Basic Theoretical Physics: A Concise Overview P43 potx

Index 439 moment 133 representation (of magnetic bodies) 140 fields 132 Dirac ’s abstract bra-ket formalism 215 ’s abstract quantum mechanics 210 ’s delta function 112, 133, 138, 214 ’s equivalent spin operator (Heitler- London theory) 273 ’s relativistic quantum theory (not described) 251 representation (interaction) 245 Dirichlet’s problem 127, 128 discrete-line radiation 179 dispersion phenomena 346 relation ω(k) 179, 399, 402 double pendulum 85, 87 droplet radius: critical value 425 Drude theory (of the electric resistance) 164 dual space 78 duality (wave-particle) 208 Dulong-Petit’s law 402 dumbbell approximation (for a dipole) 132, 140 dynamic conserved quantities 50 dynamic susceptibility 410 Dyson series (interaction representa- tion) 247 Dyson’s time-ordering operator 247 E=mc 2 101 E¨otv¨os 8 eavesdropping (spy, quantum crypto- graphy) 289 effective -charge representation (versus polarization representation) 137 charges (i.e., not compensated) 138 currents 149 electric charges 146 mass 286 currents 146 Ehrenfest (n-th order phase transitions) 403 Ehrenfest theorem 245, 247 eigenfrequencies (normal frequencies) 63 eigenzeit (proper time) 55, 59 eikonal function 186 approximation (from Schr¨odinger to Hamilton) 187 approximation (from waves to rays) 186, 187 Einstein 174, 183, 184, 203 ’s light-quantum hypothesis (photons) 208, 209, 220 ’s principle of “maximal proper time” 55, 60 ’s principle of relativity 58 ’s summation convention 176 ’s “annus mirabilis” (1905) 208 ’s contributions to quantum mechanics 4 ’s equivalence principle 96 ’s general theory of relativity 60, 96 ’s special theory of relativity 58 ’s theory of gravity 33 and de Haas 151 hidden variables 279 Einstein, Podolski, and Rosen 279, 290, 291, 295 Einstein, Podolski, and Rosen’s “ac- ceptability” postulates (reality, locality, retardation) 280 electric dipole radiation 173 octupole radiation 173 polarization P 133, 134, 139 electro-mechanical equivalent of heat 306 electromagnetic waves 163 waves (planar, spherical) 166 energy theorem 163 electromotive force 153 electron equilibrium in neutron stars 417 electrons in metals (Sommerfeld formalism) 383 electrostatics 119 elevator force 96 ellipsoid of revolution 77, 190 440 Index emission (induced versus spontaneous) 269 Emperor’s bell in Cologne 75 encoding (decoding) of a computer message 284 energy -momentum four-vector 103 dissipation 164 conservation 14 conservation for the electromagnetic field 163 density of the electric field 142 density of the magnetic field 142 theorem (electromagnetic) 163 theorem (mechanics) 27, 28 conservation 50 entanglement 279, 291, 295 enthalpy I(T,p,N) 319, 320, 428 entropy S(T,V,N) 301, 315, 350, 357, 427, 429 constant 415 from the microcanonical ensemble 363 of mixing 413, 414, 416, 421 equations of motion Hamilton 53 Lagrange 53 Newton 5, 53 Lagrange 45 equipartition of energy 352 equivalence of inertial frames 58 of mass and energy (Einstein) 104 principle (Einstein) 96 ergodic systems 367, 411, 412 Euler ’s relation (Moivre) 159 -Lagrange equations 48 angles ϕ, ϑ, ψ 81, 82, 91 equations of a rigid body 79 even normal modes 67 even parity 67 exchange integral 273 expansion theorem 215 expectation value in quantum mechanics 218 extensive versus intensive variables (thermodynamics) 314 extensive versus intensive work 320, 354 external force 23 external torque 24 extraordinary versus ordinary beam (uniaxial birefringence) 191 extremal paths (Einstein) 96 Fahrenheit, R´eaumur (or Reamur) 303 far-field range 170, 173 Faraday’s induction law 124, 153, 155 Fermat’s variational principle of the shortest optical path 55, 185, 187 Fermi ’s “golden rules” 179, 267, 269 and Bose gases 335, 337, 379 and Bose statistics 337 energy ε F 383, 394 liquid 