiquity were preserved
for us by a chance event that occurred in Italy in 1417: a manuscript by
Lucretius, a Roman poet and philosopher, was discovered. In this script,
De rerum natura, which is composed in a measured hexameter,
Lucretius not only describes the ideas of Leucippus and Democritus but
also further develops them. In the work of Lucretius, the atomism of
antiquity reaches its highest form. This book was one of the first to be
viii THE FUNDAMENTAL CONSTANTS
made after the invention of printing. Copies spread all over Europe and
have influenced scientists ever since.
In the work of Lucretius, one finds the best and most detailed
description of the atomic theory of antiquity, but, in the end, the theory
could not prevail against Plato’ and Aristotle’s system of ideal forms.
His work combined scientific questioning and the demystification of
nature with a deep respect for nature and its immutable rules.
Had Lucretius’s teachings prevailed two millennia ago, the course of
world history would have been different. It would have been less marked
by religious excesses and the religious wars in Europe and Asia. Alas,
the reality was different.
After the collapse of the Roman Empire, the western world sank
into intellectual oblivion for more than a millennium, dominated
by religious fanaticism and superstition. It was not until the Italian
Renaissance that the brilliant intellectual clarity of Greek thought
came to wide parts of Europe again, after being lost for more than a
thousand years. The scientific epoch commenced then, led by such
heroes of the mind as Copernicus, Leonardo da Vinci, Johannes Kepler,
and Galileo Galilee.
In the 17th century, the atomism of the philosophers of antiquity was
combined with scientific ideals for the first time. At this time, scientists
came to the conclusion that chemical elements such as hydrogen, oxygen,
and copper were composed of similar atoms. Isaac Newton, the inventor
of mechanics and thereby also the inventor of theoretical physics, was
even of the opinion that the consistency of materials, the hardness of a
metal for example, was somehow linked to the forces between atoms.
In the second half of the 19th century, atomism was applied with
success to the field of chemistry. Chemists found that chemical reactions
were best understood if one assumed the substances involved to be made
up of small, indivisible building blocks, or atoms. A chemical element,
hydrogen say, was thought to consist of a single type of atom. And
chemical methods made it possible to determine the approximate size of
PREFACE ix >>>
such an atom: 10−8 cm. One billion hydrogen atoms stacked on top of
one another would reach a height of around 10 cm.
Nowadays, we know of 110 different elements, that is to say, 110
different kinds of atom. This fact would have posed a serious problem
for the ancient Greek philosophers: they would certainly never have
considered it possible that more than 100 different atoms exist. For
the first time, there were doubts about whether the atom was truly an
indivisible entity. In the end, it was physicists, not chemists, who
realized, at the start of the 20th century, that atoms are not indivisible in
the sense understood by the ancient Greeks. They discovered that atoms
are made up of smaller components: electrons, the particles that make up
the atomic shells, and the atomic nucleus, which constitutes the greatest
part of the mass of an atom. Atoms became complicated systems.
In the 1920s and 1930s, atomic physics had its great breakthrough.
With the help of the newly developed field of quantum mechanics, it was
possible for the first time to understand atoms, and hence the structure
of atomic matter, both quantitatively and qualitatively on the basis of
a small number of principles. Most of the problems that physicists and
chemists had grappled with in the previous century could now be solved
elegantly.
Physicists then applied the same principles to the atomic nucleus,
in the hope that they would quickly reach a similar, more profound
understanding of the nucleus. But this hope was not to be fulfilled.
They discovered that the atomic nucleus, far from being an indivisible
object, is made up of nuclear particles called protons and neutrons.
This knowledge, however, hardly helped to reveal very much about the
properties of the atomic nucleus itself.
Soon it was observed that when particles are collided at very high
energy, new particles are created. Nobody had anticipated this. Einstein’s
now famous equation E = mc2, which states that matter and energy are
interchangeable, was seen in full effect. A whole zoo of new particles
x THE FUNDAMENTAL CONSTANTS
was discovered. Some physicists despaired at this confusing diversity
and compared subnuclear physics to botany.
Finally, between 1960 and 1980, a breakthrough was achieved that
brought order to the chaotic world of subnuclear physics. The 20th
century will enter into history as the epoch in which the substructure
of matter was largely elucidated. Today we know that normal matter is
made up of quarks, which are the building blocks of the atomic nucleus,
and electrons. In the 1970s, a clear picture of the microstructure of
matter finally evolved, often prosaically named the Standard Model of
particle physics. This model also describes the fundamental interactions
qualitatively and quantitatively in a simple form. The interactions are the
chromodynamic force, which acts between quarks, and the electroweak
force, which acts between quarks and leptons, such as electrons.
The Standard Model is, however, far more than a theoretical model of
the elementary particles and their interactions. Its claim to fame is that of
a complete unified theory for all the observed phenomena associated with
elementary particles. For specialists, the whole theory can be reduced
to a couple of lines. This makes it something like the Weltformel that
theoretical physicists such as Albert Einstein and Werner Heisenberg
had unsuccessfully looked for in the past.
Could this theory prove to be a last and thereby final truth? Are
electrons and quarks indeed nature’s elementary objects, meaning that
physicists have finally found the atoms of Democritus and Lucretius?
The answer to this question remains undecided. The Standard Model
has a number of unsatisfactory characteristics, so many physicists today
assume that it is merely an approximation, albeit a very well-functioning
approximation, to a more comprehensive theory. If so, physicists should
soon find evidence in their experiments for phenomena not explained by
the Standard Model, perhaps even evidence for a new substructure of
matter.
Electrons and quarks are not simple building blocks that can be
combined at will. They are subject to forces such as electromagnetic
PREFACE xi >>>
forces which are, in turn, transmitted by small particles. That is why
in particle physics we should avoid speaking of fundamental forces
acting on particles, rather we should speak of interactions between
particles.
It turns out that interactions in the Standard Model are governed by
very specific laws that are based on symmetry. Symmetry in nature and
elementary particles are closely connected. Plato had already referred to
such a connection in antiquity. Werner Heisenberg, one of the founders
of quantum mechanics and one of the most important physicists of the
20th century, had the following to say: “For Plato is the elementary
particle not an unchangeable and indivisible object. The elementary
particle is reduced to mathematics. The roots of the observed phenomena
are not matter, but the mathematical law and the underlying symmetry.”
Are ideas finally more important than matter? Or does the distinction
between matter and ideas disappear as we attempt to describe the limits
of particle physics? Until today, the answer to this question is undecided;
it is even unclear whether this is a valid question at all.
Our lives are full of change. Everything is in flux, nothing stays as it
was. But this is not the whole truth. There is a continuity in the world
that allows us to predict objects and occurrences. We scientists discover
that some things stay the same, the laws of nature, for instance. These
laws, however, depend on strange numbers that we call the constants
of nature. Experiments allow us to determine these numbers with ever
increasing precision. But the more precise our answer, the stranger these
numbers seem.
The constants of nature reflect a profound knowledge of the universe.
They characterize our universe. The fact that they exist tells us that, in
other regions of our universe, laws similar to our own apply. But other
universes could exist and could have different natural constants. Some
physicists don’t even speak of “the” universe anymore, because it may
not exist in such a form. They speak of a whole collection of universes,
called the multiverse.
xii THE FUNDAMENTAL CONSTANTS
At the same time, these natural constants stand for our knowledge
of the universe and they stand for our ignorance. We do not know how
their values come to be. Introducing values that cannot be derived but
which must be fixed by experimentation is not satisfactory. Scientists
cannot be reconciled to this situation.
Are the values of the natural constants a consequence of some hitherto
unknown natural laws, or are they merely a random product of the Big
Bang? We do not know. To date, no one has been able to explain the
value of any of the natural constants. This is one of the mysteries of
science, perhaps the biggest mystery in our world.
Life in the universe is only possible for certain very special values of
these natural constants. Why do the natural constants take precisely these
values here in our universe? We do not know. What is clear, however, is
that if they had other values, life could not exist and there would be no
one around to ask the question.
The problem of the natural constants arose due to the increased
precision in determining their values that particle physicists have been
capable of since the 1970s. That is why the discussions in this book
center around particle physics. Quantum mechanics is also discussed
along with questions concerning the Big Bang and astrophysics.
It was the Scottish physicist James Clerk Maxwell who first
suggested basing certain standards on microscopic objects that are to be
found everywhere, such as molecules. Up to that point, standards had
been set using objects specially designed for the purpose. Maxwell was
impressed by the fact that hydrogen molecules, for example, are the same
everywhere, unlike large bodies which have their own particularities. As
president of the British Association for the Advancement of Science,
a society formed in the 19th century and modeled on the Gesellschaft
Deutscher Naturforscher und Ärzte (a society of German scientists and
medical doctors), Maxwell wrote: “If, then, we wish to obtain standards
of length, time, and mass which shall be absolutely permanent, we must
seek them not in the dimensions, or the motion, or the mass of our planet,
PREFACE xiii >>>
but in the wavelength, the period of vibration, and the absolute mass of
these imperishable and unalterable and perfectly similar molecules.”
And indeed these ideas of Maxwell are now followed. Our measure
of length, for instance, is set by the wavelength of light emitted by atoms
of krypton-86. Time is measured by caesium clocks using an atomic
transition of caesium.
A large portion of this book deals with questions related to the natural
constants that have been introduced in today’s Standard Model, or rather,
that have had to be introduced in this model. These constants define, to
a large degree, the structure of our world. This gives me the opportunity
to report on some questions on which I worked with Murray Gell-Mann
at the California Institute of Technology in the ’70s.
Towards the end of the book, a different question is addressed: are the
natural constants really constant? Or are they, however minimally, time
dependent? By examining the light emitted by distant quasars, which
takes billions of years to reach us, it is possible to study the natural
constants in the past. People have found a small yet measurable time
dependency of the fine structure constant. Should these measurements
truly be correct, the consequences are not yet foreseen. The natural
constants could also be minimally dependent on the location in space.
That means that the natural constants could have different values in
different parts of the universe.
The natural constants confront us with one of the most profound
riddles of our universe. Where do they come from? Are they really
absolutely constant? Are they the same everywhere? Are they dependent
on one another? At this point in time, we can give no definitive answers
to these questions. Albert Einstein believed the natural constants to be
fixed by the interactions, thus excluding any freedom. To date, however,
we have not seen any way of verifying this. Presumably there is freedom
in the choice of the constants. It may take hundreds of years to be sure.
At the beginning of the new millennium, particle physics is
faced with new challenges, and in all probability physics stands at
xiv THE FUNDAMENTAL CONSTANTS
Murray Gell-Mann (right) and the author Harald Fritzsch (left) in Berlin in 1995.
the threshold of new and important discoveries. In the 20th century,
physicists delved ever deeper into the inner workings of matter. New,
previously unknown worlds were accessed, new horizons opened up.
The structure of the microphysical world became visible. This complex
world can be described by a surprisingly simple theory, a theory that can
be formulated mathematically.
The questions to be answered become ever more fundamental.
Where does matter come from? What happens to it in the distant future?
Where do the natural constants come from? Particle physics remains
a great adventure. When a new experiment begins, usually after years
of preparation, the physicists involved start on a journey into terra
incognita.
The abyss between the world of particle physics and the everyday
world has become huge. This book aims to reduce this abyss. Why
should one set oneself the goal of exploring particles and phenomena
that have practically nothing to do with our everyday life? The reasons
PREFACE xv >>>
are the same as those that drive scientists to explore outer space, to push
ever further into the depths of our oceans, or to overcome other frontiers.
As with all fundamental research, particle physics is part of our culture,
part of our effort to gain a rational understanding of the cosmic order.
Fundamental research in physics is, to a large extent, particle physics.
The considerable investments made in this field have played a big role
in the development of the open and enlightened societies that we find in
most parts of the world today. The fascinating insights into the world of
the microcosmos that particle physics has allowed us can be counted as
one of the lasting achievements of the last century.
One of the essential goals of this book is to inform the general public,
not primarily physicists, about the problem of the natural constants.
To this end, I use a format that has been used with success before: an
imagined dialogue between real and fictitious persons. In this book the
persons are Isaac Newton, Albert Einstein, and a modern-day physicist
named Adrian Haller, who comes from the University of Bern and is
serving as a guest professor at the California Institute of Technology
(Caltech) in Pasadena.
The dialogue form has been known since Plato’s dialogues in ancient
Greece. The famous dialogue Timaeus, for example, features the three
figures of Critias, Socrates, and Timaeus. Another example is Galileo
Galilee’s dialogues published in the famous Discorsi from the Middle
Ages. The format is useful because the reader is often confronted with
questions that he may have wanted to ask, and these questions are then
answered. The discussions in this book take place in California, in places
where I was active years ago.
I wish the reader pleasure, new insights and success in understanding
the problem of the fundamental constants.

SUPERSYMMETRY AND COSMOLOGY By Jonathan L. Feng free download

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Contents
1 Introduction 3
2 Supersymmetry Essentials 4
2.1 A New Spacetime Symmetry . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Supersymmetry and the Weak Scale . . . . . . . . . . . . . . . . . . . 5
2.3 The Neutral Supersymmetric Spectrum . . . . . . . . . . . . . . . . . . 7
2.4 R-Parity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.5 Supersymmetry Breaking and Dark Energy . . . . . . . . . . . . . . . 9
2.6 Minimal Supergravity . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3 Neutralino Cosmology 15
3.1 Freeze Out and WIMPs . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2 Thermal Relic Density . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2.1 Bulk Region . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2.2 Focus Point Region . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2.3 A Funnel Region . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2.4 Co-annihilation Region . . . . . . . . . . . . . . . . . . . . . . 22
3.3 Direct Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.4 Indirect Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.4.1 Positrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.4.2 Photons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.4.3 Neutrinos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4 Gravitino Cosmology 34
4.1 Gravitino Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.2 Thermal Relic Density . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.3 Production during Reheating . . . . . . . . . . . . . . . . . . . . . . . 38
4.4 Production from Late Decays . . . . . . . . . . . . . . . . . . . . . . . 39
4.5 Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.5.1 Energy Release . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.5.2 Big Bang Nucleosynthesis . . . . . . . . . . . . . . . . . . . . 43
4.5.3 The Cosmic Microwave Background . . . . . . . . . . . . . . . 47
2
4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5 Prospects 49
5.1 The Particle Physics/Cosmology Interface . . . . . . . . . . . . . . . . 49
5.2 The Role of Colliders . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.3 Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
6 Acknowledgments 55
References 55

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Lecture Notes in Physics, Stochastic Processes in Physics, Chemistry, and Biology by Jan A. Freund Thorsten Poschel free download

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QUANTUM MECHANICS A Conceptual Approach By HENDRIK F. HAMEKA free download

