Energy Conversion Table

J eV cm- 1

1 Joule (J) = 1 6.24146· 1018 5.03404. 1022

1 eVolt (eV) = 1.60219.10- 19 1 8.06548.103

= 1.98648.10- 23 1.23985.10-4 1

K

7.24290.1022

1.16045.104

1.43879

1 K = 1.38066.10- 23 8.61735.10- 5 6.95030.10- 1 1

Explanation

The energy E is quoted in Joule (1) or watt-seconds (Ws)

1 J = 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 k T, the absolute tem­perature is a measure of the energy. From E = kT we obtain the conversion factor

EIT= k = 1.38066.10- 23 J/K.

Raken . Wolf The Physics of Atoms and Quanta

Springer Berlin Heidelberg New York Barcelona Budapest Hong Kong London Milan Paris Santa Clara Singapore Tokyo

Hermann Haken Hans Christoph Wolf

The Physics of Atoms and Quanta Introduction to Experiments and Theory

Translated by William D. Brewer

Fifth Edition With 279 Figures, 167 Problems and Solutions

Springer

Professor Dr. Dr. h. c. Hermann Haken

Institut fur Theoretische Physik, Universitiit Stuttgart, Pfaffenwaldring 57 0-70550 Stuttgart, Germany

Professor Dr. Hans Christoph Wolf

Physikalisches Institut, Universitiit Stuttgart, Pfaffenwaldring 57 0-70550 Stuttgart, Germany

Translator:

Professor Dr. William D. Brewer

Freie Universitiit Berlin, Fachbereich Physik, Arnimallee 14 0-14195 Berlin, Germany

Title of the German original edition: H. Haken, H. C. Wolf: Atom- und Quantenphysik. Einfuhrung in die experimentellen und theoretischen Grundlagen. (Sechste, verbesserte und erweiterte Auflage) © Springer-Verlag Berlin Heidelberg 1980, 1983, 1987, 1990, 1993, 1996

ISBN-13: 978-3-642-97691-9 e-ISBN-13: 978-3-642-97689-6 DOl: 10.1007/978-3-642-97689-6

Die Deutsche Bibliothek - CIP-Einheitsaufnahme Haken, Hermann: The physics of atoms and quanta: introduction to experiments and theory; with 167 problems and solutions / Hermann Haken ; Hans Christoph Wolf. Trans!. by William D. Brewer. - 5. ed. -Berlin; Heidelberg; New York; Barcelona; Budapest; Hong Kong; London; Milan; Paris; Santa Clara; Singapore; Tokyo: Springer, 1996 ISBN-13: 978-3-642-97691-9 NE: Wolf, Hans C.:

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 microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9,1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law.

© Springer-Verlag Berlin Heidelberg 1984, 1987, 1993, 1994, 1996 Softcover reprint of the hardcover 5th edition 1996

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 therefore free for general use.

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Preface to the Fifth Edition

Thanks to the favourable reception of this book by students and teachers, a new edition has become necessary in a relatively short time. We have taken advantage of this to in­clude some important new developments in the book, for example the recent detection of antihydrogen and the production of very highly ionised atoms. It is remarkable how high-energy physics has repeatedly succeeded in providing new impulses for atomic physics.

Once again, colleagues, coworkers, and students have helped us to eliminate minor errors, in particular typographical errors. We thank them and also the personnel of Springer-Verlag, especially Dr. H. J. Kolsch and C.-D. Bachem, for their as always ex­cellent cooperation in preparing this edition.

Stuttgart, May 1996 H. Haken H. C. Wolf

Preface to the Fourth Edition

This fourth edition contains a few additional figures. Otherwise only typographical er­rors have been removed.

The final chapter on Fundamentals of the Quantum Theory of Chemical Bonding is continued in an extended way in the textbook Molecular Physics and Elements of Quantum Chemistry by the same authors. This book contains, in particular, a profound presentation of group theory as applied to atoms and molecules. Furthermore, the in­teraction between atoms and molecules and light is treated in detail.

We thank again Springer-Verlag, in particular Dr. H. J. Kolsch and Mr. C.-D. Bachem for their excellent cooperation as always, and Prof. W. D. Brewer for his con­tinuous support in translating our German text.

Stuttgart, February 1994 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 micro world of atoms caused a revolution of physical thought, and that 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 further 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 Atom- and 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 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

VIII Preface to the First Edition

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 1.2 Short Historical Review ........................................ 1

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 ...................................................... . 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

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 and Other Particles ................ . 6.6 Interferometry with Atoms .................................... . Problems ........................................................ .

