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Igor N. Toptygin: Foundations of Classical and Quantum Electrodynamics — 2013/10/8 — page 709 — le-tex

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709

Index

Symbols4-vector of current density, 320, 339

Aaction for an electromagnetic field, 320adiabatic invariants, 306amplitude, complex, 143angular momentum tensor of an

electromagnetic field, 389annihilation eC C e�, 672annihilation and creation operators, 517anomalous magnetic moment, 668antiparticles, 563asymptotic value

– of cylindrical functions, 45– of modified Bessel functions, 46– of spherical Bessel functions, 47

Bbasis, mutual, 11Bessel function

– integral representation, 43– recurrent relations, 43

big canonical distribution of Gibbs, 695bispinors of a free particle, 572Bohr magneton, 174boundary conditions

– for Maxwell’s equations, 134– in magnetostatics, 118– periodic, 150

Breit formula, 367bremsstrahlung, 425

CCasimir effect, 520causality principle, 398charge

– electric, 281– elementary, 92– magnetic, 312

Christoffel symbols– of the first kind, 33– of the second kind, 32

chronological operator, 651classical electron radius, 270coefficient

– of mutual induction, 126– of self-induction, 127

combinational (Raman) scattering, 612completeness of a system of functions, 65Compton effect, 229, 573Compton wavelength, 270condition of completeness (closeness), 66conduction current, 133continuity equation, 115coordinate system, affine, 11critical (Eddington) luminosity, 447

Dd’Alembert’s wave equation, 142degree of depolarization, 146degree of polarization, 146density

– of electric current, 113– of electromagnetic energy flux, 137– of the energy of electromagnetic field,

137derivative

– covariant, 31– from delta function, 58

differential operations– in cylindrical coordinates, 37– in spherical coordinates, 37

dipole moment– electric, 101– magnetic, 120

dipole moment of the transition, 545Dirac equation, 561Dirac matrices, 561

Foundations of Classical and Quantum Electrodynamics, First Edition. Igor N. Toptygin.©2014 WILEY-VCH Verlag GmbH & Co. KGaA. Published 2014 by WILEY-VCH Verlag GmbH & Co. KGaA.

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Igor N. Toptygin: Foundations of Classical and Quantum Electrodynamics — 2013/10/8 — page 710 — le-tex

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710 Index

Dirac γ matrices, 631Dirac’s monopoles, 312Dirac–Lorentz equation, 440direction of rotation of electrical vector, 185dispersion relations, 149displacement current, 133distribution

– Gibbs canonical, 694– Planck, 521– Poisson, 524

divergence, covariant, 35Doppler effect, 210

– in a refractive medium, 231Doppler width of the spectral line, 496double electric layer, 101drift

– centrifugal, 308– electric, 302– gradient, 303

dual tensors, 276

EEarnshaw

– theorem, 111Einstein coefficients, 544electric charge, 91electric dipole transition, 545electric field, 93electric polarization vector, 408electric quadrupole moment, 101electromagnetic potentials, 138electromotive force of induction, 132electrostatic Gauss theorem, 95electrostatic potential, 94elliptic point, 305emission

– of pulsar, 413– spontaneous and stimulated, 543

energy– kinetic, 217– relativistic total, 216, 217

energy efficiency of a reaction, 220energy–momentum tensor of electromagnetic

field, 328equation

– Bessel, 44– Helmholtz, 31– inhomogeneous d’Alembert, 139– of Legendre, 52

equation for the density matrix, 694equivalent photons, 577Euclidean geometry of three-dimensional

space, 194

Euclidean space, 1evolution operator, 648expansion in multipoles, 101, 102

FFeynman diagrams, 574, 655, 663, 666, 671,

672Feynman’s electron propagator, 659Feynman’s photon propagator, 662field spin, 388field vacuum fluctuations, 517fine structure constant, 540Fizeau experiment, 213formula, Klein–Nishina–Tamm, 577formulas, Sokhotskii, 62four-dimensional

– force, 285, 286– photon polarization, 646– pseudotensors, 275– space–time, 196– vector, 205

Fourier image of a function, 69frequency, circular, 143F-sum rule, 558full set of observables, 688function

– Bessel, 41– Bessel modified, 46– generating Bessel functions, 41– Legendre, spherical, 54– Neuman’s, Weber’s, Hankel’s, 45– step, Heaviside, 58

function generating Legendre polynomials, 49

GGalilean transformations, 195gauge transformation of potentials, 138, 290Gauss–Ostrogradskii theorem

– generalized, 80general properties of wave functions, 687generalized currents Noether, 325Green’s function

– advanced, 401– retarded, 397

Green’s identities, 30gyromagnetic ratio, 298

HHamiltonian function, 287

– of field oscillators, 153Hamilton–Jacobi equation, 287hard Vavilov–Cherenkov radiation, 232Heisenberg’s uncertainty relations, 690Higgs boson, 385

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Igor N. Toptygin: Foundations of Classical and Quantum Electrodynamics — 2013/10/8 — page 711 — le-tex

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Index 711

Hilbert transforms, 149homogeneity of time, 194hyperbolic point (saddle), 306hyperfine splitting, 549

Iidentity, Parseval’s, 65inequality, Bessel’s, 65inertial frames of reference, 194integral form

– of electrostatics equations, 95– of magnetostatics equations, 117

integral form of the law of electromagneticinduction, 133

integrals of motion, 692intensity of radiation, 400interaction representation, 551, 648internal bremsstrahlung, 427interval