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CONTENTS
Preface xi
1 The Discovery of Quantum Mechanics 1
I Introduction, 1
II Planck and Quantization, 3
III Bohr and the Hydrogen Atom, 7
IV Matrix Mechanics, 11
V The Uncertainty Relations, 13
VI Wave Mechanics, 14
VII The Final Touches of Quantum Mechanics, 20
VIII Concluding Remarks, 22
2 The Mathematics of Quantum Mechanics 23
I Introduction, 23
II Differential Equations, 24
III Kummer’s Function, 25
IV Matrices, 27
V Permutations, 30
VI Determinants, 31
vii
VII Properties of Determinants, 32
VIII Linear Equations and Eigenvalues, 35
IX Problems, 37
3 Classical Mechanics 39
I Introduction, 39
II Vectors and Vector Fields, 40
III Hamiltonian Mechanics, 43
IV The Classical Harmonic Oscillator, 44
V Angular Momentum, 45
VI Polar Coordinates, 49
VII Problems, 51
4 Wave Mechanics of a Free Particle 52
I Introduction, 52
II The Mathematics of Plane Waves, 53
III The Schro¨dinger Equation of a Free Particle, 54
IV The Interpretation of the Wave Function, 56
V Wave Packets, 58
VI Concluding Remarks, 62
VII Problems, 63
5 The Schro¨dinger Equation 64
I Introduction, 64
II Operators, 66
III The Particle in a Box, 68
IV Concluding Remarks, 71
V Problems, 72
6 Applications 73
I Introduction, 73
II A Particle in a Finite Box, 74
viii CONTENTS
III Tunneling, 78
IV The Harmonic Oscillator, 81
V Problems, 87
7 Angular Momentum 88
I Introduction, 88
II Commuting Operators, 89
III Commutation Relations of the Angular Momentum, 90
IV The Rigid Rotor, 91
V Eigenfunctions of the Angular Momentum, 93
VI Concluding Remarks, 96
VII Problems, 96
8 The Hydrogen Atom 98
I Introduction, 98
II Solving the Schro¨dinger Equation, 99
III Deriving the Energy Eigenvalues, 101
IV The Behavior of the Eigenfunctions, 103
V Problems, 106
9 Approximate Methods 108
I Introduction, 108
II The Variational Principle, 109
III Applications of the Variational Principle, 111
IV Perturbation Theory for a Nondegenerate State, 113
V The Stark Effect of the Hydrogen Atom, 116
VI Perturbation Theory for Degenerate States, 119
VII Concluding Remarks, 120
VIII Problems, 120
10 The Helium Atom 122
I Introduction, 122
CONTENTS ix
II Experimental Developments, 123
III Pauli’s Exclusion Principle, 126
IV The Discovery of the Electron Spin, 127
V The Mathematical Description of the Electron Spin, 129
VI The Exclusion Principle Revisited, 132
VII Two-Electron Systems, 133
VIII The Helium Atom, 135
IX The Helium Atom Orbitals, 138
X Concluding Remarks, 139
XI Problems, 140
11 Atomic Structure 142
I Introduction, 142
II Atomic and Molecular Wave Function, 145
III The Hartree-Fock Method, 146
IV Slater Orbitals, 152
V Multiplet Theory, 154
VI Concluding Remarks, 158
VII Problems, 158
12 Molecular Structure 160
I Introduction, 160
II The Born-Oppenheimer Approximation, 161
III Nuclear Motion of Diatomic Molecules, 164
IV The Hydrogen Molecular Ion, 169
V The Hydrogen Molecule, 173
VI The Chemical Bond, 176
VII The Structures of Some Simple Polyatomic Molecules, 179
VIII The Hu¨ckel Molecular Orbital Method, 183
IX Problems, 189

PREFACE
The physical laws and mathematical structure that constitute the basis of quantum
mechanics were derived by physicists, but subsequent applications became of interest
not just to the physicists but also to chemists, biologists, medical scientists,
engineers, and philosophers. Quantum mechanical descriptions of atomic and molecular
structure are now taught in freshman chemistry and even in some high school
chemistry courses. Sophisticated computer programs are routinely used for predicting
the structures and geometries of large organic molecules or for the indentification
and evaluation of new medicinal drugs. Engineers have incorporated the
quantum mechanical tunneling effect into the design of new electronic devices,
and philosophers have studied the consequences of some of the novel concepts
of quantum mechanics. They have also compared the relative merits of different
axiomatic approaches to the subject.
In view of the widespread applications of quantum mechanics to these areas
there are now many people who want to learn more about the subject. They may,
of course, try to read one of the many quantum textbooks that have been written,
but almost all of these textbooks assume that their readers have an extensive background
in physics and mathematics; very few of these books make an effort to
explain the subject in simple non-mathematical terms.
In this book we try to present the fundamentals and some simple applications of
quantum mechanics by emphasizing the basic concepts and by keeping the mathematics
as simple as possible. We do assume that the reader is familiar with elementary
calculus; it is after all not possible to explain the Scho¨dinger equation to
someone who does not know what a derivative or an integral is. Some of the mathematical
techniques that are essential for understanding quantum mechanics, such as
matrices and determinants, differential equations, Fourier analysis, and so on are
xi
described in a simple manner. We also present some applications to atomic and
molecular structure that constitute the basis of the various molecular structure computer
programs, but we do not attempt to describe the computation techniques in
detail.
Many authors present quantum mechanics by means of the axiomatic approach,
which leads to a rigorous mathematical representation of the subject. However, in
some instances it is not easy for an average reader to even understand the axioms,
let alone the theorems that are derived from them. I have always looked upon quantum
mechanics as a conglomerate of revolutionary new concepts rather than as a
rigid mathematical discipline. I also feel that the reader might get a better understanding
and appreciation of these concepts if the reader is familiar with the background
and the personalities of the scientists who conceived them and with the
reasoning and arguments that led to their conception. Our approach to the presentation
of quantum mechanics may then be called historic or conceptual but is perhaps
best described as pragmatic. Also, the inclusion of some historical background
makes the book more readable.
I did not give a detailed description of the various sources I used in writing the
historical sections of the book because many of the facts that are presented were
derived from multiple sources. Some of the material was derived from personal
conversations with many scientists and from articles in various journals. The
most reliable sources are the original publications where the new quantum mechanical
ideas were first proposed. These are readily available in the scientific literature,
and I was intrigued in reading some of the original papers. I also read various
biographies and autobiographies. I found Moore’s biography of Schro¨edinger, Constance
Reid’s biographies of Hilbert and Courant, Abraham Pais’ reminiscences,
and the autobiographies of Elsasser and Casimir particularly interesting. I should
mention that Kramers was the professor of theoretical physics when I was a student
at Leiden University. He died before I finished my studies and I never worked under
his supervision, but I did learn quantum mechanics by reading his book and by
attending his lectures.
Finally I wish to express my thanks to Mrs. Alice Chen for her valuable help in
typing and preparing the manuscript.
HENDRIK F. HAMEKA

Quantum Mechanics By Professor John W. Norbury free download

2011-11-19T09:48:00.000-08:00

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Contents
1 WAVE FUNCTION 7
1.1 Probability Theory . . . . . . . . . . . . . . . . . . . . . . . . 8
1.1.1 Mean, Average, Expectation Value . . . . . . . . . . . 8
1.1.2 Average of a Function . . . . . . . . . . . . . . . . . . 10
1.1.3 Mean, Median, Mode . . . . . . . . . . . . . . . . . . 10
1.1.4 Standard Deviation and Uncertainty . . . . . . . . . . 11
1.1.5 Probability Density . . . . . . . . . . . . . . . . . . . 14
1.2 Postulates of Quantum Mechanics . . . . . . . . . . . . . . . 14
1.3 Conservation of Probability (Continuity Equation) . . . . . . 19
1.3.1 Conservation of Charge . . . . . . . . . . . . . . . . . 19
1.3.2 Conservation of Probability . . . . . . . . . . . . . . . 22
1.4 Interpretation of the Wave Function . . . . . . . . . . . . . . 23
1.5 Expectation Value in Quantum Mechanics . . . . . . . . . . . 24
1.6 Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.7 Commutation Relations . . . . . . . . . . . . . . . . . . . . . 27
1.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.9 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2 DIFFERENTIAL EQUATIONS 35
2.1 Ordinary Di®erential Equations . . . . . . . . . . . . . . . . . 35
2.1.1 Second Order, Homogeneous, Linear, Ordinary Di®erential
Equations with Constant Coe±cients . . . . . . 36
2.1.2 Inhomogeneous Equation . . . . . . . . . . . . . . . . 39
2.2 Partial Di®erential Equations . . . . . . . . . . . . . . . . . . 42
2.3 Properties of Separable Solutions . . . . . . . . . . . . . . . . 44
2.3.1 General Solutions . . . . . . . . . . . . . . . . . . . . . 44
2.3.2 Stationary States . . . . . . . . . . . . . . . . . . . . . 44
2.3.3 De¯nite Total Energy . . . . . . . . . . . . . . . . . . 45
1
2 CONTENTS
2.3.4 Alternating Parity . . . . . . . . . . . . . . . . . . . . 46
2.3.5 Nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.3.6 Complete Orthonormal Sets of Functions . . . . . . . 46
2.3.7 Time-dependent Coe±cients . . . . . . . . . . . . . . 49
2.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.5 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3 INFINITE 1-DIMENSIONAL BOX 53
3.1 Energy Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.2 Wave Function . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.4 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4 POSTULATES OF QUANTUM MECHANICS 65
4.1 Mathematical Preliminaries . . . . . . . . . . . . . . . . . . . 65
4.1.1 Hermitian Operators . . . . . . . . . . . . . . . . . . . 65
4.1.2 Eigenvalue Equations . . . . . . . . . . . . . . . . . . 66
4.2 Postulate 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.3 Expansion Postulate . . . . . . . . . . . . . . . . . . . . . . . 68
4.4 Measurement Postulate . . . . . . . . . . . . . . . . . . . . . 69
4.5 Reduction Postulate . . . . . . . . . . . . . . . . . . . . . . . 70
4.6 Summary of Postulates of Quantum Mechanics (Simple Version) 71
4.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.8 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
I 1-DIMENSIONAL PROBLEMS 77
5 Bound States 79
5.1 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 80
5.2 Finite 1-dimensional Well . . . . . . . . . . . . . . . . . . . . 81
5.2.1 Regions I and III With Real Wave Number . . . . . . 82
5.2.2 Region II . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.2.3 Matching Boundary Conditions . . . . . . . . . . . . . 84
5.2.4 Energy Levels . . . . . . . . . . . . . . . . . . . . . . . 87
5.2.5 Strong and Weak Potentials . . . . . . . . . . . . . . . 88
5.3 Power Series Solution of ODEs . . . . . . . . . . . . . . . . . 89
5.3.1 Use of Recurrence Relation . . . . . . . . . . . . . . . 91
5.4 Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . 92
CONTENTS 3
5.5 Algebraic Solution for Harmonic Oscillator . . . . . . . . . . 100
5.5.1 Further Algebraic Results for Harmonic Oscillator . . 108
6 SCATTERING STATES 113
6.1 Free Particle . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.1.1 Group Velocity and Phase Velocity . . . . . . . . . . . 117
6.2 Transmission and Re°ection . . . . . . . . . . . . . . . . . . . 119
6.2.1 Alternative Approach . . . . . . . . . . . . . . . . . . 120
6.3 Step Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.4 Finite Potential Barrier . . . . . . . . . . . . . . . . . . . . . 124
6.5 Quantum Description of a Colliding Particle . . . . . . . . . . 126
6.5.1 Expansion Coe±cients . . . . . . . . . . . . . . . . . . 128
6.5.2 Time Dependence . . . . . . . . . . . . . . . . . . . . 129
6.5.3 Moving Particle . . . . . . . . . . . . . . . . . . . . . . 130
6.5.4 Wave Packet Uncertainty . . . . . . . . . . . . . . . . 131
7 FEW-BODY BOUND STATE PROBLEM 133
7.1 2-Body Problem . . . . . . . . . . . . . . . . . . . . . . . . . 133
7.1.1 Classical 2-Body Problem . . . . . . . . . . . . . . . . 134
7.1.2 Quantum 2-Body Problem . . . . . . . . . . . . . . . . 137
7.2 3-Body Problem . . . . . . . . . . . . . . . . . . . . . . . . . 139
II 3-DIMENSIONAL PROBLEMS 141
8 3-DIMENSIONAL SCHRÄODINGER EQUATION 143
8.1 Angular Equations . . . . . . . . . . . . . . . . . . . . . . . . 144
8.2 Radial Equation . . . . . . . . . . . . . . . . . . . . . . . . . 147
8.3 Bessel's Di®erential Equation . . . . . . . . . . . . . . . . . . 148
8.3.1 Hankel Functions . . . . . . . . . . . . . . . . . . . . . 150
9 HYDROGEN-LIKE ATOMS 153
9.1 Laguerre Associated Di®erential Equation . . . . . . . . . . . 153
9.2 Degeneracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
10 ANGULAR MOMENTUM 159
10.1 Orbital Angular Momentum . . . . . . . . . . . . . . . . . . . 159
10.1.1 Uncertainty Principle . . . . . . . . . . . . . . . . . . 162
10.2 Zeeman E®ect . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
10.3 Algebraic Method . . . . . . . . . . . . . . . . . . . . . . . . . 164
4 CONTENTS
10.4 Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
10.4.1 Spin 1
2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
10.4.2 Spin-Orbit Coupling . . . . . . . . . . . . . . . . . . . 167
10.5 Addition of Angular Momentum . . . . . . . . . . . . . . . . 169
10.5.1 Wave Functions for Singlet and Triplet Spin States . . 171
10.5.2 Clebsch-Gordon Coe±cients . . . . . . . . . . . . . . . 172
10.6 Total Angular Momentum . . . . . . . . . . . . . . . . . . . . 172
10.6.1 LS and jj Coupling . . . . . . . . . . . . . . . . . . . 173
11 SHELL MODELS 177
11.1 Atomic Shell Model . . . . . . . . . . . . . . . . . . . . . . . 177
11.1.1 Degenerate Shell Model . . . . . . . . . . . . . . . . . 177
11.1.2 Non-Degenerate Shell Model . . . . . . . . . . . . . . 178
11.1.3 Non-Degenerate Model with Surface E®ects . . . . . . 178
11.1.4 Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . 179
11.2 Hartree-Fock Self Consistent Field Method . . . . . . . . . . 180
11.3 Nuclear Shell Model . . . . . . . . . . . . . . . . . . . . . . . 181
11.3.1 Nuclear Spin . . . . . . . . . . . . . . . . . . . . . . . 181
11.4 Quark Shell Model . . . . . . . . . . . . . . . . . . . . . . . . 182
12 DIRAC NOTATION 183
12.1 Finite Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . 183
12.1.1 Real Vector Space . . . . . . . . . . . . . . . . . . . . 183
12.1.2 Complex Vector Space . . . . . . . . . . . . . . . . . . 185
12.1.3 Matrix Representation of Vectors . . . . . . . . . . . . 188
12.1.4 One-Forms . . . . . . . . . . . . . . . . . . . . . . . . 188
12.2 In¯nite Vector Spaces . . . . . . . . . . . . . . . . . . . . . . 189
12.3 Operators and Matrices . . . . . . . . . . . . . . . . . . . . . 191
12.3.1 Matrix Elements . . . . . . . . . . . . . . . . . . . . . 191
12.3.2 Hermitian Conjugate . . . . . . . . . . . . . . . . . . . 194
12.3.3 Hermitian Operators . . . . . . . . . . . . . . . . . . . 195
12.3.4 Expectation Values and Transition Amplitudes . . . . 197
12.4 Postulates of Quantum Mechanics (Fancy Version) . . . . . . 198
12.5 Uncertainty Principle . . . . . . . . . . . . . . . . . . . . . . . 198
13 TIME-INDEPENDENT PERTURBATION THEORY, HYDROGEN
ATOM, POSITRONIUM, STRUCTURE OF HADRONS201
13.1 Non-degenerate Perturbation Theory . . . . . . . . . . . . . . 204
13.2 Degenerate Perturbation Theory . . . . . . . . . . . . . . . . 208
CONTENTS 5
13.2.1 Two-fold Degeneracy . . . . . . . . . . . . . . . . . . . 209
13.2.2 Another Approach . . . . . . . . . . . . . . . . . . . . 211
13.2.3 Higher Order Degeneracies . . . . . . . . . . . . . . . 212
13.3 Fine Structure of Hydrogen . . . . . . . . . . . . . . . . . . . 212
13.3.1 1-Body Relativistic Correction . . . . . . . . . . . . . 212
13.3.2 Two-Body Relativistic Correction . . . . . . . . . . . 216
13.3.3 Spin-Orbit Coupling . . . . . . . . . . . . . . . . . . . 217
13.4 Zeeman e®ect . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
13.5 Stark e®ect . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
13.6 Hyper¯ne splitting . . . . . . . . . . . . . . . . . . . . . . . . 221
13.7 Lamb shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
13.8 Positronium and Muonium . . . . . . . . . . . . . . . . . . . 221
13.9 Quark Model of Hadrons . . . . . . . . . . . . . . . . . . . . . 221
14 VARIATIONAL PRINCIPLE, HELIUM ATOM, MOLECULES223
14.1 Variational Principle . . . . . . . . . . . . . . . . . . . . . . . 223
14.2 Helium Atom . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
14.3 Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
15 WKB APPROXIMATION, NUCLEAR ALPHA DECAY 225
15.1 Generalized Wave Functions . . . . . . . . . . . . . . . . . . . 225
15.2 Finite Potential Barrier . . . . . . . . . . . . . . . . . . . . . 230
15.3 Gamow's Theory of Alpha Decay . . . . . . . . . . . . . . . . 231
16 TIME-DEPENDENT PERTURBATION THEORY, LASERS235
16.1 Equivalent SchrÄodinger Equation . . . . . . . . . . . . . . . . 236
16.2 Dyson Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 240
16.3 Constant Perturbation . . . . . . . . . . . . . . . . . . . . . . 241
16.4 Harmonic Perturbation . . . . . . . . . . . . . . . . . . . . . . 244
16.5 Photon Absorption . . . . . . . . . . . . . . . . . . . . . . . . 247
16.5.1 Radiation Bath . . . . . . . . . . . . . . . . . . . . . . 247
16.6 Photon Emission . . . . . . . . . . . . . . . . . . . . . . . . . 249
16.7 Selection Rules . . . . . . . . . . . . . . . . . . . . . . . . . . 249
16.8 Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
17 SCATTERING, NUCLEAR REACTIONS 251
17.1 Cross Section . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
17.2 Scattering Amplitude . . . . . . . . . . . . . . . . . . . . . . . 252
17.2.1 Calculation of cl . . . . . . . . . . . . . . . . . . . . . 255
6 CONTENTS
17.3 Phase Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
17.4 Integral Scattering Theory . . . . . . . . . . . . . . . . . . . . 259
17.4.1 Lippman-Schwinger Equation . . . . . . . . . . . . . . 259
17.4.2 Scattering Amplitude . . . . . . . . . . . . . . . . . . 261
17.4.3 Born Approximation . . . . . . . . . . . . . . . . . . . 262
17.5 Nuclear Reactions . . . . . . . . . . . . . . . . . . . . . . . . 264
18 SOLIDS AND QUANTUM STATISTICS 265
18.1 Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
18.2 Quantum Statistics . . . . . . . . . . . . . . . . . . . . . . . . 265
19 SUPERCONDUCTIVITY 267
20 ELEMENTARY PARTICLES 269
21 chapter 1 problems 271
21.1 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
21.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
21.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
22 chapter 2 problems 281
22.1 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
22.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
22.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
23 chapter 3 problems 287
23.1 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
23.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
23.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
24 chapter 4 problems 291
24.1 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
24.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
24.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

