Atomic and Quantum Physics

Energy Conversion Table J 1 Joule (J) = 1 1 eVolt (eV) 1 cm -1 = 1K eV K 6.24146· 10 18 5.03404. 1022 7.24290.1022 1.60219· 10- 19 1 8.06548· 103 1.98648. 10 -23 1.23985. 10 -4 1 1.16045 . 104 = 1.43879 23 5 1 = 1.38066.10- 8.61735.10- 6.95030.10- 1 Explanation The energy E is quoted in Joule (J) or watt-seconds (Ws) 1J = 1 Ws. In spectroscopy, one frequently quotes the term values in wavenumbers v=Elhc. The conversion factor is Elv = hc = 1.98648 . 10- 23 J/cm -1. Another energy unit, especially in collision experiments, is the electron volt (eVolt, eV). The voltage Vis given in volts, and the energy conversion factor is obtained from E = eV: EIV = e = 1.60219.10- 19 J/V . In the case of thermal excitation with the heat energy kT, the absolute tem- perature is a measure of the energy. From E factor EIT= k = 1.38066 .10- 23 J/K. = kT we obtain the conversion H. Haken H. C. Wolf Atomic and Quantum Physics An Introduction to the Fundamentals of Experiment and Theory Translated by W D. Brewer Second Enlarged Edition With 265 Figures Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Professor Dr. Dr. h. c. Hermann Haken Institut fiir Theoretische Physik, Universitiit Stuttgart, Pfaffenwaldring 57, 0-7000 Stuttgart 80, Fed. Rep. of Germany Professor Dr. Hans Christoph Wolf Physikalisches Institut, Universitiit Stuttgart, Pfaffenwaldring 57, 0-7000 Stuttgart 80, Fed. Rep. of Germany Translator: Professor Dr. William D. Brewer Freie Universitiit Berlin, Fachbereich Physik, Arnimallee 14, 0-1000 Berlin 33 Title of the german original edition: H. Haken, H. C. Wolf: Atom- und Quantenphysik. Eine Einj'ahrung in die experimentellen und theoretischen Grundlagen. (Oritte, iiberarbeitete und erweiterte Auflage) © Springer-Verlag Berlin Heidelberg 1980, 1983, and 1987 ISBN-13: 978-3-540-17702-9 DOl: 10.1007/978-3-642-97014-6 e-ISBN-13: 978-3-642-97014-6 Library of Congress Cataloging-in-Publication Data. Haken, H. Atomic and quantum physics. Translation of: Atom- und Quantenphysik. Bibliography: p. Includes index. 1. Atoms. 2. Quantum theory. I. Wolf, H.C. (Hans Christoph), 1929·. II. Title. QC173.H17513 1987 539.7 87-9450 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1984 and 1987 The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. 2153/3150-543210 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 SchOtz 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 proof-reading. 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 .......................................... 1 1 1 2. The Mass and Size of the Atom ....................................... 2.1 What is an Atom? ............................................... 2.2 Determination of the Mass ....................................... 2.3 Methods for Determining Avogadro's Number ...................... 2.3.1 Electrolysis ............................................... 2.3.2 The Gas Constant and Boltzmann's Constant .................. 2.3.3 X-Ray Diffraction in Crystals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Determination Using Radioactive Decay. .. . . . . . . . . . . . . . .. . . . . . 2.4 Determination of the Size ofthe Atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Application of the Kinetic Theory of Gases .................... 2.4.2 The Interaction Cross Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Experimental Determination of Interaction Cross Sections ....... 2.4.4 Determining the Atomic Size from the Covolume ............... 2.4.5 Atomic Sizes from X-Ray Diffraction Measurements on Crystals. . 2.4.6 Can Individual Atoms Be Seen? .............................. Problems .......................................................... 5 5 5 7 7 7 8 9 10 10 11 14 15 15 20 23 3. Isotopes ........ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 The Periodic System of the Elements ............................... 3.2 Mass Spectroscopy .............................................. 3.2.1 Parabola Method. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Improved Mass Spectrometers ............................... 3.2.3 Results of Mass Spectrometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Modern Applications of the Mass Spectrometer ................ 3.2.5 Isotope Separation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems .......................................................... 25 25 27 27 30 31 32 33 34 4. The Nucleus of the Atom ............................................ 4.1 Passage of Electrons Through Matter .............................. 