Class Notes Introduction to Modern Physics Physics 321 Plan II Under Construction Austin M. Gleeson 1 Department of Physics University of Texas at Austin Austin, TX 78712 January 15, 2010 1 gleeson@physics.utexas.edu
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Contents 1 Introduction 11 1.1 Purpose of This Course..................... 11 1.2 Physics that you should know.................. 13 1.2.1 Introduction....................... 13 1.2.2 Kinematics........................ 14 1.2.3 Dynamics......................... 15 1.3 The Role of Mathematics.................... 19 1.3.1 Mathematics and Symbols That You Should Know.. 19 1.4 First Day Handout........................ 23 1.4.1 Fermi Problems...................... 24 1.4.2 Things Everyone Should Know............. 24 1.4.3 Order of Magnitude Estimates............. 26 1.4.4 Home Experiments.................... 27 1.4.5 Review Syllabus..................... 27 1.4.6 Text............................ 28 1.5 What is Physics?......................... 28 1.5.1 Range of Phenomena................... 30 1.5.2 Reductionism and General Principles.......... 34 1.6 Problems............................. 35 2 Measurement 37 2.1 The Role of Measurement.................... 37 2.2 Measurability........................... 38 2.3 Role of Standards......................... 42 2.3.1 The Story of Length................... 43 2.3.2 Accuracy and Precision of Standards.......... 46 2.4 Quantities of Physics....................... 46 2.5 Dimensional Analysis....................... 47 2.5.1 Uses of Dimensional Analysis.............. 48 3
4 CONTENTS 2.5.2 Scaling Laws....................... 50 2.6 Fundamental Dimensional Constants.............. 51 2.6.1 Sizes............................ 51 2.6.2 Modern Standards.................... 53 2.7 Problems............................. 55 3 Pre 19th Century Physics 57 3.1 Introduction............................ 57 3.2 Least Time Formulation of Light Propagation......... 59 3.2.1 Speculation on the form of Fermat s Theory...... 63 3.3 Applications of Fermat s Principle............... 65 3.3.1 Light Travels in Straight Lines............. 65 3.3.2 Refraction & Snell s Law................ 65 3.3.3 Lenses........................... 69 3.3.4 Total Internal Reflection................. 70 3.3.5 Rays in a General Inhomogeneous Space and Mirages. 71 3.3.6 Reflection and Mirrors.................. 72 3.3.7 Mathematical Digression................. 76 3.4 Newton and Color........................ 78 3.5 Fresnel/Young/Huygens Theory................. 81 3.5.1 Recapitulation of Fermat s Least time principal.... 81 3.5.2 Problems with Fermat s Least Time.......... 83 3.5.3 Huygens.......................... 84 3.5.4 Thomas Young and Interference............ 86 3.5.5 Detail of the Analysis of Interference for the Double Slit 91 3.5.6 Phasers.......................... 96 3.5.7 Example of Three Slits and More............ 100 3.5.8 The Theory of How Light Or Any Other Wavelike Disturbance Propagates................. 103 3.5.9 How do we get least time from Fresnel s Theory?... 115 3.5.10 Polarization........................ 118 3.5.11 The Field......................... 119 4 19th Century Physics 121 4.1 Action at a Distance and Field Dynamics........... 121 4.1.1 Action at a Distance................... 122 4.1.2 Local Field Theory.................... 124 4.2 The Stretched String....................... 125 4.3 Maxwell s Theory of Electromagnetism............. 136 4.4 Dynamics and Action...................... 146
CONTENTS 5 4.4.1 Background on Formulation of Action......... 147 4.4.2 Introduction to Action.................. 148 4.4.3 Definition of Action................... 150 4.4.4 Trajectory of a Free Particle.............. 152 4.4.5 Proof that the Least Action Reproduces Newtonian Physics.......................... 154 4.4.6 Examples of action gravitation near a flat earth... 154 4.4.7 Same Example done another way........... 158 4.4.8 Digression on averages and slicing........... 159 4.4.9 More Examples of Actions................ 163 5 Basic Principles of Physics 169 5.1 Symmetry............................. 169 5.2 The Nature of Symmetry in Physics.............. 174 5.2.1 Discrete Transformations................ 175 5.2.2 Continuous Transformations............... 175 5.2.3 Identity Transformation................. 176 5.2.4 Examples of symmetry in situations like physics... 