Einstein’s Physics Atoms, Quanta, and Relativity Derived, explained, and appraised Ta-Pei Cheng University of Missouri - St. Louis Portland State University To be published by Oxford University Press 2013 Contents I ATOMIC NATURE OF MATTER 1 1 Molecular size from classical fluid 3 1.1 Two relations of molecular size and the Avogadro number . 4 1.2 The relation for the effective viscosity . 6 1.2.1 The equation of motion for a viscous fluid . 7 1.2.2 Viscosity and heat loss in a fluid . 8 1.2.3 Volume fraction in terms of molecular dimensions . 11 1.3 The relation for the diffusion coefficient . 11 1.3.1 Osmoticforce....................... 12 1.3.2 Frictional drag force — the Stokes law . 13 1.4 SuppMat: Basics of fluid mechanics . 15 1.4.1 The equation of continuity . 15 1.4.2 The Euler equation for an ideal fluid . 16 1.5 SuppMat: Calculating the effective viscosity . 18 1.5.1 The induced velocity field v′ ............... 18 1.5.2 The induced pressure field p′ .............. 20 1.5.3 Heat dissipation in a fluid with suspended particles . 20 1.6 SuppMat: The Stokes formula for viscous force . 24 2 The Brownian motion 27 2.1 Diffusion and Brownian motion . 29 2.1.1 Einstein’s statistical derivation of the diffusion equation 29 2.1.2 The solution of the diffusion equation and the mean- squaredisplacement . 32 2.2 Fluctuations of a particle system . 32 2.2.1 Randomwalk....................... 33 2.2.2 Brownian motion as a random walk . 34 2.3 The Einstein-Smoluchowski relation . 34 2.3.1 Fluctuation and dissipation . 36 iii iv CONTENTS 2.3.2 Mean-square displacement and molecular dimensions . 36 2.4 Perrin’s experimental verification . 37 II QUANTUM THEORY 41 3 Blackbody radiation: From Kirchhoff to Planck 43 3.1 Radiation as a collection of oscillators . 45 3.2 Thermodynamics of blackbody radiation . 47 3.2.1 Radiation energy density is an universal function . 47 3.2.2 The Stefan-Boltzmann law . 49 3.2.3 Wien’s displacement law . 50 3.2.4 Planck’s distribution proposed . 54 3.3 Planck’s investigation of cavity oscillator entropy . 54 3.3.1 Relating the oscillator energy to the radiation density 55 3.3.2 The mean entropy of an oscillator . 55 3.4 Planck’s statistical analysis . 57 3.4.1 Calculating the complexion of Planck’s distribution . 57 3.4.2 The Planck’s constant and the Boltzmann’s constant . 61 3.4.3 Planck’s energy quantization proposal — a summary . 62 3.5 SuppMat: Radiation oscillator energy and frequency . 63 3.5.1 Ratio of the oscillator energy and frequency is an adi- abaticinvariant. .. .. .. .. .. .. 64 3.5.2 The thermodynamic derivation of the relation between radiation pressure and energy density . 67 4 Einstein’s proposal of light quanta 69 4.1 TheRayleigh-Jeanslaw . 71 4.1.1 Einstein’s derivation of the Rayleigh-Jeans law . 71 4.1.2 The history of the Rayleigh-Jeans law and "Planck’s fortunatefailure". 73 4.1.3 An excursion to Rayleigh’s calculation of the density ofwavestates....................... 74 4.2 Radiation entropy and complexion á la Einstein . 76 4.2.1 The entropy and complexion of radiation in the Wien limit............................ 77 4.2.2 The entropy and complexion of an ideal gas . 79 4.2.3 Radiation as a gas of light quanta . 80 4.2.4 Photons as quanta of radiation . 81 4.3 Thephotoelectriceffect . 81 CONTENTS v 4.4 SuppMat: The equipartition theorem . 83 5 Quantum theory of specific heat 85 5.1 The quantum postulate: Einstein vs. Planck.......... 86 5.1.1 Einstein’s derivation of Planck’s distribution . 87 5.2 Specific heat and the equipartition theorem . 88 5.2.1 The study of heat capacity in the pre-quantum era . 88 5.2.2 Einstein’s quantum insight . 90 5.3 TheEinsteinsolid ........................ 91 5.4 TheDebyesolidandphonons . 94 6 Waves, particles, and quantum jumps 99 6.1 Wave-particleduality. 101 6.1.1 Fluctuation theory (Einstein 1904) . 101 6.1.2 Energy fluctuation of radiation (Einstein 1909a) . 102 6.2 Bohr’satom............................105 6.2.1 Spectroscopy: Balmer and Rydberg . 106 6.2.2 Atomic structure: Thomson and Rutherford . 107 6.2.3 Bohr’s quantum model and the hydrogen spectrum . 107 6.3 Einstein’s A and B coefficients . 111 6.3.1 Probability introduced in quantum dynamics . 112 6.3.2 Stimulated emission and the idea of laser . 114 6.4 Looking ahead to quantum field theory . 115 6.4.1 Oscillators in matrix mechanics . 116 6.4.2 Quantum jumps: from emission and absorption of ra- diation to creation and annihilation of particles . 119 6.4.3 Resolving the riddle of wave-particle duality in radia- tionfluctuation. 122 6.5 SuppMat: Fluctuations of a wave system . 