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 ........................................................ .

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 Muonic Atoms ............................................... . 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 .............................................. . 8.13 Positronium, Muonium, and Antihydrogen ...................... . Problems ........................................................ .

9. The Mathematical Framework of Quantum Theory ................... . 9.1 The Particle in a Box ......................................... . 9.2 The Schrodinger Equation ..................................... .

49 49 51 51 53 54 58 60 60 62 64

69 69 69 70 71 74 78 79

81 81 85 87 89 90 93

95 95 97

100 104 105 107 109 111

114 115

116 117 119 121

125 125 129

Contents XI

9.3 The Conceptual Basis of Quantum Theory ..................... 131 9.3.1 Observations, Values of Measurements and Operators..... 131 9.3.2 Momentum Measurement and Momentum Probability. .... 132 9.3.3 Average Values and Expectation Values ................. 133 9.3.4 Operators and Expectation Values ...................... 136 9.3.5 Equations for Determining the Wave function ..... . . . . . . . . 137 9.3.6 Simultaneous Observability and Commutation Relations ... 139

9.4 The Quantum Mechanical Oscillator ........................... 142 Problems ........................................................ 148

10. Quantum Mechanics of the Hydrogen Atom ......................... 153 10.1 Motion in a Central Field .................................... 153 10.2 Angular Momentum Eigenfunctions ........................... 155 10.3 The Radial Wave functions in a Central Field * .................. 161 10.4 The Radial Wavefunctions of Hydrogen ....................... 163 Problems ........................................................ 169

11. Lifting of the Orbital Degeneracy in the Spectra of Alkali Atoms ....... 171 11.1 Shell Structure .............................................. 171 11.2 Screening................................................... 173 11.3 The Term Diagram .......................................... 174 11.4 Inner Shells ................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 179 Problems ........................................................ 179

12. Orbital and Spin Magnetism. Fine Structure ........ . . . . . . . . . . . . . . . . .. 181 12.1 Introduction and Overview ................................... 181 12.2 Magnetic Moment of the Orbital Motion. . . . . . . . .. . . . . .. . . . . . .. 182 12.3 Precession and Orientation in a Magnetic Field ................. 184 12.4 Spin and Magnetic Moment of the Electron .................... 186 12.5 Determination of the Gyromagnetic Ratio

by the Einstein-de Haas Method .............................. 188 12.6 Detection of Directional Quantisation by Stern and Gerlach ...... 189 12.7 Fine Structure and Spin-Orbit Coupling: Overview .............. 191 12.8 Calculation of Spin-Orbit Splitting in the Bohr Model ........... 192 12.9 Level Scheme of the Alkali Atoms ............................ 196 12.10 Fine Structure in the Hydrogen Atom ......................... 197 12.11 The Lamb Shift ............................................. 198 Problems ........................................................ 202

13. Atoms in a Magnetic Field: Experiments and Their Semiclassical Description ...................... 205 13.1 Directional Quantisation in a Magnetic Field ................... 205 13.2 Electron Spin Resonance ..................................... 205 13.3 The Zeeman Effect .......................................... 208

13.3.1 Experiments .......................................... 208 13.3.2 Explanation of the Zeeman Effect from the Standpoint

of Classical Electron Theory ........................... 210 13.3.3 Description of the Ordinary Zeeman Effect

by the Vector Model .................................. 212

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 ....................................................... .

214 215 217 218 220

14. Atoms in a Magnetic Field: Quantum Mechanical Treatment ........... 223 14.1 Quantum Theory of the Ordinary Zeeman Effect ................ 223 14.2 Quantum Theoretical Treatment of the Electron and Proton Spins . 225

14.2.1 Spin as Angular Momentum............................. 225 14.2.2 Spin Operators, Spin Matrices and Spin Wavefunctions .... 226 14.2.3 The Schr6dinger Equation of a Spin in a Magnetic Field ... 228 14.2.4 Description of Spin Precession by Expectation Values ...... 230

14.3 Quantum Mechanical Treatment of the Anomalous Zeeman Effect with Spin-Orbit Coupling * .................................... 232

14.4 Quantum Theory of a Spin in Mutually Perpendicular Magnetic Fields, One Constant and One Time Dependent ......... 236

14.5 The Bloch Equations ......................................... 241 14.6 The Relativistic Theory of the Electron. The Dirac Equation ...... 243 Problems ........................................................ 249

15. Atoms in an Electric Field ......................................... 251 15.1 Observations of the Stark Effect ............................... 251 15.2 Quantum Theory of the Linear and Quadratic Stark Effects ....... 253

15.2.1 The Hamiltonian. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 253 15.2.2 The Quadratic Stark Effect.

Perturbation Theory Without Degeneracy * ................ 254 15.2.3 The Linear Stark Effect.