Quantum Gravity By Carol Rovelli free download

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Entropy and Partial Differential Equations By Lawrence C. Evans free download

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CONTENTS
Introduction
A. Overview
B. Themes
I. Entropy and equilibrium
A. Thermal systems in equilibrium
B. Examples
1. Simple fluids
2. Other examples
C. Physical interpretations of the model
1. Equilibrium
2. Positivity of temperature
3. Extensive and intensive parameters
4. Concavity of S
5. Convexity of E
6. Entropy maximization, energy minimization
D. Thermodynamic potentials
1. Review of Legendre transform
2. Definitions
3. Maxwell relations
E. Capacities
F. More examples
1. Ideal gas
2. Van der Waals fluid
II. Entropy and irreversibility
A. A model material
1. Definitions
2. Energy and entropy
a. Working and heating
b. First Law, existence of E
c. Carnot cycles
d. Second Law
e. Existence of S
3. Efficiency of cycles
4. Adding dissipation, Clausius inequality
B. Some general theories
1. Entropy and efficiency
1
a. Definitions
b. Existence of S
2. Entropy, temperature and separating hyperplanes
a. Definitions
b. Second Law
c. Hahn–Banach Theorem
d. Existence of S, T
III. Continuum thermodynamics
A. Kinematics
1. Definitions
2. Physical quantities
3. Kinematic formulas
4. Deformation gradient
B. Conservation laws, Clausius–Duhem inequality
C. Constitutive relations
1. Fluids
2. Elastic materials
D. Workless dissipation
IV. Elliptic and parabolic equations
A. Entropy and elliptic equations
1. Definitions
2. Estimates for equilibrium entropy production
a. A capacity estimate
b. A pointwise bound
3. Harnack’s inequality
B. Entropy and parabolic equations
1. Definitions
2. Evolution of entropy
a. Entropy increase
b. Second derivatives in time
c. A differential form of Harnack’s inequality
3. Clausius inequality
a. Cycles
b. Heating
c. Almost reversible cycles
V. Conservation laws and kinetic equations
A. Some physical PDE
2
1. Compressible Euler equations
a. Equations of state
b. Conservation law form
2. Boltzmann’s equation
a. A model for dilute gases
b. H-Theorem
c. H and entropy
B. Single conservation law
1. Integral solutions
2. Entropy solutions
3. Condition E
4. Kinetic formulation
5. A hydrodynamical limit
C. Systems of conservation laws
1. Entropy conditions
2. Compressible Euler equations in one dimension
a. Computing entropy/entropy flux pairs
b. Kinetic formulation
VI. Hamilton–Jacobi and related equations
A. Viscosity solutions
B. Hopf–Lax formula
C. A diffusion limit
1. Formulation
2. Construction of diffusion coefficients
3. Passing to limits
VII. Entropy and uncertainty
A. Maxwell’s demon
B. Maximum entropy
1. A probabilistic model
2. Uncertainty
3. Maximizing uncertainty
C. Statistical mechanics
1. Microcanonical distribution
2. Canonical distribution
3. Thermodynamics
VIII. Probability and differential equations
A. Continuous time Markov chains
3
1. Generators and semigroups
2. Entropy production
3. Convergence to equilibrium
B. Large deviations
1. Thermodynamic limits
2. Basic theory
a. Rate functions
b. Asymptotic evaluation of integrals
C. Cramer’s Theorem
D. Small noise in dynamical systems
1. Stochastic differential equations
2. Itˆo’s formula, elliptic PDE
3. An exit problem
a. Small noise asymptotics
b. Perturbations against the flow
Appendices:
A. Units and constants
B. Physical axioms
References

Electromagnetic Fields and Waves Including Electric Circuits 3rd ed. By Paul Lorrain free download

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Differential Geometry, Analysis and Physics By Jeffrey M. Lee free download