176 5.2.5 Physics transformations:................. 177 5.3 Examples of Symmetry in physics................ 177 5.3.1 Physics transformations:................. 177 5.4 Symmetry and Action...................... 180 5.4.1 Introduction....................... 180 5.4.2 Galilean invariance.................... 183 5.4.3 More on Symmetry and Action............. 184 5.4.4 Noether s Theorem.................... 184 6 Special Classical Physical Systems 191 6.1 Introduction............................ 191 6.2 The Harmonic Oscillator..................... 191 6.2.1 Importance........................ 191 6.2.2 Dynamics......................... 192 6.2.3 Examples of harmonic oscillator systems........ 195 6.2.4 Normal Modes...................... 196 6.3 The Stretched String Revisited................. 198 6.3.1 Distributed Systems................... 198 6.3.2 Concluding Remarks................... 201
6 CONTENTS 7 The Special Theory of Relativity 205 7.1 Pre-History of concepts about light............... 205 7.2 Galilean Invariance........................ 206 7.3 Implications of and for Maxwell s Equations.......... 211 7.4 Pursuit of a special frame.................... 214 7.5 Michelson-Morley Experiment.................. 214 8 Kinematics of special relativity 219 8.1 Special Relativity......................... 219 8.1.1 Principles of Relativity.................. 219 8.2 Harry and Sally and Space Time Diagrams.......... 222 8.2.1 Introduction....................... 222 8.2.2 The Paradox of Harry and Sally............ 222 8.3 The Relativity of Simultaneity................. 223 8.3.1 Harry and Sally s Movements in a Diagram...... 224 9 The Nature of Space-Time 231 9.1 The Problem of Coordinates................... 231 9.2 The Lorentz Transformations.................. 236 9.2.1 The Relatively Moving Clock.............. 241 9.2.2 Derivation of the Lorentz Transformation....... 244 9.2.3 Details of the Derivation of the Lorentz Transformations245 9.3 Using Lorentz Transformations................. 247 9.3.1 Time Dilation....................... 247 9.3.2 Length contraction.................... 249 9.3.3 The Doppler Effect.................... 250 9.3.4 Addition of velocities................... 252 9.3.5 Time for Different Travelers............... 255 9.3.6 Visual Appearence of Rapidly Moving Objects.... 256 10 Events, Worldlines, Intervals 259 10.1 Introduction............................ 259 10.2 Place and Path in the Two Dimensional Plane........ 259 10.3 Minkowski Space-time...................... 270 10.3.1 Future, Past, and Elsewhere............... 271 10.4 Causality and Trajectories.................... 277 10.5 The Hyperbolic Hangle...................... 279 10.5.1 The same result directly using calculus......... 284 10.6 Four Vectors and Invariants................... 284 10.7 Harry, Dorothy, and Sally Revisited.............. 288
CONTENTS 7 11 Paradoxes of Relativity 293 11.1 The Twin Paradox........................ 293 11.1.1 The Problem....................... 293 11.1.2 The Solution....................... 293 11.2 The Boy in the Barn....................... 294 11.2.1 The Problem....................... 294 11.2.2 The Solution....................... 294 11.3 The Bandits and the Bullet Train................ 294 12 Uniform Acceleration 295 12.1 Events at the same proper distance from some event..... 295 12.2 Uniformly accelerated motion.................. 296 12.2.1 Details of the calculation of the acceleration...... 297 12.3 The proper time along the trajectory.............. 300 12.3.1 Timelike Trajectories and Accelerated Motion..... 300 12.4 Examples using accelerated motion............... 301 12.4.1 Deceleration........................ 301 12.4.2 Accelerated Rocket.................... 302 12.4.3 John Bell s Problem................... 304 12.5 The Accelerated Reference Frame................ 308 13 Relativistic Dynamics 317 13.1 Relativistic Action........................ 317 13.1.1 The Action for a Free Particle............. 317 13.2 Energy and momentum of a single free particle........ 320 13.3 Mass................................ 322 13.4 Kinetic Energy of a Single Particle............... 323 13.5 Transformations of Momentum and Energy.......... 324 13.6 The Energy, Momentum, and Mass of Light.......... 325 13.7 Interactions............................ 326 13.8 Multi-particle Systems...................... 326 13.9 Rest energy of composite and elementary systems....... 326 13.10Applications of Energy Momentum............... 326 14 Introduction to General Relativity 329 14.1 The Problem........................... 329 14.2 Free Fall Observers and the Equivalence Principle....... 330 14.3 The Equivalence Principle.................... 332 14.3.1 The Monkey and the Hunter.............. 333 14.4 Direct Effects from the Equivalence Principle......... 334