383 fermions 252 fermions and bosons 379, 383 ferroelectric systems 136 ferromagnetism ˆ= interplay of Coulomb interaction and Pauli’s principle 274 ideal 304 in solids 275 Feynman diagram for induced absorption 269 Fick’s laws (heat diffusion) 306, 307 fictitious (=inertial) forces 95 field -energy conservation 163 energy 141 tensor (skew-symmetric) in electro- magnetism 176 figure axis 84 fine structure (of the spectra) 251, 256, 294 First and Second Laws of Thermody- namics: Equivalent Formulations 315 First Law of Thermodynamics 313 first-order phase transitions 403 Index 441 fixed cartesian vectors for a rigid body 79 fluctuation-dissipation theorem 410 fluctuations 332, 333, 408, 428 fluid (liquid or gas) 305 flux tubes (vortex lattice) 398 Fock-Darwin theory 287 force -power four-vector 102 and torque on electric and magnetic dipoles 25, 139 of constraint 13, 90, 91 and torque on electric and magnetic dipoles 140 Foucault pendulum 98 fountain effect (superfluidity) 394 four-vectors (Minkowski) 58 Fourier methods (heat conduction) 308 Fourier transformation (Fraunhofer diffraction) 197 Fraunhofer diffraction at rectangular and circular apertures 197 free energy (Gibbs) F g (T,V,H,N) 366 energy (Helmholtz) F (T,V,N) 301, 353, 366, 396, 397, 427 energy (Helmholtz) of droplets 423 energy for Landau’s theory 404 enthalpy per particle (≡ μ in fluids) 365, 416, 418 frequency dispersion 179 Fresnel ’s integrals 195 ellipsoid (E-ellipsoid) 189 diffraction at an edge 193 diffraction versus Fraunhofer diffraction 193 frictional force 17, 27, 45 frictional heat 305 fringes (in Fresnel’s diffraction from an edge) 196 from reversibility (quantum mechanics) to irreversibility: Fermi’s “golden rules” 268 fundamental solutions (→ Green’s functions) 168 g-factor 250, 294 Gabor 199 Galilean transformations 56, 90, 175 Galileo 5 gamma function 393 gas thermometer 303 gate voltage (of a quantum dot) 286 gauge function 62, 165, 210 invariance of the Aharonov-Bohm effect 282 transformations 61, 147, 148, 165 transformations in quantum mechanics 217, 245 invariance 396 transformations in quantum mechanics 216 Gauss ’s interface box 143 ’s law 121, 153 ’s geodesic triangulation in 1831 4 ’s integral 115 ’s integral theorem 130, 308 Gay-Lussac 335 Gay-Lussac’s experiment 317, 318, 324, 371, 430 general theory of relativity (Einstein) 4, 60, 97 generalized coordinates 45, 47 forces of constraint 90 irrotational behavior 28 generator voltage (induction) 154 geoid model of the earth (oblate spheroid) 99 geometrical optics (ray optics) 185 Gibbs ’ phase rule 419 ’ grand canonical potential Φ(T,V,μ) 367, 428 ’ phase rule 419 free energy F g (T,V,H,N) 354, 366 Ginzburg, Abrikosov, Legget 395 Ginzburg-Landau theory of supercon- ductivity 395 glasses (C V ∝ T ) 402 glassy behavior (non-ergodic) 412 global inertial frames (Mach) 96 Global Positioning System (GPS) 97 442 Index Golden Rules (Fermi) 179, 267, 269 Goldstone modes 64 good quantum numbers n, l, s, j, m j (for H) 250 general remarks (if at all) 255 N, L, S, J, M J (non-H atoms, shell model) 255 n, l, s, m l ,m s (for H, without spin-orbit coupling) 250 N, L, S, M L ,M S (for non-H atoms, shell model, without spin-orbit coupling) 255 parity 224 grand canonical Boltzmann-Gibbs distribution 367 ensembles 366, 428 partition function Z(T,V,μ) 367, 428 potential Φ(T,V,μ) 428 gravitational forces 8, 39 gravitational mass 8, 119 Green ’s 2nd integral theorem 192 ’s function 21 ’s function methods (heat diffusion) 308, 310 ’s integral theorem 308 group velocity (≤ c) 183 velocity (meaning) 183 velocity versus phase velocity 179, 182 Grover’s algorithm (quantum comput- ing) 285 gyromagnetic ratio 151 Hamilton ’s variational principle of “least action” 48, 55 formalism 248 operator (quantum mechanics) ver- sus Hamilton function (classical