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Contents
0.1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
1 Preliminaries and Local Theory 1
1.1 Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Chain Rule, Product rule and Taylor’s Theorem . . . . . . . . . 11
1.3 Local theory of maps . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 Differentiable Manifolds 15
2.1 Rough Ideas I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Topological Manifolds . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3 Differentiable Manifolds and Differentiable Maps . . . . . . . . . 17
2.4 Pseudo-Groups and Models Spaces . . . . . . . . . . . . . . . . . 22
2.5 Smooth Maps and Diffeomorphisms . . . . . . . . . . . . . . . . 27
2.6 Coverings and Discrete groups . . . . . . . . . . . . . . . . . . . 30
2.6.1 Covering spaces and the fundamental group . . . . . . . . 30
2.6.2 Discrete Group Actions . . . . . . . . . . . . . . . . . . . 36
2.7 Grassmannian manifolds . . . . . . . . . . . . . . . . . . . . . . . 39
2.8 Partitions of Unity . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.9 Manifolds with boundary. . . . . . . . . . . . . . . . . . . . . . . 43
3 The Tangent Structure 47
3.1 Rough Ideas II . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.2 Tangent Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.3 Interpretations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.4 The Tangent Map . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.5 The Tangent and Cotangent Bundles . . . . . . . . . . . . . . . . 55
3.5.1 Tangent Bundle . . . . . . . . . . . . . . . . . . . . . . . . 55
3.5.2 The Cotangent Bundle . . . . . . . . . . . . . . . . . . . . 57
3.6 Important Special Situations. . . . . . . . . . . . . . . . . . . . . 59
4 Submanifold, Immersion and Submersion. 63
4.1 Submanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.2 Submanifolds of Rn . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.3 Regular and Critical Points and Values . . . . . . . . . . . . . . . 66
4.4 Immersions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
iii
iv CONTENTS
4.5 Immersed Submanifolds and Initial Submanifolds . . . . . . . . . 71
4.6 Submersions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.7 Morse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.8 Problem set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5 Lie Groups I 81
5.1 Definitions and Examples . . . . . . . . . . . . . . . . . . . . . . 81
5.2 Lie Group Homomorphisms . . . . . . . . . . . . . . . . . . . . . 84
6 Fiber Bundles and Vector Bundles I 87
6.1 Transitions Maps and Structure . . . . . . . . . . . . . . . . . . . 94
6.2 Useful ways to think about vector bundles . . . . . . . . . . . . . 94
6.3 Sections of a Vector Bundle . . . . . . . . . . . . . . . . . . . . . 97
6.4 Sheaves,Germs and Jets . . . . . . . . . . . . . . . . . . . . . . . 98
6.5 Jets and Jet bundles . . . . . . . . . . . . . . . . . . . . . . . . . 102
7 Vector Fields and 1-Forms 105
7.1 Definition of vector fields and 1-forms . . . . . . . . . . . . . . . 105
7.2 Pull back and push forward of functions and 1-forms . . . . . . . 106
7.3 Frame Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
7.4 Lie Bracket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
7.5 Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
7.6 Action by pullback and push-forward . . . . . . . . . . . . . . . . 112
7.7 Flows and Vector Fields . . . . . . . . . . . . . . . . . . . . . . . 114
7.8 Lie Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
7.9 Time Dependent Fields . . . . . . . . . . . . . . . . . . . . . . . 123
8 Lie Groups II 125
8.1 Spinors and rotation . . . . . . . . . . . . . . . . . . . . . . . . . 133
9 Multilinear Bundles and Tensors Fields 137
9.1 Multilinear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 137
9.1.1 Contraction of tensors . . . . . . . . . . . . . . . . . . . . 141
9.1.2 Alternating Multilinear Algebra . . . . . . . . . . . . . . . 142
9.1.3 Orientation on vector spaces . . . . . . . . . . . . . . . . 146
9.2 Multilinear Bundles . . . . . . . . . . . . . . . . . . . . . . . . . 147
9.3 Tensor Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
9.4 Tensor Derivations . . . . . . . . . . . . . . . . . . . . . . . . . . 149
10 Differential forms 153
10.1 Pullback of a differential form. . . . . . . . . . . . . . . . . . . . 155
10.2 Exterior Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . 156
10.3 Maxwell’s equations. . . . . . . . . . . . . . . . . . . . . . . . . 159
10.4 Lie derivative, interior product and exterior derivative. . . . . . . 161
10.5 Time Dependent Fields (Part II) . . . . . . . . . . . . . . . . . . 163
10.6 Vector valued and algebra valued forms. . . . . . . . . . . . . . . 163
CONTENTS v
10.7 Global Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . 165
10.8 Orientation of manifolds with boundary . . . . . . . . . . . . . . 167
10.9 Integration of Differential Forms. . . . . . . . . . . . . . . . . . . 168
10.10Stokes’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
10.11Vector Bundle Valued Forms. . . . . . . . . . . . . . . . . . . . . 172
11 Distributions and Frobenius’ Theorem 175
11.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
11.2 Integrability of Regular Distributions . . . . . . . . . . . . . . . 175
11.3 The local version Frobenius’ theorem . . . . . . . . . . . . . . . . 177
11.4 Foliations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
11.5 The Global Frobenius Theorem . . . . . . . . . . . . . . . . . . . 183
11.6 Singular Distributions . . . . . . . . . . . . . . . . . . . . . . . . 185
12 Connections on Vector Bundles 189
12.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
12.2 Local Frame Fields and Connection Forms . . . . . . . . . . . . . 191
12.3 Parallel Transport . . . . . . . . . . . . . . . . . . . . . . . . . . 193
12.4 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
13 Riemannian and semi-Riemannian Manifolds 201
13.1 The Linear Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 201
13.1.1 Scalar Products . . . . . . . . . . . . . . . . . . . . . . . 201
13.1.2 Natural Extensions and the Star Operator . . . . . . . . . 203
13.2 Surface Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
13.3 Riemannian and semi-Riemannian Metrics . . . . . . . . . . . . . 214
13.4 The Riemannian case (positive definite metric) . . . . . . . . . . 220
13.5 Levi-Civita Connection . . . . . . . . . . . . . . . . . . . . . . . . 221
13.6 Covariant differentiation of vector fields along maps. . . . . . . . 228
13.7 Covariant differentiation of tensor fields . . . . . . . . . . . . . . 229
13.8 Comparing the Differential Operators . . . . . . . . . . . . . . . 230
14 Formalisms for Calculation 233
14.1 Tensor Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
14.2 Covariant Exterior Calculus, Bundle-Valued Forms . . . . . . . . 234
15 Topology 235
15.1 Attaching Spaces and Quotient Topology . . . . . . . . . . . . . 235
15.2 Topological Sum . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
15.3 Homotopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
15.4 Cell Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
16 Algebraic Topology 245
16.1 Axioms for a Homology Theory . . . . . . . . . . . . . . . . . . . 245
16.2 Simplicial Homology . . . . . . . . . . . . . . . . . . . . . . . . . 246
16.3 Singular Homology . . . . . . . . . . . . . . . . . . . . . . . . . . 246
vi CONTENTS
16.4 Cellular Homology . . . . . . . . . . . . . . . . . . . . . . . . . . 246
16.5 Universal Coefficient theorem . . . . . . . . . . . . . . . . . . . . 246
16.6 Axioms for a Cohomology Theory . . . . . . . . . . . . . . . . . 246
16.7 De Rham Cohomology . . . . . . . . . . . . . . . . . . . . . . . . 246
16.8 Topology of Vector Bundles . . . . . . . . . . . . . . . . . . . . . 246
16.9 de Rham Cohomology . . . . . . . . . . . . . . . . . . . . . . . . 248
16.10The Meyer Vietoris Sequence . . . . . . . . . . . . . . . . . . . . 252
16.11Sheaf Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . 253
16.12Characteristic Classes . . . . . . . . . . . . . . . . . . . . . . . . 253
17 Lie Groups and Lie Algebras 255
17.1 Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
17.2 Classical complex Lie algebras . . . . . . . . . . . . . . . . . . . . 257
17.2.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . 258
17.3 The Adjoint Representation . . . . . . . . . . . . . . . . . . . . . 259
17.4 The Universal Enveloping Algebra . . . . . . . . . . . . . . . . . 261
17.5 The Adjoint Representation of a Lie group . . . . . . . . . . . . . 265
18 Group Actions and Homogenous Spaces 271
18.1 Our Choices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
18.1.1 Left actions . . . . . . . . . . . . . . . . . . . . . . . . . . 272
18.1.2 Right actions . . . . . . . . . . . . . . . . . . . . . . . . . 273
18.1.3 Equivariance . . . . . . . . . . . . . . . . . . . . . . . . . 273
18.1.4 The action of Diff(M) and map-related vector fields. . . 274
18.1.5 Lie derivative for equivariant bundles. . . . . . . . . . . . 274
18.2 Homogeneous Spaces. . . . . . . . . . . . . . . . . . . . . . . . . 275
19 Fiber Bundles and Connections 279
19.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
19.2 Principal and Associated Bundles . . . . . . . . . . . . . . . . . . 282
20 Analysis on Manifolds 285
20.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
20.1.1 Star Operator II . . . . . . . . . . . . . . . . . . . . . . . 285
20.1.2 Divergence, Gradient, Curl . . . . . . . . . . . . . . . . . 286
20.2 The Laplace Operator . . . . . . . . . . . . . . . . . . . . . . . . 286
20.3 Spectral Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 289
20.4 Hodge Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
20.5 Dirac Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
20.5.1 Clifford Algebras . . . . . . . . . . . . . . . . . . . . . . . 291
20.5.2 The Clifford group and Spinor group . . . . . . . . . . . . 296
20.6 The Structure of Clifford Algebras . . . . . . . . . . . . . . . . . 296
20.6.1 Gamma Matrices . . . . . . . . . . . . . . . . . . . . . . . 297
20.7 Clifford Algebra Structure and Representation . . . . . . . . . . 298
20.7.1 Bilinear Forms . . . . . . . . . . . . . . . . . . . . . . . . 298
20.7.2 Hyperbolic Spaces And Witt Decomposition . . . . . . . . 299
CONTENTS vii
20.7.3 Witt’s Decomposition and Clifford Algebras . . . . . . . . 300
20.7.4 The Chirality operator . . . . . . . . . . . . . . . . . . . 301
20.7.5 Spin Bundles and Spin-c Bundles . . . . . . . . . . . . . . 302
20.7.6 Harmonic Spinors . . . . . . . . . . . . . . . . . . . . . . 302
21 Complex Manifolds 303
21.1 Some complex linear algebra . . . . . . . . . . . . . . . . . . . . 303
21.2 Complex structure . . . . . . . . . . . . . . . . . . . . . . . . . . 306
21.3 Complex Tangent Structures . . . . . . . . . . . . . . . . . . . . 309
21.4 The holomorphic tangent map. . . . . . . . . . . . . . . . . . . . 310
21.5 Dual spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310
21.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312
21.7 The holomorphic inverse and implicit functions theorems. . . . . 312
22 Classical Mechanics 315
22.1 Particle motion and Lagrangian Systems . . . . . . . . . . . . . . 315
22.1.1 Basic Variational Formalism for a Lagrangian . . . . . . . 316
22.1.2 Two examples of a Lagrangian . . . . . . . . . . . . . . . 319
22.2 Symmetry, Conservation and Noether’s Theorem . . . . . . . . . 319
22.2.1 Lagrangians with symmetries. . . . . . . . . . . . . . . . . 321
22.2.2 Lie Groups and Left Invariants Lagrangians . . . . . . . . 322
22.3 The Hamiltonian Formalism . . . . . . . . . . . . . . . . . . . . . 322
23 Symplectic Geometry 325
23.1 Symplectic Linear Algebra . . . . . . . . . . . . . . . . . . . . . . 325
23.2 Canonical Form (Linear case) . . . . . . . . . . . . . . . . . . . . 327
23.3 Symplectic manifolds . . . . . . . . . . . . . . . . . . . . . . . . . 327
23.4 Complex Structure and K¨ahler Manifolds . . . . . . . . . . . . . 329
23.5 Symplectic musical isomorphisms . . . . . . . . . . . . . . . . . 332
23.6 Darboux’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 332
23.7 Poisson Brackets and Hamiltonian vector fields . . . . . . . . . . 334
23.8 Configuration space and Phase space . . . . . . . . . . . . . . . 337
23.9 Transfer of symplectic structure to the Tangent bundle . . . . . . 338
23.10Coadjoint Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . 340
23.11The Rigid Body . . . . . . . . . . . . . . . . . . . . . . . . . . . 341
23.11.1 The configuration in R3N . . . . . . . . . . . . . . . . . . 342
23.11.2Modelling the rigid body on SO(3) . . . . . . . . . . . . . 342
23.11.3 The trivial bundle picture . . . . . . . . . . . . . . . . . . 343
23.12The momentum map and Hamiltonian actions . . . . . . . . . . . 343
24 Poisson Geometry 347
24.1 Poisson Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 347
viii CONTENTS
25 Quantization 351
25.1 Operators on a Hilbert Space . . . . . . . . . . . . . . . . . . . . 351
25.2 C*-Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
25.2.1 Matrix Algebras . . . . . . . . . . . . . . . . . . . . . . . 354
25.3 Jordan-Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . 354
26 Appendices 357
26.1 A. Primer for Manifold Theory . . . . . . . . . . . . . . . . . . . 357
26.1.1 Fixing a problem . . . . . . . . . . . . . . . . . . . . . . . 360
26.2 B. Topological Spaces . . . . . . . . . . . . . . . . . . . . . . . . 361
26.2.1 Separation Axioms . . . . . . . . . . . . . . . . . . . . . . 363
26.2.2 Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 364
26.3 C. Topological Vector Spaces . . . . . . . . . . . . . . . . . . . . 365
26.3.1 Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . 367
26.3.2 Orthonormal sets . . . . . . . . . . . . . . . . . . . . . . . 368
26.4 D. Overview of Classical Physics . . . . . . . . . . . . . . . . . . 368
26.4.1 Units of measurement . . . . . . . . . . . . . . . . . . . . 368
26.4.2 Newton’s equations . . . . . . . . . . . . . . . . . . . . . . 369
26.4.3 Classical particle motion in a conservative field . . . . . . 370
26.4.4 Some simple mechanical systems . . . . . . . . . . . . . . 375
26.4.5 The Basic Ideas of Relativity . . . . . . . . . . . . . . . . 380
26.4.6 Variational Analysis of Classical Field Theory . . . . . . . 385
26.4.7 Symmetry and Noether’s theorem for field theory . . . . . 386
26.4.8 Electricity and Magnetism . . . . . . . . . . . . . . . . . . 388
26.4.9 Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . 390
26.5 E. Calculus on Banach Spaces . . . . . . . . . . . . . . . . . . . . 390
26.6 Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391
26.7 Differentiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395
26.8 Chain Rule, Product rule and Taylor’s Theorem . . . . . . . . . 400
26.9 Local theory of maps . . . . . . . . . . . . . . . . . . . . . . . . . 405
26.9.1 Linear case. . . . . . . . . . . . . . . . . . . . . . . . . . 411
26.9.2 Local (nonlinear) case. . . . . . . . . . . . . . . . . . 412
26.10The Tangent Bundle of an Open Subset of a Banach Space . . . 413
26.11Problem Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415
26.11.1 Existence and uniqueness for differential equations . . . . 417
26.11.2 Differential equations depending on a parameter. . . . . . 418
26.12Multilinear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 418
26.12.1Smooth Banach Vector Bundles . . . . . . . . . . . . . . . 435
26.12.2Formulary . . . . . . . . . . . . . . . . . . . . . . . . . . . 441
26.13Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444
26.14Group action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444
26.15Notation and font usage guide . . . . . . . . . . . . . . . . . . . . 445
27 Bibliography 453
0.1. PREFACE ix
0.1 Preface
In this book I present differential geometry and related mathematical topics with
the help of examples from physics. It is well known that there is something
strikingly mathematical about the physical universe as it is conceived of in
the physical sciences. The convergence of physics with mathematics, especially
differential geometry, topology and global analysis is even more pronounced in
the newer quantum theories such as gauge field theory and string theory. The
amount of mathematical sophistication required for a good understanding of
modern physics is astounding. On the other hand, the philosophy of this book
is that mathematics itself is illuminated by physics and physical thinking.
The ideal of a truth that transcends all interpretation is perhaps unattain-
able. Even the two most impressively objective realities, the physical and the
mathematical, are still only approachable through, and are ultimately insepa-
rable from, our normative and linguistic background. And yet it is exactly the
tendency of these two sciences to point beyond themselves to something tran-
scendentally real that so inspires us. Whenever we interpret something real,
whether physical or mathematical, there will be those aspects which arise as
mere artifacts of our current descriptive scheme and those aspects that seem to
be objective realities which are revealed equally well through any of a multitude
of equivalent descriptive schemes-“cognitive inertial frames” as it were. This
theme is played out even within geometry itself where a viewpoint or interpre-
tive scheme translates to the notion of a coordinate system on a differentiable
manifold.
A physicist has no trouble believing that a vector field is something beyond
its representation in any particular coordinate system since the vector field it-
self is something physical. It is the way that the various coordinate descriptions
relate to each other (covariance) that manifests to the understanding the pres-
ence of an invariant physical reality. This seems to be very much how human
perception works and it is interesting that the language of tensors has shown up
in the cognitive science literature.

earn money

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A Brief History of Time By Stephen Hawking free download

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FOREWARD
I didn’t write a foreword to the original edition of A Brief History of Time. That was done by Carl Sagan. Instead,
I wrote a short piece titled “Acknowledgments” in which I was advised to thank everyone. Some of the
foundations that had given me support weren’t too pleased to have been mentioned, however, because it led to
a great increase in applications.
I don’t think anyone, my publishers, my agent, or myself, expected the book to do anything like as well as it did.
It was in the London Sunday Times best-seller list for 237 weeks, longer than any other book (apparently, the
Bible and Shakespeare aren’t counted). It has been translated into something like forty languages and has sold
about one copy for every 750 men, women, and children in the world. As Nathan Myhrvold of Microsoft (a
former post-doc of mine) remarked: I have sold more books on physics than Madonna has on sex.
The success of A Brief History indicates that there is widespread interest in the big questions like: Where did
we come from? And why is the universe the way it is?
I have taken the opportunity to update the book and include new theoretical and observational results obtained
since the book was first published (on April Fools’ Day, 1988). I have included a new chapter on wormholes
and time travel. Einstein’s General Theory of Relativity seems to offer the possibility that we could create and
maintain wormholes, little tubes that connect different regions of space-time. If so, we might be able to use
them for rapid travel around the galaxy or travel back in time. Of course, we have not seen anyone from the
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future (or have we?) but I discuss a possible explanation for this.
I also describe the progress that has been made recently in finding “dualities” or correspondences between
apparently different theories of physics. These correspondences are a strong indication that there is a complete
unified theory of physics, but they also suggest that it may not be possible to express this theory in a single
fundamental formulation. Instead, we may have to use different reflections of the underlying theory in different
situations. It might be like our being unable to represent the surface of the earth on a single map and having to
use different maps in different regions. This would be a revolution in our view of the unification of the laws of
science but it would not change the most important point: that the universe is governed by a set of rational laws
that we can discover and understand.
On the observational side, by far the most important development has been the measurement of fluctuations in
the cosmic microwave background radiation by COBE (the Cosmic Background Explorer satellite) and other
collaborations. These fluctuations are the finger-prints of creation, tiny initial irregularities in the otherwise
smooth and uniform early universe that later grew into galaxies, stars, and all the structures we see around us.
Their form agrees with the predictions of the proposal that the universe has no boundaries or edges in the
imaginary time direction; but further observations will be necessary to distinguish this proposal from other
possible explanations for the fluctuations in the background. However, within a few years we should know
whether we can believe that we live in a universe that is completely self-contained and without beginning or
end.
Stephen Hawking

A Guided Tour of Mathematical Physics By Roel Snieder free download

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Introduction
The topic of this course is the application of mathematics to physical problems In practice
mathematics and physics are taught separately Despite the fact that education in physics
relies on mathematics it turns out that students consider mathematics to be disjoint
from physics Although this point of view may strictly be correct it re ects an erroneous
opinion when it concerns an education in physics or geophysics The reason for this is that
mathematics is the only language at our disposal for quantifying physical processes One
cannot learn a language by just studying a textbook In order to truly learn how to use a
language one has to go abroad and start using a language By the same token one cannot
learn how to use mathematics in physics by just studying textbooks or attending lectures
the only way to achieve this is to venture into the unknown and apply mathematics to
physical problems
It is the goal of this course to do exactly that
a number of problems is presented
in order to apply mathematical techniques and knowledge to physical concepts These
examples are not presented as welldeveloped theory Instead these examples are presented
as a number of problems that elucidate the issues that are at stake In this sense this
book o ers a guided tour
material for learning is presented but true learning will only
take place by active exploration
Since this book is written as a set of problems you may frequently want to consult
other material to refresh or deepen your understanding of material In many places we
will refer to the book of Boas In addition the books of Butkov and Arfken are
excellent When you are a physics of geophysics student you should seriously consider
buying a comprehensive textbook on mathematical physics it will be of great bene
t to
you
In addition to books colleagues either in the same
eld or other
elds can be a great
source of knowledge and understanding Therefore don t hesitate to work together with
others on these problems if you are in the fortunate positions to do so This may not
only make the work more enjoyable it may also help you in getting unstuck at di cult
moments and the di erent viewpoints of others may help to deepen yours
This book is set up with the goal of obtaining a good working knowledge of mathe
matical geophysics that is needed for students in physics or geophysics A certain basic
knowledge of calculus and linear algebra is needed for digesting the material presented
here For this reason this book is meant for upperlevel undergraduate students or lower
level graduate students depending on the background and skill of the student

CHAPTER 􀀀 INTRODUCTION
At this point the book is still under construction
New sections are reg
ularly added and both corrections and improvements will be made
If you
are interested in this material therefore regularly check the latest version at
Samizdat Press
The feedback of both teachers and students who use this
material is vital in improving this manuscript please send you remarks to

ABC OF RELATIVITY Fourth Revised Edition Edited by Felix Pirani free download

2011-11-18T08:08:00.000-08:00

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Preface to the
Fourth Edition
This book first appeared in 1925. The basic principles of
relativity have not changed since then, but both the theory
and its applications have been much extended, and some
revision has been necessary for the second and subsequent
editions. For the second and third editions I carried out these
revisions with Bertrand Russell's approval. The revisions for
this fourth edition are entirely my responsibility. I have again
altered a number of passages to agree with present knowledge
or opinion, and I have attempted to eliminate the possessive
case, as applied to laws or theories, where it seemed to me
no longer appropriate.
I have also done my best to renounce the convention that
the masculine includes the feminine. Sixty years ago this may
have been acceptable, or at least tolerated; now it is no longer
so,'and I have little doubt that Russell, who was a pro-feminist
ahead of his time, would have approved of the renunciation.
I have not presumed to meddle with the substance of the
last two chapters, which are largely philosophical, rather than
physical, in character, although there is much in them which
I disagree with.
P.P.

