Haken . Wolf The Physics of Atoms and Quanta

Hermann Haken Hans Christoph Wolf

The Physics of Atoms and Quanta Introduction to Experiments and Theory

Translated by William D. Brewer

Third Corrected and Enlarged Edition With 273 Figures, 160 Problems and Solutions

Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest

Professor Dr. Dr. h. c. Hermann Haken

Institut fUr Theoretische Physik, Universitat Stuttgart, Pfaffenwaldring 57 W -7000 Stuttgart 80, Fed. Rep. of Germany

Professor Dr. Hans Christoph Wolf

Physikalisches Institut, Universitat Stuttgart, Pfaffenwaldring 57 W-7000 Stuttgart 80, Fed. Rep. of Germany

Translator:

Professor Dr. William D. Brewer

Freie Universitat Berlin, Fachbereich Physik, Arnimallee 14 W-IOOO Berlin 33, Fed. Rep. of Germany

Previous editions were published as: Atomic and Quantum Physics. An Introduction to the Fundamentals of Experiment and Theory.

Title of the German original edition: H. Haken, H. C. Wolf: Atom- und Quantenphysik. Einfiihrung in die experimentellen und theoretischen Grundlagen. (Vierte, erweiterte AufJage) © Springer-Verlag Berlin Heidelberg 1980, 1983, 1987, 1990

ISBN-13: 978-3-540-56312-9 e-ISBN-13: 978-3-642-97468-7 DOl: 10.1007/978-3-642-97468-7

Library of Congress Cataloging-in-Publication Data. Haken, H. [Atom- und Quantenphysik. English] The physics of atoms and quanta: introduction to experiments and theory / H. Haken, H. C. Wolf; translated by W. D. Brewer. - 3rd corrected and enlarged ed. p. cm. Translation of: Atom- und Quantenphysik. Includes bibliographical references and index. ISBN 3-540-56312-1 (Berlin). - ISBN 0-387-56312-1 (New York) 1. Atoms. 2. Quantum theory. I. Wolf, H. C. (Hans Christoph), 1929- . II. Title. QC173.HI7513 1993 539.7-dc20 92-44416

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Preface to the Third Edition

The second edition of this book again enjoyed a very positive reception from both uni­versity teachers and students. In this edition we have removed all of the typographical errors that came to our attention. In order to keep the book as current as possible, new developments in the direct observation of individual atoms in electromagnetic traps (Paul traps) and of atoms in molecules on solid surfaces using the scanning tunnel microscope have been added to this edition.

Furthermore, new experiments in atomic interferometry have been included; they have become possible by making use of modern microstructure technology. Finally, the possibility of cooling atomic beams using laser radiation has taken on increasing impor­tance; therefore we have treated this subject as well. This third English Edition cor­responds to a future Fifth German Edition.

We again thank all those colleagues and students who have pointed out to us errors in previous editions.

It is a pleasure for us to thank again Springer-Verlag, in particular Dr. H. Latsch, Dr. H. J. Kolsch and C.-D. Bachem for their always excellent cooperation.

Stuttgart, January 1993 H Haken H. C. Wolf

Preface to the Second Edition

The excellent critique and very positive response to the first edition of this book have encouraged us to prepare this second edition, in which we have tried to make improve­ments wherever possible. We have profited much from the suggestions of professors and students as well as from our own experience in teaching atomic and quantum physics at our university.

Following a widespread request, we have now included the solutions to the exercises and present these at the end of the book. Among the major new sections to be found in this second edition are the following:

We now include the derivation of the relativistic Klein-Gordon equation and of the Dirac equation because the latter, in particular, appears in atomic physics whenever relativistic effects must be taken into account. Our derivation of the Schrodinger equa­tion allowed us to present this extension in a straightforward manner.

The high precision methods of modern spectroscopy allow the atomic physicist to measure extremely small but important shifts of the atomic lines. A very important effect of this kind is the Lamb shift, for which a detailed theoretical derivation is given in a new section. In order to put this in an adequate framework, the basic ideas of the quantization of the electromagnetic field as used in quantum electrodynamics are given. Again it turned out that all the concepts and methods needed to discuss these seemingly advanced theories had already been presented in previous chapters so that again the reader may easily follow these theoretical explanations.