mechanics) 294 operator for the Lorentz force 217 operator with spin 249 operator 216 Hamiltonian 45, 50 Hamiltonian for the Lorentz force 61 handedness 71 harmonic functions 128 oscillator (algebraic treatment) 241 oscillator (Schr¨odinger’s wave mechanics) 231 functions 127 Hartree-Fock approximation 254, 259, 408 He 3 (superfluidity) 278 He 4 (superfluidity) 278 He 4 versus He 3 (Bose versus Fermi) 338 heat bath 366 capacity 305, 409 capacity of solids (Debye theory) 399 diffusion constant D w 307 diffusion equation 307 general remarks (if at all) 305 of transition 403 production 164 electro-mechanical equivalent 306 Heaviside function 138 heavy fermions 386 heavy roller, i.e., a circular cylinder rolling down an inclined plane 74, 91 heavy symmetric top 83 Heisenberg ’s “matrix mechanics”209 ’s uncertainty principle 219 ’s uncertainty relation (general form) 294 general remarks (if at all) 4, 169, 241 model (magnetism) 274, 405, 407 representation (aspect) 245 Heitler and London (diamagnetism of H 2 ) 271, 275 Heitler-London theory 287 Helmholtz free energy 366 Hermite polynomials 233 Hermitian operators 213 Hertz 173 Hertz’s oscillating electric dipole 170 Hertzsprung-Russel diagram (stars) 388 Index 443 hidden variables (Einstein) 279 Higgs boson (analogy with Cooper pairs) 398 Higgs-Kibble mechanism (analogy with Meissner-Ochsenfeld effect) 399 high-temperature superconductors 278 Hilbert space 211, 212, 293 history of quantum mechanics 207 holography 187, 199 holonomous constraints 45, 89, 93 holonomy 316 homogeneity 50 homogeneous differential equation 17 homogeneous fields 142 Hund’s rule (maximal multiplicity) 275 Hund’s-rule exchange 276 Huygens’s principle 169 Huyghens’s principle 184 hydrogen atom (bound states) 237 atoms (product ansatz; radial functions) 235 problem 31 atom (positive energies) 238 atoms (in Schr¨odinger’s wave mechanics) 235 hyperbolas 37 hyperfine structure (of the spectra) 254, 256 ideal ferromagnet 304 gas 303 heat machine (impossible, Thomson) 355 heat pump (impossible, Clausius) 355 paramagnets 304 identical particles 349, 379 impact parameter 41 incoherency (addition of intensities instead of amplitudes) 344 incoherency of thermal averages 344 incoherent versus coherent radiation 268 independent partial systems 413 index ellipsoid (D-ellipsoid) 189 indistinguishability of identical particles 337, 350 induced dipole moment 136 induced emission and absorption 269 inductances (self-inductance and mutual inductances) 158 induction law (Faraday) 124, 153 voltage 154 charge 129 inertia ellipsoid 77 inertia tensor 73, 189 inertial forces 95 frames 5, 56, 58, 96, 98 infinitesimal Carnot process (Clausius- Clapeyron) 370 infrared (spectral range) 179 inhibited reaction equilibrium 416 inhomogeneous differential equation 17 integrable systems 85 integral theorem of Gauss 130, 441 intensive versus extensive variables (thermodynamics) 314 intensive versus extensive work 320, 354 interaction representation (Dirac) 245 interaction-free quantum measurement 221, 287, 288 interface curl 143 interface divergence 143 interface-current density (for supercon- ductors) 146 interference 283, 291, 295 internal energy U (T,V,N) 301, 315, 319, 353, 427, 428 internal force 23 inverse β-decay 418 inversion temperature 323 irreversibility 315 irrotational vector fields 125 Ising model 332, 333, 354, 409, 425 model (lattice-gas interpretation) 333, 425 model (magnetic interpretation) 333 444 Index isothermal changes of state 324 isotherms versus adiabatics 325 isotropy 50 Joule losses 164 Joule’s heat (Ohmic processes) 305 Joule-Thomson effect 321, 324, 372 Joule-Thomson process 319, 322 Kelvin temperature T 303, 427, 429 Kelvin thermometer (ideal gas, ideal paramagnet) 304 Kepler ’s laws 32 ’s problem (hyperbola orbits) 238 problems 50 