Perturbation Theory in the Presence of Degeneracy * ....... 257 15.3 The Interaction of a Two-Level Atom

with a Coherent Radiation Field ............................... 260 15.4 Spin and Photon Echoes ...................................... 263 15.5 A Glance at Quantum Electrodynamics * ........................ 266

15.5.1 Field Quantization ..................................... 266 15.5.2 Mass Renormalization and Lamb Shift ................... 271

Problems ........................................................ 278

16. General Laws of Optical Transitions ................................ 281 16.1 Symmetries and Selection Rules ................................ 281

16.1.1 Optical Matrix Elements ................................ 281 16.1.2 Examples of the Symmetry Behaviour of Wavefunctions .... 281 16.1.3 Selection Rules ........................................ 286 16.1.4 Selection Rules and Multipole Radiation * ................. 289

16.2 Linewidths and Lineshapes .................................... 292

17. Many-Electron Atoms ............................................. 297 17.1 The Spectrum of the Helium Atom ............................. 297 17.2 Electron Repulsion and the Pauli Principle ...................... 299

Contents XIII

17.3 Angular Momentum Coupling ................................. 300 17.3.1 Coupling Mechanism ................................... 300 17.3.2 LS Coupling (Russell-Saunders Coupling) ................. 300 17.3.3 jj Coupling... . . . . .. . . . . ... . . ... . . . . . . . .. .... . . . . . ... .. 304

17.4 Magnetic Moments of Many-Electron Atoms .................... 306 17.5 Multiple Excitations .......................................... 307 Problems ........................................................ 307

18. X-Ray Spectra, Internal Shells ...................................... 309 18.1 Introductory Remarks ........................................ 309 18.2 X-Radiation from Outer Shells ................................. 309 18.3 X-Ray Bremsstrahlung Spectra. .. . . . . . . . . .. . . . . . . . . . . . . . . . . . . .. 310 18.4 Emission Line Spectra: Characteristic Radiation . . . . . . . . . . . . . . . . .. 312 18.5 Fine Structure of the X-Ray Spectra. . . . . . . .. . . . . . . . . . .. . . . . . . .. 314 18.6 Absorption Spectra. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . .. 316 18.7 The Auger Effect ............................................ 318 18.8 Photoelectron Spectroscopy (XPS), ESCA ....................... 320 Problems ........................................................ 322

19. Structure of the Periodic System. Ground States of the Elements ....... 323 19.1 Periodic System and Shell Structure ............................ 323 19.2 Ground States of Atoms ...................................... 330 19.3 Excited States and Complete Term Scheme ...................... 333 19.4 The Many-Electron Problem. Hartree-Fock Method * ............. 335

19.4.1 The Two-Electron Problem ............................. 335 19.4.2 Many Electrons Without Mutual Interactions .............. 340 19.4.3 Coulomb Interaction of Electrons.

Hartree and Hartree-Fock Methods ...................... 341 Problems ........................................................ 344

20. Nuclear Spin, Hyperfine Structure .................................. 347 20.1 Influence of the Atomic Nucleus on Atomic Spectra .............. 347 20.2 Spins and Magnetic Moments of Atomic Nuclei .................. 348 20.3 The Hyperfine Interaction ..................................... 350 20.4 Hyperfine Structure in the Ground State of the Hydrogen Atom,

the Sodium Atom, and the Hydrogen-like Ion 83Bi82+ ............ 354 20.5 Hyperfine Structure in an External Magnetic Field,

Electron Spin Resonance ...................................... 356 20.6 Direct Measurements of Nuclear Spins and Magnetic Moments,

Nuclear Magnetic Resonance .................................. 361 20.7 Applications of Nuclear Magnetic Resonance .................... 364 20.8 The Nuclear Electric Quadrupole Moment. . . . . . . . . . . . . . . . . . . . . .. 369 Problems ........................................................ 371

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 ....................................................... .