Contents
Preface to the Fourth Edition page 5
1 Touch and Sight: The Earth and the Heavens 9
2 What Happens and What is Observed 17
3 The Velocity of Light 26
4 Clocks and Foot-rules 34
5 Space-Time 45
6 The Special Theory of Relativity 53
7 Intervals in Space-Time 66
8 Einstein's Law of Gravitation 78
9 Proofs of Einstein's Law of Gravitation 91
10 Mass, Momentum, Energy, and Action 100
11 The Expanding Universe 113
12 Conventions and Natural Laws 124
13 The Abolition of'Force' 133
14 What is Matter? 141
15 Philosophical Consequences 148

Charged Particle Beams By Stanley Humphries, Jr. free download

2011-11-18T08:03:00.000-08:00

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Contents
________
1. Introduction 1
1.1. Charged particle beams 1
1.2. Methods and organization 6
1.3. Single-particle dynamics 9
2. Phase space description of charged particle beams 20
2.1. Particle trajectories in phase space 22
2.2. Distribution functions 28
2.3. Numerical calculation of particle orbits with beam-generated forces 32
2.4. Conservation of phase space volume 36
2.5. Density and average velocity 46
2.6. Maxwell distribution 49
2.7. Collisionless Boltzmann equation 52
2.8. Charge and current density 56
2.9. Computer simulations 60
2.10. Moment equations 65
2.11. Pressure force in collisionless distributions 71
2.12. Relativistic particle distributions 76
Charged Particle Beams
I-6
3. Introduction to beam emittance 79
3.1. Laminar and non-laminar beams 80
3.2. Emittance 87
3.3. Measurement of emittance 93
3.4. Coupled beam distributions, longitudinal emittance, normalized
emittance, and brightness 101
3.5 Emittance force 107
3.6. Non-laminar beams in drift regions 109
3.7. Non-laminar beams in linear focusing systems 113
3.8. Compression and expansion of non-laminar beams 128
4. Beam emittance - advanced topics 133
4.1. Linear transformations of elliptical distributions 134
4.2. Transport parameters from particle orbit theory 145
4.3. Beam matching 150
4.4. Non-linear focusing systems 157
4.5. Emittance in storage rings 167
4.6. Beam cooling 174
5. Introduction to beam-generated forces 187
5.1. Electric and magnetic fields of beams 188
5.2. One-dimensional Child law for non-relativistic particles 195
5.3. Longitudinal transport limits for a magnetically-confined electron beams 204
5.4. Space-charge expansion of a drifting beam 211
5.5. Transverse forces in relativistic beams 216
Charged Particle Beams
I-7
6. Beam-generated forces - advanced topics 224
6.1. Space-charge-limited flow with an initial injection energy 225
6.2. Space-charge-limited flow from a thermionic cathode 227
6.3. Space-charge-limited flow in spherical geometry 232
6.4. Bipolar flow 239
6.5. Space-charge-limited flow of relativistic electrons 242
6.6. One-dimensional self-consistent equilibrium 246
6.7. KV distribution 256
7. Electron and ion guns 262
7.1. Pierce method for gun design 263
7.2. Medium perveance guns 271
7.3. High perveance guns and ray tracing codes 277
7.4. High current electron sources 283
7.5. Extraction of ions at a free plasma boundary 289
7.6. Plasma ion sources 300
7.7. Charged-particle extraction from grid-controlled plasmas 315
7.8. Ion extractors 322
8. High power pulsed electron and ion diodes 328
8.1. Motion of electrons in crossed electric and magnetic fields 329
8.2. Pinched electron beam diodes 337
8.3. Electron diodes with strong applied magnetic fields 346
8.4. Magnetic insulation of high power transmission lines 351
8.5. Plasma erosion 356
8.6. Reflex triode 364
8.7. Low-impedance reflex triode 370
8.8. Magnetically-insulated ion diode 377
Charged Particle Beams
I-8
8.9. Ion flow enhancement in magnetically-insulated diodes 388
9. Paraxial beam transport with space-charge 395
9.1. Envelope equation for sheet beams 396
9.2. Paraxial ray equation 400
9.3. Envelope equation in a quadrupole lens array 407
9.4 Limiting current for paraxial beams 412
9.5. Multi-beam ion transport 419
9.6. Longitudinal space-charge limits in RF accelerators and induction linacs 423
10. High current electron beam transport under vacuum 432
10.1. Motion of electrons through a magnetic cusp 433
10.2. Propagation of beams from an immersed cathode 439
10.3. Brillouin equilibrium of a cylindrical electron beam 445
10.4. Interaction of electrons with matter 451
10.5. Foil focusing of relativistic electron beams 457
10.6. Walle-charge and return-current for a beam in a pipe 470
10.7. Drifts of electron beams in a solenoidal field 477
10.8. Guiding electron beams with solenoidal fields 482
10.9. Electron beam transport in magnetic cusps 490
11. Ion beam neutralization 501
11.1. Neutralization by comoving electrons 502
11.2. Transverse neutralization 511
11.3. Current neutralization in vacuum 517
11.4. Focal limits for neutralized ion beams 522
11.5. Acceleration and transport of neutralized ion beams 528
Charged Particle Beams
I-9
12. Electron beams in plasmas 535
12.1. Space-charge neutralization in equilibrium plasmas 536
12.2. Oscillations of an un-magnetized plasma 540
12.3. Oscillations of a neutralized electron beam 546
12.4 Injection of a pulsed electron beam into a plasma 552
12.5. Magnetic skin depth 563
12.6. Return current in a resistive plasma 569
12.7. Limiting current for neutralized electron beams 577
12.8. Bennett equilibrium 583
12.9. Propagation in low-density plasmas and weakly-ionized gases 587
13. Transverse instabilities 592
13.1. Instabilities of space-charge-dominated beams in periodic
focusing systems 594
13.2. Betatron waves on a filamentary beam 610
13.3. Frictional forces and phase mixing 615
13.4. Transverse resonant modes 622
13.5. Beam breakup instability 631
13.6. Transverse resistive wall instability 640
13.7. Hose instability of an electron beam in an ion channel 645
13.8. Resistive hose instability 655
13.9. Filamentation instability of neutralized electron beams 664
14. Longitudinal instabilities 674
14.1. Two-stream instability 675
14.2. Beam-generated axial electric fields 687
14.3. Negative mass instability 697
14.4. Longitudinal resistive wall instability 704
Charged Particle Beams
I-10
15. Generation of radiation with electron beams 720
15.1. Inverse diode 722
15.2. Driving resonant cavities with electron beams 736
15.3. Longitudinal beam bunching 749
15.4. Klystron 762
15.5. Traveling wave tube 772
15.6. Magnetron 781
15.7. Mechanism of the free-electron laser 796
15.8. Phase dynamics in the free-electron laser 803

Preface
________
Charged Particle Beams is the product of a two-term course sequence that I taught on
accelerator technology and beam physics at the University of New Mexico and at Los Alamos
National Laboratory. The material for the two terms was divided into the dynamics of single
charged particles and the description of large groups of particles (the collective behavior of
beams).
In writing Charged Particle Beams my goal was to create a unified description that would be
useful to a broad audience: accelerator designers, accelerator users, industrial engineers, and
physics researchers. I organized the material to provide beginning students with the background
to understand advanced literature and to use accelerators effectively. This book can serve as an
independent reference. Combining Charged Particle Beams with Principles of Charged
Particle Acceleration gives a programmed introduction to the field of particle acceleration. I
began my research on particle beams with a background in plasma physics. This change in
direction involved a difficult process of searching for material, learning from experts, and
seeking past insights. Although I found excellent advanced references on specialized areas, no
single work covered the topics necessary to understand high-power accelerators and
high-brightness beams. The difficulties I faced encouraged me to write Charged Particle
Beams. The book describes the basic ideas behind modern beam applications such as stochastic
cooling, high-brightness injectors and the free-electron laser.
Charged Particle Beams
I was fortunate to have abundant help creating this book. Richard Cooper of Los Alamos
National Laboratory applied his proofreading ability to the entire manuscript. In additional to
mechanical corrections, his suggestions on technical points and emphasis were invaluable. The
creation of this book was supported in part by a sabbatical leave from the Department of
Electrical and Computer Engineering at the University of New Mexico. David Woodall. former
Chairman of the Department of Chemical and Nuclear Engineering at the University of New
Mexico, suggested the idea of the accelerator course sequence. I am grateful for his support
during the development of the courses.
Several people contributed advice on specific sections of the book. Commentators included
Kevin O'Brien of Sandia National Laboratories, John Creedon of Physics International
Company, Brendan Godfrey of the Air Force Weapons Laboratory, Edward Lee of Lawrence
Berkeley Laboratory, William Herrmannsfeldt of the Stanford Linear Accelerator Center, and
Carl Ekdahl of Los Alamos National Laboratory. I would also like to thank A. V. Tollestrup of
Fermi National Accelerator Laboratory for permission to paraphrase his article (coauthored by
G. Dugan) on Elementary Stochastic Cooling.
I want to express appreciation to the students in my beam physics course at the University of
New Mexico and at the Los Alamos Graduate Center. Through their contributions, I clarified and
expanded the material over several years. Los Alamos National Laboratory supported the
courses since their inception. I want to thank Robert Jameson and Alan Wadlinger of the
Accelerator Technology Division for their encouragement. The efforts of the Instructional
Television Center of UNM made it feasible to present classes at Los Alamos. I have also taught
the material in short course format. I am grateful to Thomas Roberts and Stanley Pruett for
organizing a course at the Strategic Defense Command.
Several accelerator science groups helped in the development of material for the book. I have
worked closely with the Heavy Ion Fusion Accelerator Research Group at Lawrence Berkeley
Laboratory for several years. I want to thank Henry Rutkowski, Thomas Fessenden, Denis Keefe
and Edward Lee for their suggestions on the book and for providing the opportunity to work in
the field of accelerator inertial fusion. The long-term support of Charles Roberson of the Office
of Naval Research has been critical for accelerator research at the University of New Mexico.
The University has also received generous research support from Groups CLS-7 and P-14 of the
Los Alamos National Laboratory. I am grateful to Roger Bangerter and the late Kenneth Riepe
who initiated the UNM program on vacuum arc plasma sources. I would also like to thank Carl
Ekdahl – much of the material in this book evolved from spirited discussions on high-current
beam physics.