The section on photoelectron spectroscopy has been enlarged and revised. Further­more, the two-electron problem has been made more explicit by treating the difference between triplet and singlet states in detail. Finally, our previous presentation of nuclear spin resonance has been considerably enlarged because this method is finding wide­spread and very important applications, not only in chemistry but also in medicine, for instance in NMR tomography, which is an important new tool in medical diagnostics. This is only one example of the widespread and quite often unanticipated application of atomic and quantum physics in modern science and technology.

It goes without saying that we have not only corrected a number of misprints but have also tried to include the most recent developments in each area. This second English edition corresponds to the third German edition, which is published at about the same time. We wish to thank R. Seyfang, J. U. von Schlitz and V. Weberruss for their help in preparing the second edition. It is again a pleasure for us to thank Springer-Verlag, in particular Dr. H. Lotsch and C.-D. Bachem for their always excellent cooperation.

Stuttgart, March 1987 H. Haken H. C. Wolf

Preface to the First Edition

A thorough knowledge of the physics of atoms and quanta is clearly a must for every student of physics but also for students of neighbouring disciplines such as chemistry and electrical engineering. What these students especially need is a coherent presenta­tion of both the experimental and the theoretical aspects of atomic and quantum physics. Indeed, this field could evolve only through the intimate interaction between ingenious experiments and an equally ingenious development of bold new ideas.

It is well known that the study of the microworld of atoms caused a revolution of physical thought, and fundamental ideas of classical physics, such as those on measur­ability, had to be abandoned. But atomic and quantum physics is not only a fascinating field with respect to the development of far-reaching new physical ideas. It is also of enormous importance as a basis for other fields. For instance, it provides chemistry with a conceptual basis through the quantum theory of chemical bonding. Modern solid-state physics, with its numerous applications in communication and computer technology, rests on the fundamental concepts first developed in atomic and quantum physics. Among the many other important technical applications we mention just the laser, a now widely used light source which produces light whose physical nature is quite different from that of conventional lamps.

In this book we have tried to convey to the reader some of the fascination which atomic and quantum physics still gives a physicist studying this field. We have tried to elaborate on the fundamental facts and basic theoretical methods, leaving aside all superfluous material. The text emerged from lectures which the authors, an experi­mentalist and a theoretician, have given at the University of Stuttgart for many years. These lectures were matched with respect to their experimental and theoretical con­tents.

We have occasionally included in the text some more difficult theoretical sections, in order to give a student who wants to penetrate thoroughly into this field a self-con­tained presentation. The chapters which are more difficult to read are marked by an asterisk. They can be skipped on a first reading of this book. We have included chapters important for chemistry, such as the chapter on the quantum theory of the chemical bond, which may also serve as a starting point for studying solid-state physics. We have further included chapters on spin resonance. Though we explicitly deal with electron spins, similar ideas apply to nuclear spins. The methods of spin re­sonance playa fundamental role in modern physical, chemical and biological investiga­tions as well as in medical diagnostics (nuclear spin tomography). Recent developments in atomic physics, such as studies on Rydberg atoms, are taken into account, and we elaborate the basic features of laser light and nonlinear spectroscopy. We hope that readers will find atomic and quantum physics just as fascinating as did the students of our lectures.

The present text is a translation of the second German edition A tom- und Quanten­physik. We wish to thank Prof. W. D. Brewer for the excellent translation and the most valuable suggestions he made for the improvement of the book. Our thanks also go to

VIII Preface to the First Edition

Dr. J. v. Schutz and Mr. K. Zeile for the critical reading of the manuscript, to Ms. S. Schmiech and Dr. H. Ohno for the drawings, and to Mr. G. Haubs for the careful proofreading. We would like to thank Mrs. U. Funke for her precious help in typing new chapters. Last, but not least, we wish to thank Springer-Verlag, and in particular H. Lotsch and G.M. Hayes, for their excellent cooperation.

