Theory and Experiment Mark Beck OXPORD UNIVERSITY PRESS
Contents Table of Symbols Preface xiii xix 1 MATHEMATICAL PRELIMINARIES 3 1.1 Probability and Statistics 3 1.2 LinearAlgebra 9 1.3 References 17 1.4 Problems 17 2 CLASSICAL DESCRIPTION OF POLARIZATION 21 2.1 Polarization 21 2.2 Birefringence 26 2.3 Modifying the Polarization 29 2.4 Jones Vectors and Jones Matrices 31 2.5 Polarization Interferometer 36 2.6 References 39 2.7 Problems 40 Complement to Chapter 2: 2.A Coherence and Interference 42 3 QUANTUM STATES 47 3.1 State Vectors 47 3.2 Basis States 49 3.3 Other States 51 3.4 Probabilities 53 3.5 Complex Probability Amplitudes 55 3.6 Row and Column Vector Notation 57
3.7 Interference 59 3.8 Problems 62 4 OPERATORS 65 4.1 Operators 65 4.2 The Adjoint Operator 67 4.3 The Projection Operator 69 4.4 The Matrix Representation of Operators 70 4.5 Changing Bases 75 4.6 Hermitian Operators 79 4.7 References 80 4.8 Problems 81 Complement to Chapter 4: 4.A Similarity Transformations 83 5 MEASUREMENT 89 5.1 Measuring Polarization 89 5.2 The Postulates of Quantum Mechanics 91 5.3 Expectation Values 94 5.4 Operators and Measurements 96 5.5 Commutation and Indeterminacy Relations 97 5.6 Complementarity 101 5.7 References 103 5.8 Problems 103 Complement to Chapter 5 5.A "Measuring" a Quantum State 106 6 SPIN-1/2 HI 6.1 The Stern-Gerlach Experiment 111 6.2 Spin States 114 6.3 More Spin States 116 6.4 Commutation Relations 120 6.5 Particle Interference 121 6.6 References 124 6.7 Problems 124 7 ANGULAR MOMENTUM AND ROTATION 127 7.1 Commuting Observables 127 7.2 Angular Momentum Operators 128 7.3 Eigenvalues and Eigenstates 130 7.4 Spin-1 133 7.5 Rotation 134 viii CONTENTS
7.6 Spin of a Photon 137 7.7 References 138 7.8 Problems 139 Complements to Chapter 7 7.A Compatible Observables 141 7.B Eigenvalues and Eigenstates of Angular Momentum 146 8» TWO-PARTICLE SYSTEMS AND ENTANGLEMENT 153 8.1 Pairs of Photons 153 8.2 Entangled States 159 8.3 Mixed States 163 8.4 Testing Local Realism 165 8.5 References 171 8.6 Problems 171 Complements to Chapter 8: 8.A The Density Operator 175 8.B The Bell-Clauser-Horne Inequality 182 8.C Two Spin-1/2 Particles 188 9 TIME EVOLUTION AND THE SCHRODINGER EQUATION 193 9.1 The Time-Evolution Operator 193 9.2 The Schrodinger Equation 194 9.3 Expectation Values 196 9.4 Spin-1/2 Particle in a Magnetic Field 197 9.5 Neutrino Oscillations 200 9.6 References 203 9.7 Problems 203 Complement to Chapter 9: 9.A Magnetic Resonance 206 10 POSITION AND MOMENTUM 215 10.1 Position 215 10.2 Momentum 221 10.3 The Momentum Basis 227 10.4 Problems 231 Complement to Chapter 10: 10.A Useful Mathematics 233 11* WAVE MECHANICS AND THE SCHRODINGER EQUATION 241 11.1 The Schrodinger Equation Revisited 241 11.2 Constant Potential-the Free Particle 246 11.3 Potential Step 247 CONTENTS Ix