124 7 Bose-Einstein statistics and condensation 127 7.1 PhotonandtheComptoneffect . 129 7.2 Towards Bose-Einstein statistics . 130 7.2.1 Boltzmann statistics . 131 7.2.2 Bose’s counting of photon states . 133 7.2.3 Einstein’s elaboration of Bose’s counting . 136 7.3 Quantum mechanics and identical particles . 139 7.4 Bose-Einstein condensation . 142 7.4.1 Condensate occupancy calculated . 143 7.4.2 The condensation temperature . 144 vi CONTENTS 7.4.3 Laboratory observation of BE condensation . 146 7.5 SuppMat: Radiation pressure due to photons . 148 7.6 SuppMat: Planck’s analysis and BE statistics . 149 7.7 SuppMat: The role of indistinguishability in BEC . 150 8 Local reality and the Einstein-Bohr debate 153 8.1 QM basics — superposition and probability . 154 8.2 The Copenhagen interpretation . 154 8.3 EPR paradox : entanglement and nonlocality . 156 8.3.1 The post EPR era and Bell’s inequality . 159 8.3.2 Local reality vs. QM — the experimental outcome . 162 8.4 SuppMat: QM calculation of spin correlations . 164 III SPECIAL RELATIVITY 169 9 Prelude to special relativity 171 9.1 Relativity as a coordinate symmetry . 172 9.2 Maxwell’sequations . 174 9.2.1 The electromagnetic wave equation . 175 9.2.2 Aether as the medium for EM wave propagation . 176 9.3 Developments prior to special relativity . 177 9.3.1 Stellar aberration and Fizeau’s experiment . 177 9.3.2 Lorentz’s corresponding states and local time . 180 9.3.3 The Michelson-Morley experiment . 184 9.3.4 Length contraction and the Lorentz transformation . 187 9.3.5 Poincaré and special relativity . 188 9.4 Reconstructing Einstein’s motivation . 188 9.4.1 The magnet and conductor thought experiment . 189 9.4.2 From ‘no absolute time’ to the complete theory in five weeks ...........................191 9.4.3 Influence of prior investigators in physics and philosophy193 9.5 SuppMat: Lorentz transformation à la Lorentz . 194 9.5.1 Noncovariance under Galilean transformation . 195 9.5.2 Lorentz’s local time and noncovariance at O v2/c2 . 196 9.5.3 Maxwell’s equations are Lorentz covariant . 198 10 The new kinematics and E = mc2 201 10.1 Thenewkinematics . 203 10.1.1 Einstein’s two postulates . 203 CONTENTS vii 10.1.2 The new conception of time and the derivation of the Lorentz transformation . 204 10.1.3 Relativity of simultaneity, time dilation and length contraction ........................207 10.2 The new velocity addition rule . 212 10.2.1 The invariant space-time interval . 212 10.2.2 Adding velocities but keeping light speed constant . 212 10.3 Maxwell’s equations are covariant . 213 10.3.1 The Lorentz transformation of electromagnetic fields . 214 10.3.2 The Lorentz transformation of radiation energy . 216 10.4 TheLorentzforcelaw . 216 10.5 The E = mc2 relation ......................217 10.5.1 Work-energy theorem in relativity . 218 10.5.2 The E = mc2 paper three months later . 219 10.6 SuppMat: Relativistic wave motion . 220 10.6.1 The Fresnel formula from velocity addition rule . 220 10.6.2 The Doppler effect and aberration of light . 221 10.6.3 Derivation of the radiation energy transformation . 222 10.7 SuppMat: Relativistic momentum and force . 224 11 Geometric formulation of relativity 225 11.1 Minkowskispacetime. 227 11.1.1 Rotation in 3D space - a review . 227 11.1.2 The Lorentz transformation as a rotation in 4D space- time ............................228 11.2 Tensorsinaflatspacetime. 229 11.2.1 Tensor contraction and the metric . 230 11.2.2 Minkowski spacetime is pseudo-Euclidean . 232 11.2.3 Relativistic velocity, momentum, and energy . 233 11.2.4 The electromagnetic field tensor . 234 11.2.5 The energy-momentum-stress tensor for a field system 235 11.3 The spacetime diagram . 238 11.3.1 Basic features and invariant regions . 239 11.3.2 Lorentz transformation in the spacetime diagram . 241 11.4 The geometric formulation — a summary . 244 IV GENERAL RELATIVITY 245 12 Towards a general theory of relativity 247 viii CONTENTS 12.1 Einstein’s motivations for GR . 248 12.2 Theequivalenceprinciple . 249 12.2.1 The inertia mass vs. the gravitational mass . 249 12.2.2 ‘Myhappiestthought’ . 252 12.3 Implications of the equivalence principle . 253 12.3.1 Bendingofalightray . 254 12.3.2 Gravitational redshift . 255 12.3.3 Gravitational time dilation . 258 12.3.4 Gravity-induced index of refraction in free space . 259 12.3.5 Light ray deflection calculated . 260 12.3.6 From EP to ‘gravity as the structure of spacetime’ . 262 12.4 Elements of Riemannian geometry . 262 12.4.1 Gaussian coordinates and the metric tensor . 263 12.4.2 Geodesic equation . 265 12.4.3 Flatnesstheorem .