Stuttgart, February 1984 H. Haken H. C. Wolf

Contents

List of the Most Important Symbols Used ............................... xv

1. Introduction ...................................................... . 1.1 Classical Physics and Quantum Mechanics ........................ . 1.2 Short Historical Review ......................................... .

2. The Mass and Size of the Atom ..................................... 5 2.1 What is an Atom? .............................................. 5 2.2 Determination of the Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Methods for Determining Avogadro's Number. . . . . . . . . . . . . . . . . . . . . . 7

2.3.1 Electrolysis ............................................... 7 2.3.2 The Gas Constant and Boltzmann's Constant ................. 7 2.3.3 X-Ray Diffraction in Crystals ............................... 8 2.3.4 Determination Using Radioactive Decay ...................... 9

2.4 Determination of the Size of the Atom ............................ 10 2.4.1 Application of the Kinetic Theory of Gases ................... 10 2.4.2 The Interaction Cross Section ............................... 11 2.4.3 Experimental Determination of Interaction Cross Sections ...... 14 2.4.4 Determining the Atomic Size from the Covolume .............. 15 2.4.5 Atomic Sizes from X-Ray Diffraction Measurements on Crystals. 15 2.4.6 Can Individual Atoms Be Seen? ............................. 20

Problems .......................................................... 25

3. Isotopes ........................................................... 27 3.1 The Periodic System of the Elements .............................. 27 3.2 Mass Spectroscopy .............................................. 29

3.2.1 Parabola Method .......................................... 29 3.2.2 Improved Mass Spectrometers ............................... 32 3.2.3 Results of Mass Spectrometry ............................... 33 3.2.4 Modern Applications of the Mass Spectrometer ............... 34 3.2.5 Isotope Separation ......................................... 35

Problems .......................................................... 36

4. The Nucleus of the Atom ........................................... 37 4.1 Passage of Electrons Through Matter ............................. 37 4.2 Passage of Alpha Particles Through Matter (Rutherford Scattering) ... 39

4.2.1 Some Properties of Alpha Particles .......................... 39 4.2.2 Scattering of Alpha Particles by a Foil ....................... 39 4.2.3 Derivation of the Rutherford Scattering Formula .............. 41 4.2.4 Experimental Results ....................................... 46 4.2.5 What is Meant by Nuclear Radius? .......................... 47

Problems .......................................................... 48

x Contents

5. The Photon ....................................................... 49 5.1 Wave Character of Light. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5.2 Thermal Radiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.2.1 Spectral Distribution of Black Body Radiation ............... 51 5.2.2 Planck's Radiation Formula ................................ 53 5.2.3 Einstein's Derivation of Planck's Formula ................... 54

5.3 The Photoelectric Effect ........................................ 58 5.4 The Compton Effect ........................................... 60

5.4.1 Experiments ............................................. 60 5.4.2 Derivation of the Compton Shift ........................... 62

Problems .......................................................... 64

6. The Electron ...................................................... 67 6.1 Production of Free Electrons .................................... 67 6.2 Size of the Electron ............................................ 67 6.3 The Charge of the Electron ..................................... 68 6.4 The Specific Charge elm of the Electron ......................... 69 6.5 Wave Character of Electrons and Other Particles .................. 72 6.6 Interferometry with Atoms ...................................... 76 Problems .......................................................... 77

7. Some Basic Properties of Matter Waves ............................... 79 7.1 Wave Packets .................................................. 79 7.2 Probabilistic Interpretation ...................................... 83 7.3 The Heisenberg Uncertainty Relation ............................. 85 7.4 The Energy-Time Uncertainty Relation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 7.5 Some Consequences of the Uncertainty Relations for Bound States .. 88 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

8. Bohr's Model of the Hydrogen Atom ................................. 93 8.1 Basic Principles of Spectroscopy ................................. 93 8.2 The Optical Spectrum of the Hydrogen Atom ..................... 95 8.3 Bohr's Postulates .............................................. 98 8.4 Some Quantitative Conclusions .................................. 102 8.5 Motion of the Nucleus ......................................... 103 8.6 Spectra of Hydrogen-like Atoms ................................. 105 8.7 Muonic Atoms ................................................ 107 8.8 Excitation of Quantum Jumps by Collisions ...................... 109 8.9 Sommerfeld's Extension of the Bohr Model

and the Experimental Justification of a Second Quantum Number ... 112 8.10 Lifting of Orbital Degeneracy by the Relativistic Mass Change ...... 113 8.11 Limits of the Bohr-Sommerfeld Theory.