kidney-shape (of dipole fields) 133 kilogram 7 kilopond 7 kinetic energy 27 energy (relativistic) 103 energy: translational and rotational parts 71 momentum 61 theory of Bose (Fermi) gases 335 theory of gases 311, 335 theory of relativistic gases 335 energy 46 Kirchhoff’s law 192 Kronecker symbol δ i,j 214 Kuhn (philosophy of science) 3 ladder operators (ascending or descending) 243 Lagrange formalism 89 Lagrange multiplier 90 Lagrangian equations of the 1st kind 48, 91 equations of the 2nd kind 45, 48 equations of the 2nd kind with constraints 91 for the Lorentz force 60 function 45 equations of the 2nd kind 47 Land´efactorg J (L, S) 265, 406 Landau’s theory (second-order phase transitions) 403 Laplace identity 72 laser 199 laser cooling (ultralow temperatures) 371 latent heat (e.g., vaporization and melting) 369 latitude ϑ 126 lattice-gas interpretation (Ising model) 332, 333, 425 law of mass action (chemical reactions) 417 least action (Hamilton’s principle) 55 left-handedness 71 Legendre polynomial 239 Legendre transform 428 Legget, Abrikosov, Ginzburg 395 Lenard (photoeffect) 208 length contraction 59 Lennard-Jones potential 323, 327 Lenz vector 39, 50, 236 light quanta (photons) 379 Linde process (liquefaction) 322 linear approximation (mean field) of a Hamiltonian 334 linearity (e.g., of Maxwell’s equations) 120 linearity (of a theory) 21 liquefaction (Linde) 319, 322, 323 local inertial frames (Einstein) 5, 96 locality in quantum mechanics 279 locally orthogonal curvilinear coordi- nates 126 longitude ϕ 126 Lorentz force 27, 45, 153, 155, 217, 295 force (mksA versus cgs) 112 gauge 165, 176 transformations 56 transformations for the electromag- netic fields 177 force 60 gauge 166 transformations 175 Loschmidt’s number L 0 (the number of molecules in a mole) 303 losses of a response process 410 Mach 5, 96, 98 macroscopic versus microscopic properties 301 Index 445 magnetic dipole radiation 173 dipoles m 139, 140 dipoles (not always equivalent to Amp`erian current loops) 149 field H 139, 313 fields of steady electric currents 145 induction B 139, 149, 314 moment m 313 monopoles (non-existence) 139, 140 polarization J 139, 314 susceptibility 319, 403, 409 work 313 magnetism 271 magnetization M = J/μ 0 314 magnetocaloric effect 372 magnetostatics 119, 139 magnons (spin-wave quanta) 185, 379, 399, 402 main-sequence stars 388 MAPLE, MATHEMATICA 161 Markoff chain 412 mass defect 104 mathematical (versus physical) pendulum 75 Maxwell ’s displacement current 153, 156 ’s equations in Gaussian units 111 ’s equations in SI units 111 ’s relations (thermodynamics) 317, 318, 322, 339, 369, 370, 429 ’s theory: relativistic formulation 174 -Boltzmann ideal gases 335 -Boltzmann velocity distribution 337, 352 construction 329, 331 equations 163 mean field approximation (bi-linearized) of a Hamiltonian 334 mean field approximations 405, 408 measurable quantities (“observables”) in quantum mechanics 215 measurable quantities (observables) in quantum mechanics 213 measurement process 218 Meissner effect 146, 277, 398 Meissner-Ochsenfeld effect (analogy with Higgs-Kibble mechanism) 399 Mendeleev’s system 293 Mercury (closest planet to the sun) 35 meridian ϕ 126 metric fundamental tensor 96 Metropolis algorithm 411 Michelson 3, 56, 174 microcanonical ensemble 337, 344, 363 microstates (configurations) 359 minimal substitution (p → p − eA)62 Minkowski formulation of the continuity equation 176 four-potential ˜ A 177 four-vectors 58, 59, 102, 175 four-vectors (classification: space-like, light-like, time-like) 101 general remarks (if at all) 58, 96, 175 space (pseudo-euclidean) 175 mixing entropy 413, 414, 416, 421, 423 mobile phone 170 Moivre’s relation (Euler) 159 mole 303, 335 molecular field approximation 333 field exponents 332 field theories 405, 406, 408 polarizability 