Differential Equation for Dummies By Steven Holzenr,PhD free download

2011-11-18T07:58:00.000-08:00

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Table of Contents
Introduction..................................................................1
About This Book...............................................................................................1
Conventions Used in This Book .....................................................................1
What You’re Not to Read.................................................................................2
Foolish Assumptions .......................................................................................2
How This Book Is Organized...........................................................................2
Part I: Focusing on First Order Differential Equations.......................3
Part II: Surveying Second and Higher Order
Differential Equations.........................................................................3
Part III: The Power Stuff: Advanced Techniques ................................3
Part IV: The Part of Tens........................................................................3
Icons Used in This Book..................................................................................4
Where to Go from Here....................................................................................4
Part I: Focusing on First Order Differential Equations ......5
Chapter 1: Welcome to the World of Differential Equations . . . . . . . . .7
The Essence of Differential Equations...........................................................8
Derivatives: The Foundation of Differential Equations .............................11
Derivatives that are constants............................................................11
Derivatives that are powers................................................................12
Derivatives involving trigonometry ...................................................12
Derivatives involving multiple functions ..........................................12
Seeing the Big Picture with Direction Fields...............................................13
Plotting a direction field ......................................................................13
Connecting slopes into an integral curve .........................................14
Recognizing the equilibrium value.....................................................16
Classifying Differential Equations ................................................................17
Classifying equations by order...........................................................17
Classifying ordinary versus partial equations..................................17
Classifying linear versus nonlinear equations..................................18
Solving First Order Differential Equations ..................................................19
Tackling Second Order and Higher Order Differential Equations............20
Having Fun with Advanced Techniques ......................................................21
Chapter 2: Looking at Linear First Order Differential Equations . . . . .23
First Things First: The Basics of Solving Linear First Order
Differential Equations ................................................................................24
Applying initial conditions from the start.........................................24
Stepping up to solving differential
equations involving functions.........................................................25
Adding a couple of constants to the mix...........................................26
Solving Linear First Order Differential Equations
with Integrating Factors ............................................................................26
Solving for an integrating factor.........................................................27
Using an integrating factor to solve a differential equation ...........28
Moving on up: Using integrating factors in differential
equations with functions .................................................................29
Trying a special shortcut ....................................................................30
Solving an advanced example.............................................................32
Determining Whether a Solution for a Linear First Order
Equation Exists ...........................................................................................35
Spelling out the existence and uniqueness theorem
for linear differential equations ......................................................35
Finding the general solution ...............................................................36
Checking out some existence and uniqueness examples ...............37
Figuring Out Whether a Solution for a Nonlinear
Differential Equation Exists.......................................................................38
The existence and uniqueness theorem for
nonlinear differential equations......................................................39
A couple of nonlinear existence and uniqueness examples ...........39
Chapter 3: Sorting Out Separable First Order
Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .41
Beginning with the Basics of Separable Differential Equations ...............42
Starting easy: Linear separable equations ........................................43
Introducing implicit solutions ............................................................43
Finding explicit solutions from implicit solutions ...........................45
Tough to crack: When you can’t find an explicit solution ..............48
A neat trick: Turning nonlinear separable equations into
linear separable equations ..............................................................49
Trying Out Some Real World Separable Equations....................................52
Getting in control with a sample flow problem ................................52
Striking it rich with a sample monetary problem ............................55
Break It Up! Using Partial Fractions in Separable Equations....................59
xii Differential Equations For Dummies
Chapter 4: Exploring Exact First Order Differential
Equations and Euler’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .63
Exploring the Basics of Exact Differential Equations ................................63
Defining exact differential equations .................................................64
Working out a typical exact differential equation ............................65
Determining Whether a Differential Equation Is Exact..............................66
Checking out a useful theorem...........................................................66
Applying the theorem..........................................................................67
Conquering Nonexact Differential Equations
with Integrating Factors ............................................................................70
Finding an integrating factor...............................................................71
Using an integrating factor to get an exact equation.......................73
The finishing touch: Solving the exact equation ..............................74
Getting Numerical with Euler’s Method ......................................................75
Understanding the method .................................................................76
Checking the method’s accuracy on a computer.............................77
Delving into Difference Equations................................................................83
Some handy terminology ....................................................................84
Iterative solutions ................................................................................84
Equilibrium solutions ..........................................................................85
Part II: Surveying Second and Higher Order
Differential Equations..................................................89
Chapter 5: Examining Second Order Linear
Homogeneous Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . .91
The Basics of Second Order Differential Equations...................................91
Linear equations...................................................................................92
Homogeneous equations.....................................................................93
Second Order Linear Homogeneous Equations
with Constant Coefficients ........................................................................94
Elementary solutions ...........................................................................94
Initial conditions...................................................................................95
Checking Out Characteristic Equations ......................................................96
Real and distinct roots.........................................................................97
Complex roots.....................................................................................100
Identical real roots .............................................................................106
Getting a Second Solution by Reduction of Order...................................109
Seeing how reduction of order works..............................................110
Trying out an example.......................................................................111
Table of Contents xiii
Putting Everything Together with Some Handy Theorems ....................114
Superposition......................................................................................114
Linear independence .........................................................................115
The Wronskian....................................................................................117
Chapter 6: Studying Second Order Linear Nonhomogeneous
Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .123
The General Solution of Second Order Linear
Nonhomogeneous Equations..................................................................124
Understanding an important theorem.............................................124
Putting the theorem to work.............................................................125
Finding Particular Solutions with the Method of
Undetermined Coefficients......................................................................127
When g(x) is in the form of erx ..........................................................127
When g(x) is a polynomial of order n ..............................................128
When g(x) is a combination of sines and cosines ..........................131
When g(x) is a product of two different forms ...............................133
Breaking Down Equations with the Variation of Parameters Method ....135
Nailing down the basics of the method...........................................136
Solving a typical example..................................................................137
Applying the method to any linear equation..................................138
What a pair! The variation of parameters method
meets the Wronskian......................................................................142
Bouncing Around with Springs ’n’ Things ................................................143
A mass without friction .....................................................................144
A mass with drag force ......................................................................148
Chapter 7: Handling Higher Order Linear Homogeneous
Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .151
The Write Stuff: The Notation of Higher Order
Differential Equations ..............................................................................152
Introducing the Basics of Higher Order Linear
Homogeneous Equations.........................................................................153
The format, solutions, and initial conditions .................................153
A couple of cool theorems ................................................................155
Tackling Different Types of Higher Order Linear
Homogeneous Equations.........................................................................156
Real and distinct roots.......................................................................156
Real and imaginary roots ..................................................................161
Complex roots.....................................................................................164
Duplicate roots ...................................................................................166
xiv Differential Equations For Dummies
Chapter 8: Taking On Higher Order Linear Nonhomogeneous
Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .173
Mastering the Method of Undetermined Coefficients
for Higher Order Equations.....................................................................174
When g(x) is in the form erx ...............................................................176
When g(x) is a polynomial of order n ..............................................179
When g(x) is a combination of sines and cosines ..........................182
Solving Higher Order Equations with Variation of Parameters..............185
The basics of the method..................................................................185
Working through an example............................................................186
Part III: The Power Stuff: Advanced Techniques...........189
Chapter 9: Getting Serious with Power Series
and Ordinary Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .191
Perusing the Basics of Power Series..........................................................191
Determining Whether a Power Series Converges
with the Ratio Test ...................................................................................192
The fundamentals of the ratio test...................................................192
Plugging in some numbers ................................................................193
Shifting the Series Index..............................................................................195
Taking a Look at the Taylor Series .............................................................195
Solving Second Order Differential Equations with Power Series ...........196
When you already know the solution ..............................................198
When you don’t know the solution beforehand.............................204
A famous problem: Airy’s equation..................................................207
Chapter 10: Powering through Singular Points . . . . . . . . . . . . . . . . . .213
Pointing Out the Basics of Singular Points ...............................................213
Finding singular points ......................................................................214
The behavior of singular points .......................................................214
Regular versus irregular singular points.........................................215
Exploring Exciting Euler Equations ...........................................................219
Real and distinct roots.......................................................................220
Real and equal roots ..........................................................................222
Complex roots.....................................................................................223
Putting it all together with a theorem..............................................224
Figuring Series Solutions Near Regular Singular Points..........................225
Identifying the general solution........................................................225
The basics of solving equations near singular points ...................227
A numerical example of solving an equation
near singular points........................................................................230
Taking a closer look at indicial equations.......................................235
Table of Contents xv
Chapter 11: Working with Laplace Transforms . . . . . . . . . . . . . . . . . .239
Breaking Down a Typical Laplace Transform...........................................239
Deciding Whether a Laplace Transform Converges ................................240
Calculating Basic Laplace Transforms ......................................................241
The transform of 1..............................................................................242
The transform of eat ............................................................................242
The transform of sin at ......................................................................242
Consulting a handy table for some relief ........................................244
Solving Differential Equations with Laplace Transforms........................245
A few theorems to send you on your way.......................................246
Solving a second order homogeneous equation ............................247
Solving a second order nonhomogeneous equation .....................251
Solving a higher order equation .......................................................255
Factoring Laplace Transforms and Convolution Integrals .....................258
Factoring a Laplace transform into fractions .................................258
Checking out convolution integrals .................................................259
Surveying Step Functions............................................................................261
Defining the step function .................................................................261
Figuring the Laplace transform of the step function .....................262
Chapter 12: Tackling Systems of First Order Linear
Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .265
Introducing the Basics of Matrices ............................................................266
Setting up a matrix .............................................................................266
Working through the algebra ............................................................267
Examining matrices............................................................................268
Mastering Matrix Operations......................................................................269
Equality................................................................................................269
Addition ...............................................................................................270
Subtraction..........................................................................................270
Multiplication of a matrix and a number.........................................270
Multiplication of two matrices..........................................................270
Multiplication of a matrix and a vector ...........................................271
Identity.................................................................................................272
The inverse of a matrix......................................................................272
Having Fun with Eigenvectors ’n’ Things..................................................278
Linear independence .........................................................................278
Eigenvalues and eigenvectors ..........................................................281
Solving Systems of First-Order Linear Homogeneous
Differential Equations ..............................................................................283
Understanding the basics..................................................................284
Making your way through an example ............................................285
Solving Systems of First Order Linear Nonhomogeneous Equations .....288
Assuming the correct form of the particular solution...................289
Crunching the numbers.....................................................................290
Winding up your work .......................................................................292
xvi Differential Equations For Dummies
Chapter 13: Discovering Three Fail-Proof Numerical Methods . . . . .293
Number Crunching with Euler’s Method ..................................................294
The fundamentals of the method .....................................................294
Using code to see the method in action..........................................295
Moving On Up with the Improved Euler’s Method ..................................299
Understanding the improvements ...................................................300
Coming up with new code.................................................................300
Plugging a steep slope into the new code.......................................304
Adding Even More Precision with the Runge-Kutta Method ..................308
The method’s recurrence relation....................................................308
Working with the method in code ....................................................309
Part IV: The Part of Tens ............................................315
Chapter 14: Ten Super-Helpful Online Differential
Equation Tutorials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .317
AnalyzeMath.com’s Introduction to Differential Equations ...................317
Harvey Mudd College Mathematics Online Tutorial ...............................318
John Appleby’s Introduction to Differential Equations...........................318
Kardi Teknomo’s Page .................................................................................318
Martin J. Osborne’s Differential Equation Tutorial..................................318
Midnight Tutor’s Video Tutorial.................................................................319
The Ohio State University Physics Department’s
Introduction to Differential Equations...................................................319
Paul’s Online Math Notes ............................................................................319
S.O.S. Math ....................................................................................................319
University of Surrey Tutorial ......................................................................320
Chapter 15: Ten Really Cool Online Differential
Equation Solving Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .321
AnalyzeMath.com’s Runge-Kutta Method Applet ....................................321
Coolmath.com’s Graphing Calculator .......................................................321
Direction Field Plotter .................................................................................322
An Equation Solver from QuickMath Automatic Math Solutions...........322
First Order Differential Equation Solver....................................................322
GCalc Online Graphing Calculator .............................................................322
JavaView Ode Solver....................................................................................323
Math @ CowPi’s System Solver...................................................................323
A Matrix Inverter from QuickMath Automatic Math Solutions ..............323
Visual Differential Equation Solving Applet..............................................323

FIRST COURSE IN THE THEORY OF EQUATIONS BY LEONARD EUGENE DICKSON, Ph.D. free download

2011-11-18T07:53:00.000-08:00

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PREFACE
The theory of equations is not only a necessity in the subsequent mathematical
courses and their applications, but furnishes an illuminating sequel to
geometry, algebra and analytic geometry. Moreover, it develops anew and in
greater detail various fundamental ideas of calculus for the simple, but important,
case of polynomials. The theory of equations therefore affords a useful
supplement to differential calculus whether taken subsequently or simultaneously.
It was to meet the numerous needs of the student in regard to his earlier and
future mathematical courses that the present book was planned with great care
and after wide consultation. It differs essentially from the author’s Elementary
Theory of Equations, both in regard to omissions and additions, and since it
is addressed to younger students and may be used parallel with a course in
differential calculus. Simpler and more detailed proofs are now employed.
The exercises are simpler, more numerous, of greater variety, and involve more
practical applications.
This book throws important light on various elementary topics. For example,
an alert student of geometry who has learned how to bisect any angle
is apt to ask if every angle can be trisected with ruler and compasses and if
not, why not. After learning how to construct regular polygons of 3, 4, 5, 6,
8 and 10 sides, he will be inquisitive about the missing ones of 7 and 9 sides.
The teacher will be in a comfortable position if he knows the facts and what
is involved in the simplest discussion to date of these questions, as given in
Chapter III. Other chapters throw needed light on various topics of algebra. In
particular, the theory of graphs is presented in Chapter V in a more scientific
and practical manner than was possible in algebra and analytic geometry.
There is developed a method of computing a real root of an equation with
minimum labor and with certainty as to the accuracy of all the decimals obtained.
We first find by Horner’s method successive transformed equations
whose number is half of the desired number of significant figures of the root.
The final equation is reduced to a linear equation by applying to the constant
term the correction computed from the omitted terms of the second and
iv PREFACE
higher degrees, and the work is completed by abridged division. The method
combines speed with control of accuracy.
Newton’s method, which is presented from both the graphical and the
numerical standpoints, has the advantage of being applicable also to equations
which are not algebraic; it is applied in detail to various such equations.
In order to locate or isolate the real roots of an equation we may employ a
graph, provided it be constructed scientifically, or the theorems of Descartes,
Sturm, and Budan, which are usually neither stated, nor proved, correctly.
The long chapter on determinants is independent of the earlier chapters.
The theory of a general system of linear equations is here presented also from
the standpoint of matrices.
For valuable suggestions made after reading the preliminary manuscript of
this book, the author is greatly indebted to Professor Bussey of the University
of Minnesota, Professor Roever of Washington University, Professor Kempner
of the University of Illinois, and Professor Young of the University of Chicago.
The revised manuscript was much improved after it was read critically by
Professor Curtiss of Northwestern University. The author’s thanks are due
also to Professor Dresden of the University of Wisconsin for various useful
suggestions on the proof-sheets.

CONTENTS
Numbers refer to pages.
CHAPTER I
Complex Numbers
Square Roots, 1. Complex Numbers, 1. Cube Roots of Unity, 3. Geometrical
Representation, 3. Product, 4. Quotient, 5. De Moivre’s Theorem, 5. Cube
Roots, 6. Roots of Complex Numbers, 7. Roots of Unity, 8. Primitive Roots of
Unity, 9.
CHAPTER II
Theorems on Roots of Equations
Quadratic Equation, 13. Polynomial, 14. Remainder Theorem, 14. Synthetic
Division, 16. Factored Form of a Polynomial, 18. Multiple Roots, 18. Identical
Polynomials, 19. Fundamental Theorem of Algebra, 20. Relations between Roots
and Coefficients, 20. Imaginary Roots occur in Pairs, 22. Upper Limit to the Real
Roots, 23. Another Upper Limit to the Roots, 24. Integral Roots, 27. Newton’s
Method for Integral Roots, 28. Another Method for Integral Roots, 30. Rational
Roots, 31.
CHAPTER III
Constructions with Ruler and Compasses
Impossible Constructions, 33. Graphical Solution of a Quadratic Equation, 33.
Analytic Criterion for Constructibility, 34. Cubic Equations with a Constructible
Root, 36. Trisection of an Angle, 38. Duplication of a Cube, 39. Regular Polygon
of 7 Sides, 39. Regular Polygon of 7 Sides and Roots of Unity, 40. Reciprocal
Equations, 41. Regular Polygon of 9 Sides, 43. The Periods of Roots of Unity, 44.
Regular Polygon of 17 Sides, 45. Construction of a Regular Polygon of 17 Sides, 47.
Regular Polygon of n Sides, 48.
v
vi CONTENTS
CHAPTER IV
Cubic and Quartic Equations
Reduced Cubic Equation, 51. Algebraic Solution of a Cubic, 51. Discriminant,
53. Number of Real Roots of a Cubic, 54. Irreducible Case, 54. Trigonometric
Solution of a Cubic, 55. Ferrari’s Solution of the Quartic Equation, 56.
Resolvent Cubic, 57. Discriminant, 58. Descartes’ Solution of the Quartic Equation,
59. Symmetrical Form of Descartes’ Solution, 60.
CHAPTER V
The Graph of an Equation
Use of Graphs, 63. Caution in Plotting, 64. Bend Points, 64. Derivatives, 66.
Horizontal Tangents, 68. Multiple Roots, 68. Ordinary and Inflexion Tangents, 70.
Real Roots of a Cubic Equation, 73. Continuity, 74. Continuity of Polynomials, 75.
Condition for a Root Between a and b, 75. Sign of a Polynomial at Infinity, 77.
Rolle’s Theorem, 77.
CHAPTER VI
Isolation of Real Roots
Purpose and Methods of Isolating the Real Roots, 81. Descartes’ Rule of
Signs, 81. Sturm’s Method, 85. Sturm’s Theorem, 86. Simplifications of Sturm’s
Functions, 88. Sturm’s Functions for a Quartic Equation, 90. Sturm’s Theorem
for Multiple Roots, 92. Budan’s Theorem, 93.
CHAPTER VII
Solution of Numerical Equations
Horner’s Method, 97. Newton’s Method, 102. Algebraic and Graphical Discussion,
103. Systematic Computation, 106. For Functions not Polynomials, 108.
Imaginary Roots, 110.
CHAPTER VIII
Determinants; Systems of Linear Equations
Solution of 2 Linear Equations by Determinants, 115. Solution of 3 Linear Equations
by Determinants, 116. Signs of the Terms of a Determinant, 117. Even and
Odd Arrangements, 118. Definition of a Determinant of Order n, 119. Interchange
of Rows and Columns, 120. Interchange of Two Columns, 121. Interchange of Two
Rows, 122. Two Rows or Two Columns Alike, 122. Minors, 123. Expansion, 123.
Removal of Factors, 125. Sum of Determinants, 126. Addition of Columns or
Rows, 127. System of n Linear Equations in n Unknowns, 128. Rank, 130. System
of n Linear Equations in n Unknowns, 130. Homogeneous Equations, 134.
System of m Linear Equations in n Unknowns, 135. Complementary Minors, 137.
CONTENTS vii
Laplace’s Development by Columns, 137. Laplace’s Development by Rows, 138.
Product of Determinants, 139.
CHAPTER IX
Symmetric Functions
Sigma Functions, Elementary Symmetric Functions, 143. Fundamental Theorem,
144. Functions Symmetric in all but One Root, 147. Sums of Like Powers
of the Roots, 150. Waring’s Formula, 152. Computation of Sigma Functions, 156.
Computation of Symmetric Functions, 157.
CHAPTER X
Elimination, Resultants And Discriminants
Elimination, 159. Resultant of Two Polynomials, 159. Sylvester’s Method of
Elimination, 161. Bézout’s Method of Elimination, 164. General Theorem on
Elimination, 166. Discriminants, 167.
APPENDIX
Fundamental Theorem of Algebra
Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