The Correspondence Principle ................................... 114 8.12 Rydberg Atoms ................................................ 115 Problems .......................................................... 117

9. The Mathematical Framework of Quantum Theory ..................... 119 9.1 The Particle in a Box .......................................... 119 9.2 The SchrOdinger Equation ...................................... 123

Contents XI

9.3 The Conceptual Basis of Quantum Theory. . . . . . . . . . . . . . . . . . . . . . 125 9.3.1 Observations, Values of Measurements and Operators ...... 125 9.3.2 Momentum Measurement and Momentum Probability. . . . .. 126 9.3.3 Average Values and Expectation Values ................... 127 9.3.4 Operators and Expectation Values ....................... 130 9.3.5 Equations for Determining the Wavefunction .............. 131 9.3.6 Simultaneous Observability and Commutation Relations .... 133

9.4 The Quantum Mechanical Oscillator ........................... 136 Problems ......................................................... 142

10. Quantum Mechanics of the Hydrogen Atom .......................... 147 10.1 Motion in a Central Field .................................... 147 10.2 Angular Momentum Eigenfunctions. . . . . . . . . . . . . . . . . . . . . . . . . . .. 149 10.3 The Radial Wavefunctions in a Central Field * ................... 155 10.4 The Radial Wavefunctions of Hydrogen ........................ 157 Problems ......................................................... 163

11. Lifting of the Orbital Degeneracy in the Spectra of Alkali Atoms ....... 165 11.1 Shell Structure .............................................. 165 11.2 Screening ................................................... 167 11.3 The Term Diagram ........................................... 168 11.4 Inner Shells ................................................. 173 Problems ......................................................... 173

12. Orbital and Spin Magnetism. Fine Structure .......................... 175 12.1 Introduction and Overview .................................... 175 12.2 Magnetic Moment of the Orbital Motion ....................... 176 12.3 Precession and Orientation in a Magnetic Field .................. 178 12.4 Spin and Magnetic Moment of the Electron. . . . . . . . . . . . . . . . . . . .. 180 12.5 Determination of the Gyromagnetic Ratio

by the Einstein-de Haas Method ............................... 182 12.6 Detection of Directional Quantisation by Stern and Gerlach. . . . . .. 183 12.7 Fine Structure and Spin-Orbit Coupling: Overview ............... 185 12.8 Calculation of Spin-Orbit Splitting in the Bohr Model ........... 186 12.9 Level Scheme of the Alkali Atoms ............................. 190 12.10 Fine Structure in the Hydrogen Atom .......................... 191 12.11 The Lamb Shift ............................................. 192 Problems ......................................................... 196

13. Atoms in a Magnetic Field: Experiments and Their Semiclassical Description ...................... 199 13.1 Directional Quantisation in a Magnetic Field .................... 199 13.2 Electron Spin Resonance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 199 13.3 The Zeeman Effect .......................................... 202

13.3.1 Experiments .......................................... 202 13.3.2 Explanation of the Zeeman Effect from the Standpoint

of Classical Electron Theory ............................ 204 13.3.3 Description of the Ordinary Zeeman Effect

by the Vector Model ................................... 206

XII Contents

13.3.4 The Anomalous Zeeman Effect 13.3.5 Magnetic Moments with Spin-Orbit Coupling ............. .

13.4 The Paschen-Back Effect ..................................... . 13.5 Double Resonance and Optical Pumping ....................... . Problems ........................................................ .

