moment 133representation 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 spi
Trang 1moment 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 = mc2 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
Trang 2emission (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
Trang 3fixed 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
Trang 4Golden Rules (Fermi) 179, 267, 269
Goldstone modes 64
good quantum numbers
n, l, s, j, mj(for H) 250
general remarks (if at all) 255
N, L, S, J, MJ (non-H atoms, shell
model) 255
n, l, s, ml, ms (for H, without
spin-orbit coupling) 250
N, L, S, ML, MS (for non-H atoms,
shell model, without spin-orbit
coupling) 255
parity 224
grand canonical
Boltzmann-Gibbs distribution 367
ensembles 366, 428
partition functionZ(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
He3(superfluidity) 278
He4(superfluidity) 278
He4 versus He3 (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
H2) 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
Trang 5hidden 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
Trang 6isothermal 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´e factor g 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 L0 (the number of molecules in a mole) 303 losses of a response process 410 Mach 5, 96, 98
macroscopic versus microscopic properties 301
Trang 7dipole radiation 173
dipolesm 139, 140
dipoles (not always equivalent to
Amp`erian current loops) 149
fieldH 139, 313
fields of steady electric currents 145
inductionB 139, 149, 314
momentm 313
monopoles (non-existence) 139, 140
polarizationJ 139, 314
susceptibility 319, 403, 409
work 313
magnetism 271
magnetizationM = J/μ0 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
Trang 8radiation: 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
O2 (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 Ohmic processes (Joule’s heat) 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
Trang 9paraelectric 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 ∝ T3
401 (sound quanta) 185, 379, 399, 401, 402
, magnons, plasmons, (quasi-particles) 346
photon gases (p = 3V U) 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
Trang 10representation (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 ray optics versus wave optics 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