4.2 Passage of Alpha Particles Through Matter (Rutherford Scattering) . . . . . 4.2.1 Some Properties of Alpha Particles ........................... 4.2.2 Scattering of Alpha Particles by a Foil ........................ 4.2.3 Derivation of the Rutherford Scattering Formula ............... 4.2.4 Experimental Results ................ . . . . . . . . . . . . . . . . . . . . . . . 4.2.5 What is Meant by Nuclear Radius? ........................... Problems .......................................................... 35 35 37 37 37 39 44 45 46 x Contents 5. The Photon ........................................................ 5.1 Wave Character of Light ........................................ 5.2 Thermal Radiation ............................................. 5.2.1 Spectral Distribution of Black Body Radiation ................ 5.2.2 Planck's Radiation Formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Einstein's Derivation of Planck's Formula. . . . . . . . . . . . . . . . . . . . 5.3 The Photoelectric Effect ........................................ 5.4 The Compton Effect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Experiments ............................................. 5.4.2 Derivation of the Compton Shift ............................ Problems .......................................................... 47 47 49 49 51 52 56 58 58 60 62 6. The Electron ....................................................... 6.1 Production of Free Electrons .................................... 6.2 Size of the Electron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 The Charge of the Electron ...................................... 6.4 The Specific Charge elm of the Electron ........................... 6.5 Wave Character of Electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems .......................................................... 65 65 65 66 67 70 74 7. Some Basic Properties of Matter Waves ................................ 7 .1 Wave Packets ................................................. 7.2 Probabilistic Interpretation ...................................... 7.3 The Heisenberg Uncertainty Relation ............................. 7.4 The Energy-Time Uncertainty Relation ............................ 7.5 Some Consequences of the Uncertainty Relations for Bound States .... Problems .......................................................... 77 77 81 83 85 86 89 8. Bohr's Model of the Hydrogen Atom .................................. 8.1 Basic Principles of Spectroscopy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 The Optical Spectrum of the Hydrogen Atom. . . . . . . . . . . . . . . . . . . . . . . 8.3 Bohr's Postulates .............................................. 8.4 Some Quantitative Conclusions .................................. 8.5 Motion of the Nucleus .......................................... 8.6 Spectra of Hydrogen-like Atoms ................................. 8.7 MuonicAtoms ................................................ 8.8 Excitation of Quantum Jumps by Collisions ........................ 8.9 Sommerfeld's Extension of the Bohr Model and the Experimental Justification of a Second Quantum Number ........................ 8.10 Lifting of Orbital Degeneracy by the Relativistic Mass Change ........ 8.11 Limits of the Bohr-Sommerfeld Theory. The Correspondence Principle. 8.12 Rydberg Atoms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . 91 91 93 96 100 101 103 105 107 110 111 112 113 115 9. The Mathematical Framework of Quantum Theory ...................... 9.1 The Particle in a Box ........................................... 9.2 The SchrOdinger Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 The Conceptual Basis of Quantum Theory ......................... 9.3.1 Observations, Values of Measurements and Operators... . ...... 117 117 121 123 123 Contents XI 9.3.2 Momentum Measurement and Momentum Probability ...... 9.3.3 Average Values and Expectation Values ................... 9.3.4 Operators and Expectation Values. . . .. . .. . . . .. . . .. . . . . . . . 9.3.5 Equations for Determining the Wavefunction .............. 9.3.6 Simultaneous Observability and Commutation Relations. .. . . 9.4 The Quantum Mechanical Oscillator ............................ Problems. . . .. . .. . .. . .. . . . . .. .. . . . .. . . . .. . . . .. . .. . .. . . . . . .. . . . .. . . 124 125 128 129 131 134 140 10. Quantum Mechanics of the Hydrogen Atom ........................... 10.1 Motion in a Central Field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Angular Momentum Eigenfunctions ............................ 