14. Atoms in a Magnetic Field: Quantum Mechanical Treatment ........... . 14.1 Quantum Theory of the Ordinary Zeeman Effect ................ . 14.2 Quantum Theoretical Treatment of the Electron and Proton Spins ..

14.2.1 Spin as Angular Momentum ............................ . 14.2.2 Spin Operators, Spin Matrices and Spin Wavefunctions ..... . 14.2.3 The SchrOdinger Equation of a Spin in a Magnetic Field ... . 14.2.4 Description of Spin Precession by Expectation Values ...... .

14.3 Quantum Mechanical Treatment of the Anomalous Zeeman Effect with Spin-Orbit Coupling * .................................... .

14.4 Quantum Theory of a Spin in Mutually Perpendicular Magnetic Fields, One Constant and One Time Dependent ........ .

14.5 The Bloch Equations ......................................... . 14.6 The Relativistic Theory of the Electron. The Dirac Equation ...... . Problems ........................................................ .

15. Atoms in an Electric Field ........................................ . 15.1 Observations of the Stark Effect ............................... . 15.2 Quantum Theory of the Linear and Quadratic Stark Effects ...... .

15.2.1 The Hamiltonian ...................................... . 15.2.2 The Quadratic Stark Effect.

Perturbation Theory Without Degeneracy * ................ . 15.2.3 The Linear Stark Effect.

Perturbation Theory in the Presence of Degeneracy * ....... . 15.3 The Interaction of a Two-Level Atom

with a Coherent Radiation Field ............................... . 15.4 Spin- and Photon Echoes ..................................... . 15.5 A Glance at Quantum Electrodynamics* ....................... .

15.5.1 Field Quantization ..................................... . 15.5.2 Mass Renormalization and Lamb Shift ................... .

Problems ........................................................ .

16. General Laws of Optical Transitions ................................ . 16.1 Symmetries and Selection Rules ............................... .

16.1.1 Optical Matrix Elements ................................ . 16.1.2 Examples of the Symmetry Behaviour of Wavefunctions .... . 16.1.3 Selection Rules ........................................ . 16.1.4 Selection Rules and Multipole Radiation * ................ .

16.2 Linewidths and Lineshapes .................................... .

17. Many-Electron Atoms . ............................................ . 17.1 The Spectrum of the Helium Atom ............................ . 17.2 Electron Repulsion and the Pauli Principle ...................... .

208 209 211 212 214

215 215 217 217 218 220 222

224

228 233 235 241

243 243 245 245

246

249

252 255 258 258 263 270

273 273 273 273 278 281 284

289 289 291

Contents XIII

17.3 Angular Momentum Coupling .................................. 292 17.3.1 Coupling Mechanism. ... . . ... . . .. . . .. . . ... . ... . .. . .. . ... 292 17.3.2 LS Coupling (Russell-Saunders Coupling) .................. 292 17.3.3 jj Coupling ............................................ 296

17.4 Magnetic Moments of Many-Electron Atoms.. . .... . .. .. . . ... .. .. 298 17.5 Multiple Excitations ........................................... 298 Problems ......................................................... 299

18. X-Ray Spectra, Internal Shells ...................................... 301 18.1 Introductory Remarks ......................................... 301 18.2 X-Radiation from Outer Shells ................................. 301 18.3 X-Ray Bremsstrahlung Spectra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 302 18.4 Emission Line Spectra: Characteristic Radiation .................. 304 18.5 Fine Structure of the X-Ray Spectra. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 306 18.6 Absorption Spectra ........................................... 308 18.7 The Auger Effect ............................................. 310 18.8 Photoelectron Spectroscopy (XPS), ESCA ....................... 312 Problems. .. . .. . ... . .. . .. . .. . .. . .. . . .. . ... .. .. . .. . . .. . . .. . . . . . .... 313

19. Structure of the Periodic System. Ground States of the Elements ........ 315 19.1 Periodic System and Shell Structure ............................. 315 19.2 Ground States of Atoms ....................................... 322 19.3 Excited States and Complete Term Scheme ....................... 324 19.4 The Many-Electron Problem. Hartree-Fock Method * .............. 325

19.4.1 The Two-Electron Problem ............................... 325 19.4.2 Many Electrons Without Mutual Interactions .............. , 330 19.4.3 Coulomb Interaction of Electrons.