134 moment of inertia 73 momentum four-vector 103 Monte Carlo simulations 411 Morley 3, 56, 174 moving point-charge (electromagnetic fields) 177 multipole -moment expansion (total charge, dipole moment, quadrupole moment, ) 135 expansion 134 moments 134 potentials (monopole, dipole, quadrupole, ) 133 radiation 173 446 Index radiation: dependence on distance and/or frequency 174 radiation: right-handed rectangular trihedron 174 n-th order phase transitions (Ehrenfest) 403 Nabla operator 114 near-field microscopy 194, 197 near-field range 170 Nernst’s heat theorem (Third Law) 316, 372, 374 Neumann’s problem 129, 130 neutrino 418 neutron 418 neutron stars 97, 387, 389, 418 Newton ’s 3rd axiom, weak and strong forms 8 ’s bucket experiment 98 ’s equations of motion 5, 293 ’s particle interpretation of light 184 ’s third axiom 26, 38, 119 Newtonian mechanics 174 synthesis of Kepler’s laws 33 analysis of Kepler’s laws 33, 36, 39 nodal theorem 226, 240 Noether’s theorem 49, 50 non-additivity of the partial pressures for real gases 414 non-autonomous 86 non-degenerate perturbation theory 262 non-existence of a local maximum of a harmonic function 129 non-integrable systems 85 non-linearity (e.g. of the chromo- dynamic equations for nuclear physics) 120 non-locality of quantum mechanics (singlet decay) 281, 295 non-perturbative terms 263 non-reality of quantum mechanics (states are “prepared”, not “stated”) 281 normal fluid 394 frequencies (eigenfrequencies) 63 modes 63 north anomaly 100 nuclear transitions 180 nucleation processes 425 nuclei 180 number operator 242 numerical methods (electric fields) 131 nutation 84 Nyquist theorem 411 O 2 (why paramagnetic?) 275 object wave plus reference wave (in holography) 200 observable 429 observables (“measurable quantities”) in quantum mechanics 213, 215 octupole 173 odd normal modes 67 odd parity 67 Oersted 145 Ohmic behavior 164 Ohmicprocesses(Joule’sheat) 305 optical resolution 197 optical-path length 55, 187 optics (application of electrodynamics) 179 orbit 47 orbital angular momentum 244 order parameter 395, 397, 404 ordinary versus extraordinary beam (uniaxial birefringence) 191 orthogonal curvilinear coordinates 127 orthogonality properties (of the spherical harmonics) 136 orthonormal bases in a Hilbert space 212 orthonormalization 214 orthonormalization with δ-functions 238 osculating circle and osculating normal of a space curve 12 Osheroff, Richardson, Lee 395 osmotic pressure 420, 422 overlap integral 272, 276 pair condensate 277, 395 Index 447 paraelectric systems 136 paramagnetic resonance 68 paramagnetism 251, 275 parametric representation of an ellipse 37 parametric resonance 28 parity 67, 224 partial pressure 413, 414, 421 particle-wave duality 184, 208 partition function Z(T,V,N) 302, 344, 351, 377, 427 Passat (trade winds) 98 passive charge 119 passive gravitational mass 8 Pauli ’s exclusion principle 249, 251, 252, 271, 293, 381, 383 ’s exclusion principle (generalization: permutation behavior of an N-particle wave function) 252 general remarks (if at all) 249 matrices 244 pendulum (mathematical) 75 pendulum (physical) 75 penetration depth λ(T ) (superconduc- tivity) 397 perihelion and aphelion 34 perihelion rotation 34, 35 periodically driven oscillation 18 permafrost problem (heat conduction) 309 permanent dipole moment 136 permutation factors in front of the classical phase space element 350 perpetual motion of the second kind (impossible) 354, 355 perturbation theory (Schr¨odinger) 261 determinant 262 theory (non-degenerate versus degenerate) 262 Pfaff forms (differential forms) 29, 316 phase velocity versus “ray velocity” (birefringence) 190 space (coordinates plus momenta) 293 transitions (first-order) 403 transitions (second-order, Landau theory) 403 changes 327 phase-space integral 349 phase-space