11.4 Tunneling 253 11.5 Infinite Square Well 256 11.6 References 265 11.7 Problems 265 Complement to Chapter 11: 11. A Free Particle Propagation 269 12 THE HARMONIC OSCILLATOR 275 12.1 Why Study the Harmonic Oscillator? 275 12.2 Creation, Annihilation, and Number Operators 276 12.3 Wave Functions 281 12.4 Fock States and Photons 284 12.5 Coherent States 287 12.6 References 294 12.7 Problems 294 Complement to Chapter 12: 12.A Solving the Schrodinger Equation Directly 296 13 WAVE MECHANICS IN THREE DIMENSIONS 301 13.1 The Schrodinger Equation in Three Dimensions 301 13.2 Central Potentials 304 13.3 Orbital Angular Momentum 310 13.4 The Hydrogen Atom 314 13.5 Multielectron Atoms 323 13.6 References 325 13.7 Problems 326 Complements to Chapter 13: 13.A Quantum Dots 329 13.B Series Solution to the Radial Equation 334 14 TIME-INDEPENDENT PERTURBATION THEORY 337 14.1 Nondegenerate Theory 337 14.2 Degenerate Theory 344 14.3 Fine Structure of Hydrogen 348 14.4 Hyperfine Structure of Hydrogen 356 14.5 The Zeeman Effect 357 14.6 References 359 14.7 Problems 359 15 TIME-DEPENDENT PERTURBATION THEORY 363 15.1 Time Evolution of the State 363 15.2 Sinusoidal Perturbations 367 15.3 Atoms and Fields 369 x CONTENTS
15.4 The Photoelectric Effect 375 15.5 References 377 15.6 Problems 378 Complement to Chapter 15: 15.A Einstein's A and B Coefficients 380 16 QUANTUM FIELDS 383 16.1 The Schrodinger and Heisenberg Pictures of Quantum Mechanics 3 83 16.2 The Field Hamiltonian 385 16.3 Field Operators 387 16.4 Field States 390 16.5 Fully Quantum Mechanical Atom-Field Interactions 394 16.6 Quantum Theory of Photoelectric Detection 400 16.7 Beamsplitters 402 16.8 References 406 16.9 Problems 407 Complement to Chapter 16: 16.A Second-Order Coherence and the Grangier Experiment 410 17 QUANTUM INFORMATION 415 17.1 Qubits and Ebits 415 17.2 Quantum Cryptography 416 17.3 The No-Cloning Theorem 419 17.4 Quantum Teleportation 421 17.5 Quantum Computing 423 17.6 References 431 17.7 Problems 431 LABORATORIES 433 Getting Started 433 Before Lab 433 Important Laboratory Safety Tips 434 LAB 1 SPONTANEOUS PARAMETRIC DOWNCONVERSION 435 Lab Ticket 435 Ll.l Introduction 435 LI.2 Aligning the Crystal 438 LI.3 Aligning Detector A 440 LI.4 Aligning Detector B 443 LI.5 Angular Correlations - Momentum Conservation 444 LI.6 Polarization 445 LI.7 Timing 446 LI.8 References 447 CONTENTS xi
LAB 2 "PROOF" OF THE EXISTENCE OF PHOTONS 449 Lab Ticket 449 L2.1 Introduction 449 L2.2 Theory 451 L2.3 Aligning the Irises and the Beam Splitter 456 L2.4 Aligning the B' Detector 458 L2.5 Measuring g<2)(0) for a Single-Photon State 460 L2.6 Two-Detector Measurement of g<2)(0) 461 L2.7 References 462 LAB 3 SINGLE-PHOTON INTERFERENCE 463 Lab Ticket 463 L3.1 Introduction 463 L3.2 Aligning the Polarization Interferometer 465 L3.3 Equalizing the Path Lengths 468 L3.4 The Polarization Interferometer 470 L3.5 Single-Photon Interference and the Quantum Eraser 471 L3.6 "Experiment 6" 472 L3.7 Particles and Waves 473 L3.8 References 474 LAB 4 QUANTUM STATE MEASUREMENT 475 Lab Ticket 475 L4.1 Introduction 475 L4.2 Alignment 478 L4.3 Measurement of Linear Polarization States 479 L4.4 Measurement of Circular and Elliptical Polarization States 480 L4.5 References 481 LAB 5 TESTING LOCAL REALISM 483 Lab Ticket 483 L5.1 Introduction 483 L5.2 Theory 485 L5.3 Alignment 486 L5.4 Creating the Bell State 487 L5.5 Exploring Quantum Correlations-Entangled States and Mixed States 490 L5.6 Testing the CHSH Inequality 491 L5.7 Measuring H 492 L5.8 Optimizing Your Results 493 L5.9 Last Experiment 495 L5.10 References 495 xii CONTENTS