Hartree and Hartree-Fock Methods ....................... 331 Problems ......................................................... 334

20. Nuclear Spin, Hyperfine Structure ................................... 337 20.1 Influence of the Atomic Nucleus on Atomic Spectra .............. 337 20.2 Spins and Magnetic Moments of Atomic Nuclei .................. 338 20.3 The Hyperfine Interaction ..................................... 340 20.4 Hyperfine Structure in the Ground States of the Hydrogen

and Sodium Atoms ........................................... 344 20.5 Hyperfine Structure in an External Magnetic Field,

Electron Spin Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 346 20.6 Direct Measurements of Nuclear Spins and Magnetic Moments,

Nuclear Magnetic Resonance ................................... 350 20.7 Applications of Nuclear Magnetic Resonance. . . . . . . . . . . . . . . . . . . .. 354 20.8 The Nuclear Electric Quadrupole Moment ....................... 359 Problems ......................................................... 361

21. The Laser ........................................................ 363 21.1 Some Basic Concepts for the Laser ............................. 363 21.2 Rate Equations and Lasing Conditions .......................... 366 21.3 Amplitude and Phase of Laser Light ............................ 369 Problems ......................................................... 372

XIV Contents

22. Modern Methods of Optical Spectroscopy ............................ 375 22.1 Classical Methods ........................................... 375 22.2 Quantum Beats. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 376 22.3 Doppler-free Saturation Spectroscopy. . . . . . . . . . . . . . . . . . . . . . . . . .. 378 22.4 Doppler-free Two-Photon Absorption. . . . . . . . . . . . . . . . . . . . . . . . . .. 380 22.5 Level-Crossing Spectroscopy and the Hanle Effect ............... 382 22.6 Laser Cooling of Atoms ...................................... 384

23. Fundamentals of the Quantum Theory of Chemical Bonding ........... 389 23.1 Introductory Remarks ........................................ 389 23.2 The Hydrogen-Molecule Ion Hi ............................... 389 23.3 The Tunnel Effect ........................................... 395 23.4 The Hydrogen Molecule Hz ................................... 397 23.5 Covalent-Ionic Resonance ..................................... 404 23.6 The Hund-Mulliken-Bloch Theory of Bonding in Hydrogen ....... 405 23.7 Hybridisation ............................................... 406 23.8 The n Electrons of Benzene, C6H6 ............................. 408 Problems ......................................................... 410

Appendix ............................................................ 411 A. The Dirac Delta Function and the Normalisation

of the Wavefunction of a Free Particle in Unbounded Space ...... 411 B. Some Properties of the Hamiltonian Operator, Its Eigenfunctions

and Its Eigenvalues .......................................... 415 C. Derivation of Heisenberg's Uncertainty Relation ................. 416

Solutions to the Problems ............................................. 419

Bibliography of Supplementary and Specialised Literature . . . . . . . . . . . . . . . . .. 449

Subject Index ........................................................ 453

Fundamental Constants of Atomic Physics (Inside Front Cover)

Energy Conversion Table (Inside Back Cover)

List of the Most Important Symbols Used

The numbers of the equations in which the symbols are defined are given in parenthe-ses; the numbers in square brackets refer to the section of the book. The Greek symbols are at the end of the list.

A Vector potential .Yf Hamilton function, A Amplitude or constant Hamiltonian operator A Mass number (2.2) or area Hn Hermite polynomial a Interval factor or fine struc- h Planck's constant

ture constant (12.28) and n =hl2n hyperfine splitting (20.10) 1,1 Nuclear angular momentum

ao Bohr radius of the H atom in and corresponding quantum its ground state (8.8) number (20.1)