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Contents
1 Strings in QCD. 4
1.1 The linear trajectories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 The Veneziano formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 The classical string. 7
3 Open and closed strings. 11
3.1 The Open string. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.2 The closed string. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.3 Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.3.1 The open string. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.3.2 The closed string. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.4 The light-cone gauge. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.5 Constraints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.5.1 for open strings: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
¤Lecture notes 2003 and 2004
1
3.5.2 for closed strings: . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.6 Energy, momentum, angular momentum. . . . . . . . . . . . . . . . . . . . 17
4 Quantization. 18
4.1 Commutation rules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.2 The constraints in the quantum theory. . . . . . . . . . . . . . . . . . . . . 19
4.3 The Virasoro Algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.4 Quantization of the closed string . . . . . . . . . . . . . . . . . . . . . . . 23
4.5 The closed string spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
5 Lorentz invariance. 25
6 Interactions and vertex operators. 27
7 BRST quantization. 31
8 The Polyakov path integral. Interactions with closed strings. 34
8.1 The energy-momentum tensor for the ghost fields. . . . . . . . . . . . . . . 36
9 T-Duality. 38
9.1 Compactifying closed string theory on a circle. . . . . . . . . . . . . . . . . 39
9.2 T-duality of closed strings. . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
9.3 T-duality for open strings. . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
9.4 Multiple branes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
9.5 Phase factors and non-coinciding D-branes. . . . . . . . . . . . . . . . . . 42
10 Complex coordinates. 43
11 Fermions in strings. 45
11.1 Spinning point particles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
11.2 The fermionic Lagrangian. . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
11.3 Boundary conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
11.4 Anticommutation rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
11.5 Spin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
11.6 Supersymmetry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
11.7 The super current. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2
11.8 The light-cone gauge for fermions . . . . . . . . . . . . . . . . . . . . . . . 56
12 The GSO Projection. 58
12.1 The open string. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
12.2 Computing the spectrum of states. . . . . . . . . . . . . . . . . . . . . . . 61
12.3 String types. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
13 Zero modes 65
13.1 Field theories associated to the zero modes. . . . . . . . . . . . . . . . . . 68
13.2 Tensor fields and D-branes. . . . . . . . . . . . . . . . . . . . . . . . . . . 71
13.3 S-duality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
14 Miscelaneous and Outlook. 75
14.1 String diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
14.2 Zero slope limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
14.2.1 Type II theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
14.2.2 Type I theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
14.2.3 The heterotic theories . . . . . . . . . . . . . . . . . . . . . . . . . 77
14.3 Strings on backgrounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
14.4 Coordinates on D-branes. Matrix theory. . . . . . . . . . . . . . . . . . . . 78
14.5 Orbifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
14.6 Dualities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
14.7 Black holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
14.8 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

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Contents
1 Path Integrals 11
1.1 Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2 Choice of the Picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3 Formal Solution of the Equations of Motion . . . . . . . . . . . . . . . . . . . 16
1.4 Example: The Free Particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.5 The Feynman-Kac Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.6 The Path Integral for the Harmonic Oscillator . . . . . . . . . . . . . . . . . . 23
1.7 Some Rules for Path Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.8 The Schr¨odinger Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.9 Potential Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.10 Generating functional for Vacuum Expectation Values . . . . . . . . . . . . . 34
1.11 Bosons and Fermions, and what else? . . . . . . . . . . . . . . . . . . . . . . . 35
2 Nonrelativistic Many-Particle Theory 37
2.1 The Fock Space Representation of Quantum Mechanics . . . . . . . . . . . . 37
3 Canonical Field Quantisation 41
3.1 Space and Time in Special Relativity . . . . . . . . . . . . . . . . . . . . . . . 42
3.2 Tensors and Scalar Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.3 Noether’s Theorem (Classical Part) . . . . . . . . . . . . . . . . . . . . . . . . 51
3.4 Canonical Quantisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.5 The Most Simple Interacting Field Theory: 4 . . . . . . . . . . . . . . . . . 60
3.6 The LSZ Reduction Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.7 The Dyson-Wick Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.8 Wick’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.9 The Feynman Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4 Relativistic Quantum Fields 75
4.1 Causal Massive Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.1.1 Massive Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3
Contents
4.1.2 Massive Spin-1/2 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.2 Causal Massless Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.2.1 Massless Vector Field . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.2.2 Massless Helicity 1/2 Fields . . . . . . . . . . . . . . . . . . . . . . . . 84
4.3 Quantisation and the Spin-Statistics Theorem . . . . . . . . . . . . . . . . . . 85
4.3.1 Quantisation of the spin-1/2 Dirac Field . . . . . . . . . . . . . . . . . 85
4.4 Discrete Symmetries and the CPT Theorem . . . . . . . . . . . . . . . . . . . 89
4.4.1 Charge Conjugation for Dirac spinors . . . . . . . . . . . . . . . . . . 90
4.4.2 Time Reversal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.4.3 Parity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.4.4 Lorentz Classification of Bilinear Forms . . . . . . . . . . . . . . . . . 94
4.4.5 The CPT Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.4.6 Remark on Strictly Neutral Spin–1/2–Fermions . . . . . . . . . . . . . 97
4.5 Path Integral Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.5.1 Example: The Free Scalar Field . . . . . . . . . . . . . . . . . . . . . . 104
4.5.2 The Feynman Rules for 4 revisited . . . . . . . . . . . . . . . . . . . 106
4.6 Generating Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.6.1 LSZ Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.6.2 The equivalence theorem . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.6.3 Generating Functional for Connected Green’s Functions . . . . . . . . 111
4.6.4 Effective Action and Vertex Functions . . . . . . . . . . . . . . . . . . 113
4.6.5 Noether’s Theorem (Quantum Part) . . . . . . . . . . . . . . . . . . . 118
4.6.6 ~-Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
4.7 A Simple Interacting Field Theory with Fermions . . . . . . . . . . . . . . . . 123
5 Renormalisation 129
5.1 Infinities and how to cure them . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.1.1 Overview over the renormalisation procedure . . . . . . . . . . . . . . 133
5.2 Wick rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
5.3 Dimensional regularisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
5.3.1 The 􀀀-function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
5.3.2 Spherical coordinates in d dimensions . . . . . . . . . . . . . . . . . . 147
5.3.3 Standard-integrals for Feynman integrals . . . . . . . . . . . . . . . . 148
5.4 The 4-point vertex correction at 1-loop order . . . . . . . . . . . . . . . . . . 150
5.5 Power counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
5.6 The setting-sun diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
5.7 Weinberg’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
5.7.1 Proof of Weinberg’s theorem . . . . . . . . . . . . . . . . . . . . . . . 162
4
Contents
5.7.2 Proof of the Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
5.8 Application of Weinberg’s Theorem to Feynman diagrams . . . . . . . . . . . 170
5.9 BPH-Renormalisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
5.9.1 Some examples of the method . . . . . . . . . . . . . . . . . . . . . . . 174
5.9.2 The general BPH-formalism . . . . . . . . . . . . . . . . . . . . . . . . 176
5.10 Zimmermann’s forest formula . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
5.11 Global linear symmetries and renormalisation . . . . . . . . . . . . . . . . . . 181
5.11.1 Example: 1-loop renormalisation . . . . . . . . . . . . . . . . . . . . . 186
5.12 Renormalisation group equations . . . . . . . . . . . . . . . . . . . . . . . . . 189
5.12.1 Homogeneous RGEs and modified BPHZ renormalisation . . . . . . . 189
5.12.2 The homogeneous RGE and dimensional regularisation . . . . . . . . . 192
5.12.3 Solutions to the homogeneous RGE . . . . . . . . . . . . . . . . . . . 194
5.12.4 Independence of the S-Matrix from the renormalisation scale . . . . . 195
5.13 Asymptotic behaviour of vertex functions . . . . . . . . . . . . . . . . . . . . 195
5.13.1 The Gell-Mann-Low equation . . . . . . . . . . . . . . . . . . . . . . . 196
5.13.2 The Callan-Symanzik equation . . . . . . . . . . . . . . . . . . . . . . 197
6 Quantum Electrodynamics 203
6.1 Gauge Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
6.2 Matter Fields interacting with Photons . . . . . . . . . . . . . . . . . . . . . . 209
6.3 Canonical Path Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
6.4 Invariant Cross Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
6.5 Tree level calculations of some physical processes . . . . . . . . . . . . . . . . 219
6.5.1 Compton Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
6.5.2 Annihilation of an e−e+-pair . . . . . . . . . . . . . . . . . . . . . . . 222
6.6 The Background Field Method . . . . . . . . . . . . . . . . . . . . . . . . . . 224
6.6.1 The background field method for non-gauge theories . . . . . . . . . . 224
6.6.2 Gauge theories and background fields . . . . . . . . . . . . . . . . . . 225
6.6.3 Renormalisability of the effective action in background field gauge . . 228
7 Nonabelian Gauge fields 233
7.1 The principle of local gauge invariance . . . . . . . . . . . . . . . . . . . . . . 233
7.2 Quantisation of nonabelian gauge field theories . . . . . . . . . . . . . . . . . 237
7.2.1 BRST-Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
7.2.2 Gauge independence of the S-matrix . . . . . . . . . . . . . . . . . . . 242
7.3 Renormalisability of nonabelian gauge theories in BFG . . . . . . . . . . . . . 244
7.3.1 The symmetry properties in the background field gauge . . . . . . . . 244
7.3.2 The BFG Feynman rules . . . . . . . . . . . . . . . . . . . . . . . . . 247
5
Contents
7.4 Renormalisability of nonabelian gauge theories (BRST) . . . . . . . . . . . . 250
7.4.1 The Ward-Takahashi identities . . . . . . . . . . . . . . . . . . . . . . 250
A Variational Calculus and Functional Methods 255
A.1 The Fundamental Lemma of Variational Calculus . . . . . . . . . . . . . . . . 255
A.2 Functional Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
B The Symmetry of Space and Time 261
B.1 The Lorentz Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
B.2 Representations of the Lorentz Group . . . . . . . . . . . . . . . . . . . . . . 268
B.3 Representations of the Full Lorentz Group . . . . . . . . . . . . . . . . . . . . 269
B.4 Unitary Representations of the Poincar´e Group . . . . . . . . . . . . . . . . . 272
B.4.1 The Massive States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
B.4.2 Massless Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
B.5 The Invariant Scalar Product . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
C Formulae 283
C.1 Amplitudes for various free fields . . . . . . . . . . . . . . . . . . . . . . . . . 283
C.2 Dimensional regularised Feynman-integrals . . . . . . . . . . . . . . . . . . . 284
C.3 Laurent expansion of the 􀀀-Function . . . . . . . . . . . . . . . . . . . . . . . 284
C.4 Feynman’s Parameterisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
Bibliography 287

