II Relationship of Classical Theory to Quantum Theory A Quantum mean occupation number

Transcription

1 Appendix B Some Unifying Concepts Version 04.AppB.11.1K [including mostly Chapters 1 through 11] by Kip [This appendix is in the very early stages of development] I Physics as Geometry A Newtonian Physics as Geometry 1 Flat space as the arena: Sec Coordinate invariance of physical laws a Idea Introduced: Sec. 1.2 b Newtonian particle kinetics as an example: Sec. 1.4 c Newtonian mass conservation and force balance: Sec Elasticity in Geometric Language a Irreducible tensorial parts of strain tensor: expansion, shear and rotation: Sec. 10.2, Box 10.1 b Elastic stress tensor and force balance: Secs. 10.4, 11.2 B Special Relativistic Physics as Geometry 1 Flat spacetime as the arena: Sec Frame-invariance of physical laws a Idea introduced: Sec. 1.2 b Relativistic particle kinetics: Sec. 1.4 c Stress-energy and 4-momentum conservation: Secs. 1.4 & 1.12 d Electromagnetic theory: Secs. 1.4, 1.10 & e Kinetic theory: Chap. 2 - Secs. 2.2, 2.5 & Splits of spacetime into space plus time, and resulting relationship between frameinvariant and frame-dependent laws of physics: a Particle kinetics: Sec. 1.6 b Electromagnetic theory: Sec c Continuum mechanics; stress-energy tensor: Sec d Kinetic theory: Secs. 2.2, 2.5 & 2.7; Ex Spacetime diagrams: Secs. 1.6, 1.7, 1.8, 1.12, 2.2; Ex C General Relativistic Physics as Geometry 1 Curved spacetime as the arena: Sec Details: Part VI D Kinetic Theory in Geometric Language 1 Phase space for particles as the arena: Chap 2 2 Details: Chap 2 E Statistical Mechanics in Geometric Language 1 Phase space for ensembles as the arena: Chap 3 2 invariance under canonical transformations (change of generalized coordinates and momenta in phase space): Sec. 3.2, Ex. 3.1 II Relationship of Classical Theory to Quantum Theory A Quantum mean occupation number Page 1

2 1 As classical distribution function: Sec Determines whether particles behave like a classical wave, like classical particles, or quantum mechanically: Secs. 2.3 & 2.4; Ex. 2.1; Fig. 2.5 B Geometric optics (Eikonal or WKB approximation) 1 Geometric optics of a classical wave is particle mechanics of the wave's quanta: Sec Geometric optics limit of Schrodinger equation is classical particle mechanics: Ex. 6.6 III Conservation Laws A In relativity 1 Charge conservation: Sec particle & rest-mass conservation: Sec momentum conservation: Secs. 1.4, 1.6, 1.12 B In Newtonian physics 1 particle conservation 2 rest-mass conservation 3 momentum conservation 4 energy conservation C You get out what you put in! [use of conservation laws to deduce equations of motion] IV Statistical Physics Concepts A Systems: Sec [Give Examples] B Distribution functions & their evolution 1 For particles: Sec For photons, and its relationship to specific intensity: Sec For systems in statistical mechanics: Sec Evolution via Vlasov or Boltzmann transport equation: Sec. 2.7 a Kinetic Theory: Sec 2.7 b Statistical mechanics: Sec For plasmons: Chap For random processes: hierarchy of probability distributions: Sec. 5.2 C Thermal [statistical] equilibrium; equilibrium ensembles; representations of thermodynamics 1 Evolution into statistical equilbrium--phase mixing and coarse graining: Secs. 3.6 and In kinetic theory: Sec In statistical mechanics: a general form of distribution function in terms of quantities exchanged with environment: Sec. 3.4 b summary of ensembles and representations: Table 4.1, Sec. 3.4 c Microcanonical ensemble; Energy representation: Secs. 3.5 and 4.2 d Canonical ensemble; Free-energy representation: Sec. 4.3 e Gibbs ensemble; Gibbs representation: Sec. 4.4 f Grand canonical ensemble & representation: Sec. 3.7 and Ex. 3.6 and 3.7 D Fluctuations in statistical equilibrium Page 2