B Magnetic induction 1 Abbreviation for integrals b+,b Creation and annihilation opera- [16.13] or intensity

tors for the harmonic oscillator Imaginary unit (i = 0) b Constant, impact parameter J,J Total angular momentum of an C Constant electron shell and correspond-c Velocity of light, series expan- ing quantum number (17.5)

sion coefficient j,j Total angular momentum of c.c. Complex conjugate an electron and corresponding D Dipole moment quantum number [12.7] d Constant j Operator for the total angular dV Infinitesimal volume element momentum E Electric field strength k Boltzmann's constant, force E Energy, total energy, energy constant

eigenvalue k Wavevector Ekin Kinetic energy L,L Resultant orbital angular Epot Potential energy momentum and corresponding E tot Total energy quantum number (17.3) e Proton charge Ln Laguerre polynomial (10.81) -e Electron charge I, I Orbital angular momentum of e Exponential function an electron and corresponding F Electric field strength (14.1) quantum number F,F Total angular momentum of an i Angular momentum operator

atom, including nuclear angu- m,mo Mass lar momentum and correspond- m Magnetic quantum number ing quantum number (20.6) m, - for angular momentum

F Amplitude of the magnetic in- ms - for spin duction [14.4, 14.5] mj Magnetic quantum number for

f Spring constant total angular momentum g Lande g factor (12.10, 16, 21, mo Rest mass, especially that of

13.18,20.13) the electron

XVI List of the Most Important Symbols Used

N, n Particle number, particle num- \12 Laplace operator ber density = 82/8x2+ 82/8y2+ 82/8z2

N Normalisation factor LIE Energy uncertainty n Principal quantum number or Llk Wavenumber uncertainty

number of photons or an LIp Momentum uncertainty integer LIt Time uncertainty ( = finite

P Spectral radiation flux density measurement time) (5.2) or probability LlV Finite volume element

p? Legendre polynomial Llw Uncertainty in the angular fre-P'f (m "* 0) Associated Legendre quency

function Llx Position uncertainty p,p Momentum, expectation value t5(x) Dirac delta function (see mathe-

of momentum matics appendix) Q Nuclear quadrupole moment t5f1,v Kronecker delta symbol:

(20.20) t5f1, v = 1 for f.1 = V, t5/-l, v = 0 for Q, q Charge f.1"*v R(r) Radial part of the hydrogen e Dimensionless energy (9.S3)

wavefunction e(n) Energy contributions to pertur-r Position coordinate (three-di- bation theory

mensional vector) eo Permittivity constant of r Distance vacuum S Resultant spin (17.4) () Angle coordinate (10.2) S Symbol for orbital angular K Defined in (10.54)

momentum L = 0 A Wavelength (exception: expan-s,s Electron spin and correspond- sion parameter in [15.2.1, 2])

ing quantum number (12.15) /l,f.1 Magnetic moment (12.1) S Spin operator = (sx, Sy, sz) f.1 Reduced mass (S.15) T Absolute temperature flB Bohr magneton (12.S) T1 Longitudinal relaxation time f.1N Nuclear magnet on (20.3) T2 Transverse relaxation time v Frequency [8.1] t Time ii Wavenumber [8.1] u Spectral energy density (5.2), ~ Dimensionless coordinate (9.S3)

atomic mass unit [2.2] (} Charge density, density of V V olume, potential, electric states, mass density; or dim en-

voltage sionless distance V Expectation value of the a Scattering coefficient, interac-

potential energy tion cross section (2.16) v Velocity, particle velocity T Torque (12.2) x Particle coordinate (one- r[> Phase

dimensional) </> Phase angle, angle coordinate x Expectation value of position </>(x) Wavefunction of a particle Y/,m«(), </» Spherical harmonic func- </>r, </> t, </> Spin wavefunctions

tions (10.10, 4S-50) If/ Wavefunction Z Nuclear charge If' Wavefunction of several electrons a Fine structure constant [8.10] Q Generalised quantum

or absorption coefficient (2.22) mechanical operator

fJ Constant Q Frequency [14.4, 14.5, 15.3] r Decay constant w Angular frequency 21lv, or y Decay constant or linewidth eigenvalue [9.3.6]

gyromagnetic ratio (12.12) ~ means "corresponds to"