8 CONTENTS 14.4.1 Universality and Eötvös Dicke............. 334 14.4.2 Bending of Light Rays.................. 336 14.4.3 Clocks and Accelerations in Towers........... 337 14.5 Intrinsic Effects of Gravity.................... 339 14.5.1 Distortion of Elastic Bodies............... 339 14.5.2 Gravitation and Tidal Forces.............. 341 15 Geometry and Gravitation 347 15.1 Introduction to Geometry.................... 347 15.2 Gaussian Curvature....................... 349 15.3 Example of negative curvature: the Pringle.......... 350 15.4 Curvature and Geodesics..................... 352 15.5 The Theorema Egregium and the Line Element....... 354 15.6 Geometry in Four or More Dimensions............. 355 15.7 Coordinate Labels in General Relativity............ 355 15.8 Einstein Equations........................ 355 16 Effects of Gravitation 357 16.1 Curvature around a Massive Body............... 357 16.2 The Universe........................... 357 16.2.1 Background Ideas.................... 357 16.2.2 Copernican Principle................... 360 16.2.3 Olber s Paradox..................... 361 16.2.4 Hubble Expansion.................... 364 16.2.5 The Age of the Universe................. 367 16.2.6 Models of Expanding Universes............. 367 16.2.7 Inflationary Cosmology................. 383 16.2.8 The Space Time Structure................ 385 16.2.9 Black Body Background................. 385 16.2.10 Problems with the Expanding Universe........ 385 16.2.11 The Cosmological Constant............... 385 16.2.12 The Standard Model of the Universe.......... 385 17 Interface of Large Scale and Micro-physics 387 17.1 Structure in the Universe.................... 387 17.2 The Inflationary Universe.................... 387 17.3 String Theory........................... 387
CONTENTS 9 18 Introduction to Quantum Theory 389 18.1 Introduction............................ 389 18.2 Blackbody Radiation....................... 390 18.2.1 Thermodynamics..................... 390 18.2.2 Radiation in a Cavity.................. 393 18.2.3 Attempts to explain the spectrum........... 394 18.2.4 Planck s Explanation of the Spectrum......... 395 18.3 Photo-Electric Effect....................... 395 18.4 Young s Double Slit Experiment Revisited........... 398 18.5 Action and Quantum Mechanics................ 399 18.6 Constructing the Amplitude................... 401 18.6.1 A Mathematical Aside The Population Equation The Exponential Function................ 402 18.6.2 Even more on phasers.................. 406 18.7 The Uncertainty Relations.................... 410 18.7.1 The Uncertainty Principle and the Quantum Mechanical Harmonic Oscillator................. 412 18.7.2 Oscillator Ground State Wavefunction......... 413 18.8 An Aside on the Particle in the Box.............. 415 18.9 Returning to the Oscillator................... 417 18.10The Time Development of Quantum Systems......... 419 18.10.1 Motion in Quantum Mechanics............. 419 18.10.2 Relation between the Quantum and the Classical Oscillator........................... 420 18.10.3 Classical Motion of the Quantum Oscillator...... 422 18.10.4 An Aside on the Poisson Distribution......... 423 18.10.5 A Return to Classical Motion of the Quantum Oscillator428 19 Quantum Measurement and Bell s Theorem 429 19.1 A Two Level System....................... 429 19.2 More on polarized light as a two level system......... 431 19.3 More on Bell s Theorem..................... 434 19.3.1 What is a particle and what is the field?....... 437 20 Quantum Field Theory 441 20.1 Introduction............................ 441 20.2 The Many Photon State..................... 442 20.3 The Stretched String Revisited Again............. 443 20.4 The Quantum Stretched String................. 443 20.5 The field.............................. 443
10 CONTENTS 20.6 Elementary Particles....................... 444 20.7 Fundamental Processes...................... 445