10.3 The Radial Wavefunctions in a Central Field * .................... 10.4 The Radial Wavefunctions of Hydrogen ......................... Problems ......................................................... 145 145 147 153 155 161 11. Lifting of the Orbital Degeneracy in the Spectra of Alkali Atoms .......... 11.1 Shell Structure . .. . . .. . .. . . . .. . . . .. . . . .. . .. . .. . .. . . .. . . .. . .. . . 11.2 Screening................................................... 11.3 The Term Diagram ........................................... 11.4 Inner Shells ................................................. Problems ......................................................... 163 163 165 166 171 171 12. Orbital and Spin Magnetism. Fine Structure ........................... 12.1 Introduction and Overview .................................... 12.2 Magnetic Moment of the Orbital Motion . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Precession and Orientation in a Magnetic Field ................... 12.4 Spin and Magnetic Moment of the Electron ...................... 12.5 Determination of the Gyromagnetic Ratio by the Einstein-de Haas Method. . . . ... . .. . . .. .. .. . .. . .. . . . . . . . . 12.6 Detection of Directional Quantisation by Stern and Gerlach ........ 12.7 Fine Structure and Spin-Orbit Coupling: Overview ................ 12.8 Calculation of Spin-Orbit Splitting in the Bohr Model. . . . . . . . . . . . . . 12.9 Level Scheme ofthe Alkali Atoms .............................. 12.10 Fine Structure in the Hydrogen Atom ........................... 12.11 The Lamb Shift.............................................. Problems ......................................................... 173 173 174 176 178 13. Atoms in a Magnetic Field: Experiments and Their Semiclassical Description 13.1 Directional Quantisation in a Magnetic Field ..................... 13.2 Electron Spin Resonance ...................................... 13.3 The Zeeman Effect ........................................... 13.3.1 Experiments .......................................... 13.3.2 Explanation of the Zeeman Effect from the Standpoint of Classical Electron Theory ............................... 13.3.3 Description of the Ordinary Zeeman Effect by the Vector Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 13.3.4 The Anomalous Zeeman Effect .......................... 13.3.5 Magnetic Moments with Spin-Orbit Coupling .............. 13.4 The Paschen-Back Effect. . ... . . ... . . .. . .. . ... . .. ... .. . . .... . .. 180 181 183 184 188 189 190 194 197 197 197 200 200 202 204 206 207 209 XII Contents 13.5 Double Resonance and Optical Pumping. . . . . . . . . . . . . . . . . . . . . . . . . . Problems ......................................................... 210 212 14. Atoms in a Magnetic Field: Quantum Mechanical Treatment ............. 14.1 Quantum Theory of the Ordinary Zeeman Effect . . . . . . . . . . . . . . . . . . . 14.2 Quantum Theoretical Treatment ofthe 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 ......................................................... 213 213 215 215 216 218 220 222 226 231 233 239 15. Atoms in an Electric Field ........................................... 15.1 Observations of the Stark Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 15.2 Quantum Theory ofthe 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 ......................................................... 247 250 253 256 256 261 268 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 ..................................... 271 271 271 271 276 279 282 17. Many-Electron Atoms .............................................. 17.1 The Spectrum of the Helium Atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2 Electron Repulsion and the Pauli Principle ........................ 17.3 Angular Momentum Coupling .................................. 17.3.1 Coupling Mechanism. . . . .. . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. 17.3.2 LS Coupling (Russell-Saunders Coupling) .................. 287 287 289 290 290 290 241 241 243 243 244 Contents XIII 17.3.3 jj Coupling ............................................ 17.4 Magnetic Moments of Many-Electron Atoms ...................... 17.5 Multiple Excitations ........................................... Problems ......................................................... 294 296 296 297 18. X-Ray Spectra, Internal Shells ....................................... 18.1 Introductory Remarks ......................................... 18.2 X-Radiation from Outer Shells .................................. 