quantization (in units of h f ) 349 phonons C V ∝ T 3 401 (sound quanta) 185, 379, 399, 401, 402 , magnons, plasmons, . . . (quasi- particles) 346 photon gases (p = U 3V ) 339, 347 photon gases and ultrarelativistic behavior 338 photons (light quanta) 4, 185, 220, 379, 402 physical dimension 126 Planck ’s constant h 179, 207, 349 ’s formula (black-body radiation) 180, 184, 208, 339, 344, 345, 347 ’s reduced constant  343 general remarks (if at all) 4, 207, 361 planetary motion 31 plasmons (quanta of charge-density waves) 185 Poincar´e ’s lemma 125, 147 repetition time 268 section 86 Poinsot ’s construction (for rigid bodies) 77, 189 ’s construction (in crystal optics) 189, 191 plane 78, 80 point charge 114 point mass 5, 23 Poisson brackets 50, 53 Poisson’s equation 128 polar (versus axial) vectors 71 polarizable matter 132 polarization (electric) 133 448 Index representation (versus effective- charge representation) 137 contributions (to the effective densities of charge and current) 165 polarons 379 Popper (philosophy of science) 3 position representation (“wave mechanics”) 231 position representation (versus mo- mentum representation or matrix representations) 211 potential energy 14, 26–28, 46 power (relativistic) 102 power balance (dissipative versus reactive parts) 162 Poynting vector 163, 167, 172 pre-Einstein (aether) 58 precession (of a top) 84 principal moments of inertia 77 principal quantum number n (shell index) 237 principle of “maximal proper time” (Einstein) 60 of d’Alembert 89 of relativity (Einstein) 58 of superposition 18, 33 probability current 228 probability statements (for the results of measurements in quantum mechanics) 294 profile function (of a traveling wave) 167 projectile and target 41 proper time (eigenzeit) 55, 59 proton 418 pseudo-length 59 pseudo-perpendicular 102 pseudo-rotation of the Minkowski space (Lorentz transformation) 59, 101, 175 pseudo-scalar product 59 pulsar (neutron star) 391 quabla operator (d’Alembert) 166 quadrupoles 132, 133, 173 quality factor of a resonance 21, 161 quanta of electromagnetic waves (photons) 4 quantization (Planck) 208 quantization of the angular momentum 243, 351 quantum chromodynamics (QCD) 4 computing 283 cryptography 285, 289, 291 dots (2d) 285 electrodynamics (QED) 4 mechanics (Part III) 301 mechanics: does not “state”, but “prepare” (non-reality); plus: “non-locality” 295 well (bound states) 224 well (reflection at high energies) 229 quasi-particles 379 quasi-particles (phonons, magnons, plasmons, ) 185, 346 qubits in quantum computing 283 radiation from an oscillating electric dipole 170 radiation from an oscillating magnetic dipole 171 radius of curvature of a line in space 12 ray optics (geometrical optics) 185 rayopticsversuswaveoptics 185 Rayleigh-Jeans’ law (black-body radiation) 207, 345 rays in a complex Hilbert space (equiv- alence classes of Hilbert vectors distinguished by a complex factor) 211 Real Gases (as opposed to Ideal Gases) 304 reality in quantum mechanics 279 reconstruction wave (in holography) 200 red giant (stars) 389 reduced length of a physical pendulum 75 variables (harmonic oscillator) 231 variables (hydrogen atom) 237 reference frames 95 reference wave 199 . field approximation (bi-linearized) of a Hamiltonian 334 mean field approximations 405, 408 measurable quantities (“observables”) in quantum mechanics 215 measurable quantities (observables) in quantum. 313 magnetism 271 magnetization M = J/μ 0 314 magnetocaloric effect 372 magnetostatics 119, 139 magnons (spin-wave quanta) 185, 379, 399, 402 main-sequence stars 388 MAPLE, MATHEMATICA 161 Markoff. 136 paramagnetic resonance 68 paramagnetism 251, 275 parametric representation of an ellipse 37 parametric resonance 28 parity 67, 224 partial pressure 413, 414, 421 particle-wave duality 184, 208 partition