– and causality, 203– light like (zero), 203– space like, 203– time like, 203

invariant kinematic variables, 221inversion of coordinate systems, 3

KKlein–Gordon–Fock equation, 381, 563

LLagrange function density, 680Lagrangian form of equations of motion, 682Lagrangian function

– of a nonrelativistic particle, 215– of a relativistic particle, 216– of a relativistic particle in an

electromagnetic field, 283Lamb shift, 520, 585Lamé coefficients, 35Landau levels, 624Larmor

– formula, 406– radius (gyroradius), 347– theorem, 309

law– of the conservation of electric charge, 92,

115– Rayleigh–Jeans, 588– Wien, 589– Wien displacement, 588

Legendre polynominals, Rodrigues formula,51

Lienard–Wiechert potentials, 417long-range interaction, 282

Lorentz contour, 191Lorentz contraction of the length, 201Lorentz force, 280Lorentz transformation of a bispinor, 632Lorentz transformations, 198

Mmacroscopically small volume, 92magnetic flux, 124magnetic moment

– orbital, 298– spin, 299

magnetic polarization vector, 409mass defect, 220matrices, Pauli, 566matrix

– of Lorentz transformation for a bispinor,632

– of rotation, 2Maxwell tension tensor, 328Maxwell’s equations in 4-form, 322metric tensor, 13, 15momentum, relativistic, 216Mott formula, 657

Ooccupation numbers, 517operations of differentiation in orthogonal

coordinates, 36operator

– Hamilton’s (nabla), 19– Laplace, 28– of a derivative on time, 692– of density, 693– self-conjugate (Hermitian), 690

operators of the Dirac field, 642orbital moment of the field, 387orthogonality of Bessel functions, 48orthonormalized system of functions, 64oscillator strength, 558oscillators of the field (principal coordinates),

151

Pparticle, ultrarelativistic, 219Paschen and Buck effect, 570Pauli equation, 566phase of plane monochromatic wave, 143phase portrait, 305, 306phase trajectory, 305, 306photon, 219Planck constant, 219Planck spectral distribution, 588plane wave, 142, 184

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Igor N. Toptygin: Foundations of Classical and Quantum Electrodynamics — 2013/10/8 — page 712 — le-tex

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712 Index

Poisson equation, 95polarizability

– electrostatic, 558polarization of a plane monochromatic wave,

144polarization vector of relativistic electron, 638polynomials, Legendre, 49

– adjoint, 52potential

– function, 125– pseudoscalar of a magnetic field, 123– vector, 116

Poynting vector, 137principle

– of correspondence, 688– of relativity of classical mechanics, 194– of reversibility, 409– of the constancy of the speed of light, 196– of the superposition of states, 689

principle of the superposition of fields, 93probability current, 565probability density, 564, 686projection operator, 634, 691projection operators onto the positive and

negative energy state, 635proper time, 200pure states, 685

Qquantum number, 685

Rradiation friction force, 440radiation width of spectral line, 556Rayleigh scattering, 559Rayleigh–Jeans law, 588reference frame, center of inertia, 220relativistic factor (Lorentz factor), 219relativistic motion in a Coulomb field, 362renormalization of mass, 440rest energy, 216rule of relativistic velocities composition, 202runaway electrons, 360Rutherford formula, 356

Sscalar

– definition, 2– potential, 138

scattering matrix, 535, 649Schrödinger and Heisenberg representations,

692Schrödinger equation, 683, 692selection rules, 548

separatrix, 306singular point, 306singular points of phase plane, 306spectral line broadening

– collisional, 496– Doppler, 496– natural, 495

spectral line broadening, natural, 556spectrum of a physical value, 685spherical functions of Bessel and Hankel, 46spherical vectors, 403spin of elementary particles, 174spin–orbital interaction, 570spirality, 640Stark effect, 570state

– entangled, 538– quasi-stationary, 556– stationary, 693

states– coherent, 523– mixed, 693– squeezed, 529

statistical sum, 694Stokes theorem, 26strength

– of an electric field, 93– of magnetic field, 112

superfine splitting, 569, 619superradiance, 618superradiance effect, 618surface density of the charge, 96surfaces, equipotential, 21symmetry and antisymmetry of wave

functions, 688synchrotron radiation, 431, 432system of coordinates

– bispherical, 39– ellipsoidal, 38– extended spheroidal, 39– flattened spheroidal, 38– toroidal, 40

Ttensor, 17

– axial (pseudovector), 3– definition, 2– energy–momentum, canonical, 326– Hermitian (anti-Hermitian), 4– metric, 13, 15– of electromagnetic field, 284– of polarization, 144– polar, 3

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Igor N. Toptygin: Foundations of Classical and Quantum Electrodynamics — 2013/10/8 — page 713 — le-tex

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Index 713

– symmetric– principal directions, 9– principal values, 9

– symmetric (antisymmetric), 4theorem

– Gauss–Ostrogradskii, 25– of the uniqueness of the solution of

Maxwell’s equations, 137– Stokes, 26– Stokes’ generalized, 31– the summation of spherical functions, 54

Thomas precession, 208, 367, 372Thomson cross-section, 498, 501, 559total field moment, 388transform, Fourier, 69transitional current, 542transversality of electromagnetic waves, 142

Uundulator, 432undulator radiation, 432uniformity and isotropy of space, 194unitary operator, 691

Vvacuum fluctuations, 517Vavilov–Cherenkov effect, 231vector

– axial (pseudovector), 3– basic, 16– definition, 2– polar, 3– spherical, 403– wave, 143

vector circulation, 22vector potential, 27vector solenoidal, 27virial theorem, 354volume density of a charge, 92

Wwave zone, 400Wien’s displacement law, 588world line, 202