18.3 X-Ray Bremsstrahlung Spectra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.4 Emission Line Spectra: Characteristic Radiation ................... 18.5 Fine Structure of the X-Ray Spectra. .. . . .. . . .. . .. . . . . . . . . . . . . . . . . 18.6 Absorption Spectra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . 18.7 The Auger Effect (Inner Photoeffect) ............................ 18.8 Photoelectron Spectroscopy (XPS), ESCA ........................ Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 299 299 300 302 304 306 308 310 311 19. Structure of the Periodic System. Ground States of the Elements .......... 19.1 Periodic System and Shell Structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.2 Ground States of Atoms ....................................... 19.3 Excited States and Complete Term Scheme. . . . . . . . . . . . . . . . . . . . . . . . 19.4 The Many-Electron Problem. Hartree-Fock Method * .............. 19.4.1 The Two-Electron Problem .............................. 19.4.2 Many Electrons Without Mutual Interactions ............... 19.4.3 Coulomb Interaction of Electrons. Hartree and Hartree-Fock Methods .............................................. Problems ......................................................... 313 313 320 322 323 323 328 20. Nuclear Spin, Hyperfine Structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.1 Influence of the Atomic Nucleus on Atomic Spectra ................ 20.2 Spins and Magnetic Moments of Atomic Nuclei .................... 20.3 The Hyperfine Interaction ....... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.4 Hyperfine Structure in the Ground States of the Hydrogen and Sodium Atoms....................................................... 20.5 Hyperfine Structure in an External Magnetic Field, Electron Spin Resonance ....................................... 20.6 Direct Measurements of Nuclear Spins and Magnetic Moments, Nuclear Magnetic Resonance ................................... 20.7 Applications of Nuclear Magnetic Resonance . . . . . . . . . . . . . . . . . . . . . . 20.8 The Nuclear Electric Quadrupole Moment ........................ Problems ......................................................... 335 335 336 338 329 332 342 344 348 352 357 359 21. The Laser ........................................................ 21.1 Some Basic Concepts for the Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Rate Equations and Lasing Conditions ........................... 21.3 Amplitude and Phase of Laser Light ............................. Problems ......................................................... 361 361 364 367 370 22. Modem Methods of Optical Spectroscopy ............................. 22.1 Classical Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Quantum Beats ............................................... 373 373 374 XIV Contents 22.3 Doppler-free Saturation Spectroscopy. . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.4 Doppler-free Two-Photon Absorption ........................... 22.5 Level-Crossing Spectroscopy and the Hanle Effect ................. 376 378 380 23. Fundamentals of the Quantum Theory of Chemical Bonding ............. 23.1 Introductory Remarks ......................................... 23.2 The Hydrogen-Molecule Ion Hi ................................ 23.3 The Tunnel Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.4 The Hydrogen Molecule H2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.5 Covalent-Ionic Resonance. ... . ... . . .... . ... . ... . ... . . . . . .. . ... . 23.6 The Hund-Mulliken-Bloch Theory of Bonding in Hydrogen ......... 23.7 Hybridisation ................................................ 23.8 The 1l'Electrons of Benzene, C~6 ............................... Problems ......................................................... 383 383 383 389 391 398 399 400 402 404 Appendix ............................................................ A. The Dirac Delta Function and the Normalisation of the Wavefunction of a Free Particle in Unbounded Space ........................... B. Some Properties of the Hamiltonian Operator, Its Eigenfunctions and Its Eigenvalues ............................................... 405 Solutions to the Problems .............................................. 411 Bibliography of Supplementary and Specialised Literature .. . . . . . . . . . . . . . . . . . 