3 1 Summary: Table Particle number in a box: Ex Distribution of particles and energy inside a closed box: Sec Temperature and volume fluctuations of system interacting with a heat and volume bath: Sec Fluctuation-dissipation theorem: Sec Fokker-Planck equation: Sec Brownian motion: Sec E Entropy 1 Defined: Sec Second law (entropy increase): Secs. 3.6, Entropy per particle: Secs. 3.7, 3.8, Fig. 3.4, Exs. 3.5, Of systems in contact with thermalized baths: Table 4.1; Secs. 4.4 & 4.6; Exs Phase transitions: Secs. 4.4 & 4.6, Exs. 4.4 & Chemical reactions: Sec. 4.4, Exs. 4.5 & Of black holes and the expanding universe: Sec Connection to Information: Sec. 3.9 F Macroscopic properties of matter as integrals over momentum space: 1 In kinetic theory: Secs. 2.5, 2.6, 2,8 2 In statistical mechanics: Ex In theory of random processes: Ensemble averages: Sec. 5.2 G Random Processes: Chap 5 [extended to complex random processes in multiple dimensions: Ex. 8.7] 1 Theory of Real-valued random processes: Chap 5 2 Theory of Complex-valued random processes: Ex Some unifying tools: a Gaussian processes & central limit theorem: Secs 5.2, 5.3, 5.5 b Correlation functions, spectral densities, and Wiener-Khintchine [van Cittert-Zernike] theorem relating them: Secs , 8.7 c Filtering: Sec. 5.5, Exs. 5.2 & 5.3 d Fluctuation-dissipation theorem: Secs , 5.10, 10.5, Exs. 5.7, 5.8, 5.10, 10.6 e Fokker-Planck equation: Secs , 5.6.3, Exs 5.6, 5.9 V Optics [wave propagation] Concepts A Geometric Optics [eikonal or WKB approximation] 1 General theory: Secs. 6.3, Box Dispersion relations & their role as Hamiltonians: Secs. 6.3, Fermat's principle: Secs. 6.3, Paraxial ray optics: Secs. 6.4, 7.5 a use in analyzing optical instruments: Exs. 6.10, 6.11, Figs. 6.3, Breakdown of geometric optics: Secs. 6.3, 6.6, 7.6, Ex Caustics & Catastrophes: Secs. 6.6, Applications: a waves in solids: Chap. 11 Page 3

4 b seismic waves in Earth: Sec , Fig c gravitational lensing B Linear, Finite-wavelength phenomena 1 Wave packets: their motion, energy, and spreading 2 Diffraction a General theory (Helmholtz-Khirkhoff integral): Sec. 7.2 b Fraunhofer (distant) regime: Sec. 7.3 i Diffraction patterns: Figs. 7.4, 7.6 ii Babinet's principle: Sec iii Field near caustics: Sec. 7.6, Fig c Fresnel (near) regime: Sec Fourier optics (paraxial optics with finite wavelengths): Sec. 7.5 a use in analyzing optical instruments: Sec. 7.5, Fig , Ex. 7.8, 4 Coherence and its applications: Secs. 8.2, 8.6; Ex Interference: Chap 8 a Etalons and optical (Fabry-Perot) cavities: Sec. 8.4 b Radio interferometers: Sec. 8.3 c Gravitational-wave interferometers: Sec. 8.5 d Intensity interferometry: Sec Edge waves & wave-wave mixing at the boundary of a medium a Rayleigh waves and love waves in a solid: Sec b Wave-wave mixing in solids: Sec , Fig c Water waves: C Nonlinear wave-wave mixing 1 General theory: Chap. 9 2 Applications: a Holography: Sec. 9.3, Ex. 9.3 b Phase Conjugation: Secs. 9.4, 9.6.2; Fig. 9.10, Ex. 9.9 c Frequency doubling: Sec , 9.6.1; Ex. 9.8 d Light Squeezing: Ex Venues: a In nonlinear crystals b In fluids i solitary waves ii onset of turbulence c In plasmas VI Equilibria and their stability A Bifurcations of equilibria and the onset of instabilities 1 General discussion and examples: Secs. 10.8, Compressed beam or playing card: Secs. 10.8, VII Computational Techniques A Differential Geometry; Vectors and Tensors: Chap. 1, Sec. 10.3, Part VI Page 4

5 B Two-lengthscale expansions: Box Solution of Boltzmann transport equation in diffusion approximation: Sec Semiclosed systems in statistical mechanics: Sec Statistical independence of subsystems: Sec As foundation for geometric optics: Sec Boundary layers in fluid mechanics C Matrix and propagator techniques for linear systems 1 Paraxial geometric optics: Matrix methods: Sec Paraxial Fourier optics (finite wavelengths): Propagator methods: Sec. 7.5 D Green's functions 1 In elasticity theory: physicists' and Heaviside: Sec E Reciprocity relations and Wronskians conservation 1 For partially transmitting mirrors and beam splitters in optics: F Junction conditions at a boundary 1 In elastodynamics: Sec , Ex G Normal modes 1 In a solid: Sec , Ex Page 5