Preface
The following is a script, which tries to collect and extend some ideas about Quantum Field
Theory for the International Student Programs at GSI.
We start in the first chapter with some facts known from ordinary nonrelativistic quantum
mechanics. We emphasise the picture of the evolution of quantum systems in space and time.
The aim was to introduce the functional methods of path integrals on hand of the familiar
framework of nonrelativistic quantum theory.
In this introductory chapter it was my goal to keep the story as simple as possible. Thus
all problems concerning operator ordering or interaction with electromagnetic fields were
omitted. All these topics will be treated in terms of quantum field theory beginning with in
the third chapter.
The second chapter is not yet written completely. It will be short and is intended to contain
the vacuum many-body theory for nonrelativistic particles given as a quantum many-particle
theory. It is shown that the same theory can be obtained by using the field quantisation
method (which was often called “the second quantisation”, but this is on my opinion a very
misleading term). I intend to work out the most simple applications to the hydrogen atom
including bound states and exact scattering theory.
In the third chapter we start with the classical principles of special relativity as are Lorentz
covariance, the action principle in the covariant Lagrangian formulation but introduce only
scalar fields to keep the stuff quite easy since there is only one field degree of freedom. The
classical part of the chapter ends with a discussion of Noether’s theorem which is on the
heart of our approach to observables which are defined from conserved currents caused by
symmetries of space and time as well as by intrinsic symmetries of the fields.
After that introduction to classical relativistic field theory we quantise the free fields ending
with a sketch about the nowadays well established facts of relativistic quantum theory: It
is necessarily a many-body theory, because there is no possibility for a Schr¨odinger-like oneparticle
theory. The physical reason is simply the possibility of creation and annihilation
of particle-antiparticle pairs (pair creation). It will come out that for a local quantum field
theory the Hamiltonian of the free particles is bounded from below for the quantised field
theory only if we quantise it with bosonic commutation relations. This is a special case of
the famous spin-statistics theorem.
Then we show how to treat 4 theory as the most simple example of an interacting field theory
with help of perturbation theory, prove Wick’s theorem and the LSZ-reduction formula. The
goal of this chapter is a derivation of the perturbative Feynman-diagram rules. The chapter
ends with the sad result that diagrams containing loops do not exist since the integrals are
divergent. This difficulty is solved by renormalisation theory which will be treated later on
7
Preface
in this notes.
The fourth chapter starts with a systematic treatment of relativistic invariant theory using
appendix B which contains the complete mathematical treatment of the representation theory
of the Poincar´e group as far as it is necessary for physics. We shall treat in this chapter at
length the Dirac field which describes particles with spin 1/2. With help of the Poincar´e
group theory and some simple physical axioms this leads to the important results of quantum
field theory as there are the spin-statistics and the PCT theorem.
The rest of the chapter contains the foundations of path integrals for quantum field theories.
Hereby we shall find the methods learnt in chapter 1 helpful. This contains also the
path integral formalism for fermions which needs a short introduction to the mathematics of
Grassmann numbers.
After setting up these facts we shall rederive the perturbation theory, which we have found
with help of Wick’s theorem in chapter 3 from the operator formalism. We shall use from
the very beginning the diagrams as a very intuitive technique for book-keeping of the rather
involved (but in a purely technical sense) functional derivatives of the generating functional
for Green’s functions. On the other hand we shall also illustrate the ,,digram-less” derivation
of the ~-expansion which corresponds to the number of loops in the diagrams.
We shall also give a complete proof of the theorems about generating functionals for subclasses
of diagrams, namely the connected Green’s functions and the proper vertex functions.
We end the chapter with the derivation of the Feynman rules for a simple toy theory involving
a Dirac spin 1/2 Fermi field with the now completely developed functional (path integral)
technique. As will come out quite straight forwardly, the only difference compared to the pure
boson case are some sign rules for fermion lines and diagrams containing a closed fermion
loop, coming from the fact that we have anticommuting Grassmann numbers for the fermions
rather than commuting c-numbers for the bosons.
The fifth chapter is devoted to QED including the most simple physical applications at treelevel.
From the very beginning we shall take the gauge theoretical point of view. Gauge
theories have proved to be the most important class of field theories, including the Standard
Model of elementary particles. So we use from the very beginning the modern techniques to
quantise the theory with help of formal path integral manipulations known as Faddeev-Popov
quantisation in a certain class of covariant gauges. We shall also derive the very important
Ward-Takahashi identities. As an alternative we shall also formulate the background field
gauge which is a manifestly gauge invariant procedure.
Nevertheless QED is not only the most simple example of a physically very relevant quantum
field theory but gives also the possibility to show the formalism of all the techniques needed
to go beyond tree level calculations, i.e. regularisation and renormalisation of Quantum
Field Theories. We shall do this with use of appendix C, which contains the foundations
of dimensional regularisation which will be used as the main regularisation scheme in these
notes. It has the great advantage to keep the theory gauge-invariant and is quite easy to
handle (compared with other schemes as, for instance, Pauli-Villars). We use these techniques
to calculate the classical one-loop results, including the lowest order contribution to the
anomalous magnetic moment of the electron.
I plan to end the chapter with some calculations concerning the hydrogen atom (Lamb shift)
by making use of the Schwinger equations of motion which is in some sense the relativistic
refinement of the calculations shown in chapter 2 but with the important fact that now we
8
Preface
include the quantisation of the electromagnetic fields and radiation corrections.
There are also planned some appendices containing some purely mathematical material needed
in the main parts.
Appendix A introduces some very basic facts about functionals and variational calculus.
Appendix B has grown a little lengthy, but on the other hand I think it is useful to write
down all the stuff about the representation theory of the Poincar´e groups. In a way it may
be seen as a simplification of Wigner’s famous paper from 1939.
Appendix C is devoted to a simple treatment of dimensional regularisation techniques. It’s
also longer than in the most text books on the topic. This comes from my experience that it’s
rather hard to learn all the mathematics out of many sources and to put all this together. So
my intention in writing appendix C was again to put all the mathematics needed together. I
don’t know if there is a shorter way to obtain all this. The only things needed later on in the
notes when we calculate simple radiation corrections are the formula in the last section of the
appendix. But to repeat it again, the intention of appendix C is to derive them. The only
thing we need to know very well to do this, is the analytic structure of the 􀀀-functions well
known in mathematics since the famous work of the 18th and 19th century mathematicians
Euler and Gauss. So the properties of the 􀀀-function are derived completely using only basic
knowledge from a good complex analysis course. It cannot be overemphasised, that all these
techniques of holomorphic functions is one of the most important tools used in physics!
Although I tried not to make too many mathematical mistakes in these notes we use the physicist’s
robust calculus methods without paying too much attention to mathematical rigour.
On the other hand I tried to be exact at places whenever it seemed necessary to me. It should
be said in addition that the mathematical techniques used here are by no means the state of
the art from the mathematician’s point of view. So there is not made use of modern notation
such as of manifolds, alternating differential forms (Cartan formalism), Lie groups, fibre
bundles etc., but nevertheless the spirit is a geometrical picture of physics in the meaning of
Felix Klein’s “Erlanger Programm”: One should seek for the symmetries in the mathematical
structure, that means, the groups of transformations of the mathematical objects which leave
this mathematical structure unchanged.
The symmetry principles are indeed at the heart of modern physics and are the strongest
leaders in the direction towards a complete understanding of nature beyond quantum field
theory and the standard model of elementary particles.
I hope the reader of my notes will have as much fun as I had when I wrote them!
Last but not least I come to the acknowledgements. First to mention are Robert Roth and
Christoph Appel who gave me their various book style hackings for making it as nice looking
as it is.
Also Thomas Neff has contributed by his nice definition of the operators with the tilde below
the symbol and much help with all mysteries of the computer system(s) used while preparing
the script.
Christoph Appel was always discussing with me about the hot topics of QFT like getting
symmetry factors of diagrams and the proper use of Feynman rules for various types of
QFTs. He was also carefully reading the script and has corrected many spelling errors.
9
Preface
Literature
Finally I have to stress the fact that the lack of citations in these notes mean not that I claim
that the contents are original ideas of mine. It was just my laziness in finding out all the
references I used through my own tour through the literature and learning of quantum field
theory.
I just cite some of the textbooks I found most illuminating during the preparation of these
notes: For the fundamentals there exist a lot of textbooks of very different quality. For me the
most important were [PS95, Wei95, Wei95, Kak93]. Concerning gauge theories some of the
clearest sources of textbook or review character are [Tay76, AL73, FLS72, Kug97, LZJ72a,
LZJ72b, LZJ72c]. One of the most difficult topics in quantum field theory is the question of
renormalisation. Except the already mentioned textbooks here I found the original papers
very important, some of them are [BP57, Wei60, Zim68, Zim69, Zim70]. A very nice and
concise monograph of this topic is [Col86]. Whenever I was aware of an eprint-URL I cited it
too, so that one can access these papers as easily as possible.

INTRODUCTION TO SUPERSTRING THEORY By Elias Kiritsis free download

2011-11-18T07:22:00.000-08:00

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Contents
1 Introduction 5
2 Historical perspective 6
3 Classical string theory 9
3.1 The point particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2 Relativistic strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.3 Oscillator expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4 Quantization of the bosonic string 23
4.1 Covariant canonical quantization . . . . . . . . . . . . . . . . . . . . . . . 23
4.2 Light-cone quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.3 Spectrum of the bosonic string . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.4 Path integral quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.5 Topologically non-trivial world-sheets . . . . . . . . . . . . . . . . . . . . . 30
4.6 BRST primer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.7 BRST in string theory and the physical spectrum . . . . . . . . . . . . . . 33
5 Interactions and loop amplitudes 36
6 Conformal field theory 38
6.1 Conformal transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
6.2 Conformally invariant field theory . . . . . . . . . . . . . . . . . . . . . . . 41
6.3 Radial quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
6.4 Example: the free boson . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
6.5 The central charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
6.6 The free fermion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
6.7 Mode expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
6.8 The Hilbert space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
6.9 Representations of the conformal algebra . . . . . . . . . . . . . . . . . . . 54
6.10 Affine algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
6.11 Free fermions and O(N) affine symmetry . . . . . . . . . . . . . . . . . . . 60
1
6.12 N=1 superconformal symmetry . . . . . . . . . . . . . . . . . . . . . . . . 66
6.13 N=2 superconformal symmetry . . . . . . . . . . . . . . . . . . . . . . . . 68
6.14 N=4 superconformal symmetry . . . . . . . . . . . . . . . . . . . . . . . . 70
6.15 The CFT of ghosts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
7 CFT on the torus 75
7.1 Compact scalars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
7.2 Enhanced symmetry and the string Higgs effect . . . . . . . . . . . . . . . 84
7.3 T-duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
7.4 Free fermions on the torus . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
7.5 Bosonization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
7.6 Orbifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
7.7 CFT on higher-genus Riemann surfaces . . . . . . . . . . . . . . . . . . . . 97
8 Scattering amplitudes and vertex operators of bosonic strings 98
9 Strings in background fields and low-energy effective actions 102
10 Superstrings and supersymmetry 104
10.1 Closed (type-II) superstrings . . . . . . . . . . . . . . . . . . . . . . . . . . 106
10.2 Massless R-R states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
10.3 Type-I superstrings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
10.4 Heterotic superstrings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
10.5 Superstring vertex operators . . . . . . . . . . . . . . . . . . . . . . . . . . 117
10.6 Supersymmetric effective actions . . . . . . . . . . . . . . . . . . . . . . . . 119
11 Anomalies 122
12 Compactification and supersymmetry breaking 130
12.1 Toroidal compactifications . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
12.2 Compactification on non-trivial manifolds . . . . . . . . . . . . . . . . . . 135
12.3 World-sheet versus spacetime supersymmetry . . . . . . . . . . . . . . . . 140
12.4 Heterotic orbifold compactifications with N=2 supersymmetry . . . . . . . 145
12.5 Spontaneous supersymmetry breaking . . . . . . . . . . . . . . . . . . . . . 153
2
12.6 Heterotic N=1 theories and chirality in four dimensions . . . . . . . . . . . 155
12.7 Orbifold compactifications of the type-II string . . . . . . . . . . . . . . . . 157
13 Loop corrections to effective couplings in string theory 159
13.1 Calculation of gauge thresholds . . . . . . . . . . . . . . . . . . . . . . . . 161
13.2 On-shell infrared regularization . . . . . . . . . . . . . . . . . . . . . . . . 166
13.3 Gravitational thresholds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
13.4 Anomalous U(1)’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
13.5 N=1,2 examples of threshold corrections . . . . . . . . . . . . . . . . . . . 172
13.6 N=2 universality of thresholds . . . . . . . . . . . . . . . . . . . . . . . . . 175
13.7 Unification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
14 Non-perturbative string dualities: a foreword 179
14.1 Antisymmetric tensors and p-branes . . . . . . . . . . . . . . . . . . . . . . 183
14.2 BPS states and bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
14.3 Heterotic/type-I duality in ten dimensions. . . . . . . . . . . . . . . . . . . 186
14.4 Type-IIA versus M-theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
14.5 M-theory and the E8×E8 heterotic string . . . . . . . . . . . . . . . . . . . 196
14.6 Self-duality of the type-IIB string . . . . . . . . . . . . . . . . . . . . . . . 196
14.7 D-branes are the type-II R-R charged states . . . . . . . . . . . . . . . . . 199
14.8 D-brane actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
14.9 Heterotic/type-II duality in six and four dimensions . . . . . . . . . . . . . 205
15 Outlook 211
Acknowledgments 212
Appendix A: Theta functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
Appendix B: Toroidal lattice sums . . . . . . . . . . . . . . . . . . . . . . . . . 216
Appendix C: Toroidal Kaluza-Klein reduction . . . . . . . . . . . . . . . . . . . 219
Appendix D: N=1,2,4, D=4 supergravity coupled to matter . . . . . . . . . . . 221
Appendix E: BPS multiplets and helicity supertrace formulae . . . . . . . . . . 224
Appendix F: Modular forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
Appendix G: Helicity string partition functions . . . . . . . . . . . . . . . . . . 234
3
Appendix H: Electric-Magnetic duality in D=4 . . . . . . . . . . . . . . . . . . 240
References 243

1 Introduction
String theory has been the leading candidate over the past years for a theory that consistently
unifies all fundamental forces of nature, including gravity. In a sense, the theory
predicts gravity and gauge symmetry around flat space. Moreover, the theory is UVfinite.
The elementary objects are one-dimensional strings whose vibration modes should
correspond to the usual elementary particles.
At distances large with respect to the size of the strings, the low-energy excitations can
be described by an effective field theory. Thus, contact can be established with quantum
field theory, which turned out to be successful in describing the dynamics of the real world
at low energy.
I will try to explain here the basic structure of string theory, its predictions and problems.
In chapter 2 the evolution of string theory is traced, from a theory initially built to
describe hadrons to a “theory of everything”. In chapter 3 a description of classical bosonic
string theory is given. The oscillation modes of the string are described, preparing the scene
for quantization. In chapter 4, the quantization of the bosonic string is described. All three
different quantization procedures are presented to varying depth, since in each one some
specific properties are more transparent than in others. I thus describe the old covariant
quantization, the light-cone quantization and the modern path-integral quantization. In
chapter 6 a concise introduction is given, to the central concepts of conformal field theory
since it is the basic tool in discussing first quantized string theory. In chapter 8 the
calculation of scattering amplitudes is described. In chapter 9 the low-energy effective
action for the massless modes is described.
In chapter 10 superstrings are introduced. They provide spacetime fermions and realize
supersymmetry in spacetime and on the world-sheet. I go through quantization again,
and describe the different supersymmetric string theories in ten dimensions. In chapter 11
gauge and gravitational anomalies are discussed. In particular it is shown that the superstring
theories are anomaly-free. In chapter 12 compactifications of the ten-dimensional
superstring theories are described. Supersymmetry breaking is also discussed in this context.
In chapter 13, I describe how to calculate loop corrections to effective coupling
constants. This is very important for comparing string theory predictions at low energy
with the real world. In chapter 14 a brief introduction to non-perturbative string connections
and non-perturbative effects is given. This is a fast-changing subject and I have
just included some basics as well as tools, so that the reader orients him(her)self in the
web of duality connections. Finally, in chapter 15 a brief outlook and future problems are
presented.
I have added a number of appendices to make several technical discussions self-contained.
5
In Appendix A useful information on the elliptic ϑ-functions is included. In Appendix B,
I rederive the various lattice sums that appear in toroidal compactifications. In Appendix
C the Kaluza-Klein ansatz is described, used to obtain actions in lower dimensions after
toroidal compactification. In Appendix D some facts are presented about four-dimensional
locally supersymmetric theories with N=1,2,4 supersymmetry. In Appendix E, BPS states
are described along with their representation theory and helicity supertrace formulae that
can be used to trace their appearance in a supersymmetric theory. In Appendix F facts
about elliptic modular forms are presented, which are useful in many contexts, notably
in the one-loop computation of thresholds and counting of BPS multiplicities. In Appendix
G, I present the computation of helicity-generating string partition functions and
the associated calculation of BPS multiplicities. Finally, in Appendix H, I briefly review
electric–magnetic duality in four dimensions.
I have not tried to be complete in my referencing. The focus was to provide, in most
cases, appropriate reviews for further reading. Only in the last chapter, which covers
very recent topics, I do mostly refer to original papers because of the scarcity of relevant
reviews.