441 Subject Index ................... : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 445 Fundamental Constants of Atomic Physics (Inside Front Cover) Energy Conversion Table (Inside Back Cover) 405 409 List of the Most Important Symbols Used The numbers of the equations in which the symbols are defined are given in parentheses; the numbers in square brackets refer to the section of the book. The Greek symbols are at the end of the list. Vector potential Amplitude or constant Mass number (2.2) or area Interval factor or fine struca ture constant (12.28) and hyperfine splitting (20.10) Bohr radius of the H atom in ao its ground state (8.8) Magnetic induction B b+,b Creation and annihilation operators for the harmonic oscillator Constant, impact parameter b Constant C Velocity of light, series expanc sion coefficient Complex conjugate c.c. Dipole moment D Constant d Infinitesimal volume element dV Electric field strength E Energy, total energy, energy E eigenvalue Kinetic energy E kin Potential energy Epot Total energy E tot Proton charge e Electron charge -e Exponential function e Electric field strength (14.1) F Total angular momentum of an F,F atom, including nuclear angular momentum and corresponding quantum number (20.6) Amplitude of the magnetic inF duction [14.4, 14.5] Spring constant f Lande g factor (12.10, 16,21, g 13.18,20.13) A A A h Hamilton function, Hamiltonian operator Hermite polynomial Planck's constant h =h12n .Yf Hn Nuclear angular momentum and corresponding quantum number (20.1) Abbreviation for integrals I [16.13] or intensity i Imaginary unit (i = V=l) J,J Total angular momentum of an electron shell and corresponding quantum number (17.5) Total angular momentum of j,j an electron and corresponding quantum number [12.7] j Operator for the total angular momentum Boltzmann's constant, force k constant Wavevector k L,L Resultant orbital angular momentum and corresponding quantum number (17.3) Laguerre polynomial (10.81) Ln Orbital angular momentum of 1, I an electron and corresponding quantum number Angular momentum operator i m,mo Mass Magnetic quantum number m - for angular momentum m[ - for spin ms Magnetic quantum number for mj total angular momentum mo Rest mass, especially that of the electron 1,1 XVI List of the Most Important Symbols Used Particle number, particle number density N Normalisation factor n Principal quantum number or number of photons or an integer P Spectral radiation flux density (5.2) or probability p? Legendre polynomial P7' (m 0) Associated Legendre function p, jj Momentum, expectation value of momentum Q Nuclear quadrupole moment (20.20) Q, q Charge R(r). Radial part of the hydrogen wavefunction r Position coordinate (three-dimensional vector) r Distance S Resultant spin (17.4) S Symbol for orbital angular momentum L = 0 s, s Electron spin and corresponding quantum number (12.15) s Spin operator = (sx, sy. sz) T Absolute temperature Tl Longitudinal relaxation time T2 Transverse relaxation time t Time u Spectral energy density (5.2), atomic mass unit [2.2] V Volume, potential, electric voltage V Expectation value of the potential energy v Velocity, particle velocity x Particle coordinate (onedimensional) x Expectation value of position Yt.m((J, ¢J) Spherical harmonic functions (10.10, 48 - 50) Z Nuclear charge a Fine structure constant [8.10] or absorption coefficient (2.22) f3 Constant r Decay constant y Decay constant or linewidth gyromagnetic ratio (12.12) N, n *" \12 Laplace operator LJE LJk LJp LJt Energy uncertainty Wavenumber uncertainty Momentum uncertainty Time uncertainty ( = finite measurement time) Finite volume element Uncertainty in the angular frequency Position uncertainty Dirac delta function (see mathematics appendix) Kronecker delta symbol: J/J,v= 1 for f-l = V, J/J,v= 0 for = 82/8x 2+ 82/8y2+ 82/8z 2 LJV LJw LJx J(x) J/J,v f-l*"v Dimensionless energy (9.83) Energy contributions to perturbation theory Permittivity constant of eo vacuum (J Angle coordinate (10.2) Defined in (10.54) K Wavelength (exception: expanA. sion parameter in [15.2.1, 2]) fI.,f-l Magnetic moment (12.1) Reduced mass (8.15) f-l Bohr magneton (12.8) f-lB Nuclear magneton (20.3) f-lN Frequency [8.1] v v Wavenumber [8.1] Dimensionless coordinate (9.83) Charge density, density of (! states, mass density; or dimensionless distance (J Scattering coefficient, interaction cross section (2.16) T Torque (12.2) f/J Phase ¢J Phase angle, angle coordinate ¢J(x) Wavefunction of a particle ¢Jr, ¢JL, ¢J Spin wavefunctions If! Wavefunction If' Wavefunction of several electrons Q Generalised quantum mechanical operator D Frequency [14.4, 14.5, 15.3] w Angular frequency 2nv, or eigenvalue [9.3.6] ~ means "corresponds to" e e(n) e