373 373 376 379 382

XIV Contents

22. Modern Methods of Optical Spectroscopy ... . . . . . . . . . . . . . . . . . . . . . . . .. 385 22.1 Classical Methods ............................................ 385 22.2 Quantum Beats .............................................. 386 22.3 Doppler-free Saturation Spectroscopy ........................... 388 22.4 Doppler-free Two-Photon Absorption .......................... 390 22.5 Level-Crossing Spectroscopy and the HanIe Effect ............... 392 22.6 Laser Cooling of Atoms ...................................... 394

23. Fundamentals of the Quantum Theory of Chemical Bonding ........... 399 23.1 Introductory Remarks ........................................ 399 23.2 The Hydrogen-Molecule Ion Hi ............................... 399 23.3 The Tunnel Effect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 405 23.4 The Hydrogen Molecule H2 ................................... 407 23.5 Covalent-Ionic Resonance ..................................... 414 23.6 The Hund-Mulliken-Bloch Theory of Bonding in Hydrogen ....... 415 23.7 Hybridisation ................................................ 416 23.8 The n Electrons of Benzene, C6H6 ............................. 418 Problems ........................................................ 420

Appendix ............................................................ 421 A. The Dirac Delta Function and the Normalisation

of the Wavefunction of a Free Particle in Unbounded Space ...... 421 B. Some Properties of the Hamiltonian Operator, Its Eigenfunctions

and Its Eigenvalues ........................................... 425 C. Derivation of Heisenberg's Uncertainty Relation ................. 426

Solutions to the Problems ............................................. 429

Bibliography of Supplementary and Specialised Literature ................. 459

Subject Index ........................................................ 465

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 parenthe­ses; 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 £' 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 =h12n 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 I Abbreviation for integrals b+, b Creation and annihilation opera- [16.13] or intensity

tors for the harmonic oscillator i 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 E kin Kinetic energy L,L Resultant orbital angular Epot Potential energy momentum and corresponding Etot Total energy quantum number (17.3) e Proton charge Ln Laguerre polynomial (10.81) -e Electron charge T, 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

N, n 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 =I=- 0) Associated Legendre

function p, j5 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-di-

mensional vector) r Distance S Resultant spin (17.4) S Symbol for orbital angular

momentum L = 0 S, s Electron spin and correspond-

ing quantum number (12.15) S Spin operator = (sn SY' sz) T Absolute temperature T j 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 (one­

dimensional) x Expectation value of position Y/,m(B, ¢) Spherical harmonic func­

tions (10.10, 48-50) Z Nuclear charge a Fine structure constant [8.10]

or absorption coefficient (2.22) fJ Constant r Decay constant y Decay constant or linewidth

gyro magnetic ratio (12.12)

List of the Most Important Symbols Used

\12 Laplace operator = 82/8x 2+ 82/8y2+ 82/8z 2

LIE Energy uncertainty Llk Wavenumber uncertainty Lip Momentum uncertainty LI t Time uncertainty ( = finite

measurement time) LI V Finite volume element LI OJ Uncertainty in the angular fre­

quency Llx Position uncertainty J(x) Dirac delta function (see mathe­

matics appendix) Jj1, v Kronecker delta symbol:

Jj1,V = 1 for f.1 = V, J/l,V = 0 for f.1=1=-v e Dimensionless energy (9.83) e(n) Energy contributions to pertur­

bation theory eo Permittivity constant of

vacuum B Angle coordinate (10.2) K Defined in (10.54) A Wavelength (exception: expan­

sion parameter in [15.2.1, 2]) Mean free path [2.4.3]

p, f.1 Magnetic moment (12.1) f.1 Reduced mass (8.15) f.1B Bohr magneton (12.8) f.1N Nuclear magneton (20.3) v Frequency [8.1] v Wavenumber [8.1] ~ Dimensionless coordinate (9.83) {.! Charge density, density of

states, mass density; or dimen­sionless distance

(J Scattering coefficient, interac-tion cross section (2.16)

T Torque (12.2) ifJ Phase ¢ Phase angle, angle coordinate ¢(x) Wavefunction of a particle ¢r, ¢l, ¢ Spin wavefunctions !f/ Wavefunction If' Wavefunction of several electrons {j Generalised quantum

mechanical operator Q Frequency [14.4, 14.5, 15.3] OJ Angular frequency 2nv, or

eigenvalue [9.3.6] .!>.. means "corresponds to"