QUANTUM MECHANICS: PRINCIPLES AND FORMALISM
1 • MATHEMATICAL PRELIMINARIES 3 11 Probability and Statistics 3 12 Linear Algebra 9 13 References 17 14 Problems 17 2 • CLASSICAL DESCRIPTION OF POLARIZATION 21 21 Polarization 21 22 Birefringence 26 23 Modifying the Polarization 29 24 Jones Vectors and Jones Matrices 31 25 Polarization Interferometer 36 26 References 39 27 Problems 40 Complement to Chapter 2: 2A Coherence and Interference 42 3 • QUANTUM STATES 47 31 State Vectors 47 32 Basis States 49 33 Other States 51 34 Probabilities 53 35 Complex Probability Amplitudes 55 36 Row and Column Vector Notation 57 37 Interference 59 38 Problems 62 4 • OPERATORS 65 41 Operators 65 42 The Adjoint Operator 67 43 The Projection Operator 69 44 The Matrix Representation of Operators 70 45 Changing Bases 75 46 Hermitian Operators 79 47 References 80 48 Problems 81 Complement to Chapter 4: 4A Similarity Transformations 83 5 • MEASUREMENT 89 51 Measuring Polarization 89 52 The Postulates of Quantum Mechanics 91 53 Expectation Values 94 54 Operators and Measurements 96 55 Commutation and Indeterminacy Relations 97 56 Complementarity 101 57 References 103 58 Problems 103 Complement to Chapter 5 5A ‘‘Measuring’’ a Quantum State 106 6 • SPIN-1/2 111 61 The Stern-Gerlach Experiment 111 62 Spin States 114 63 More Spin States 116 64 Commutation Relations 120 65 Particle Interference 121 66 References 124 67 Problems 124 7 • ANGULAR MOMENTUM AND ROTATION 127 71 Commuting Observables 127 72 Angular Momentum Operators 128 73 Eigenvalues and Eigenstates 130 74 Spin-1 133 75 Rotation 134 76 Spin of a Photon 137 77 References 138 78 Problems 139 Complements to Chapter 7 7A Compatible Observables 141 7B Eigenvalues and Eigenstates of Angular Momentum 146 8 • TWO-PARTICLE SYSTEMS AND ENTANGLEMENT 153 81 Pairs of Photons 153 82 Entangled States 159 83 Mixed States 163 84 Testing Local Realism 165 85 References 171 86 Problems 171 Complements to Chapter 8: 8A The Density Operator 175 8B The Bell-Clauser-Horne Inequality 182 8C Two Spin-1/2 Particles 188 9 • TIME EVOLUTION AND THE SCHRÖDINGER EQUATION 193 91 The Time-Evolution Operator 193 92 The Schrödinger Equation 194 93 Expectation Values 196 94 Spin-1/2 Particle in a Magnetic Field 197 95 Neutrino Oscillations 200 96 References 203 97 Problems 203 Complement to Chapter 9: 9A Magnetic Resonance 206 10 • POSITION AND MOMENTUM 215 101 Position 215 102 Momentum 221 103 The Momentum Basis 227 104 Problems 231 Complement to Chapter 10: 10A Useful Mathematics 233 11 • WAVE MECHANICS AND THE SCHRÖDINGER EQUATION 241 111 The Schrödinger Equation Revisited 241 112 Constant Potential–the Free Particle 246 113 Potential Step 247 114 Tunneling 253 115 Infi nite Square Well 256 116 References 265 117 Problems 265 Complement to Chapter 11: 11A Free Particle Propagation 269 12 • THE HARMONIC OSCILLATOR 275 121 Why Study the Harmonic Oscillator? 275 122 Creation, Annihilation, and Number Operators 276 123 Wave Functions 281 124 Fock States and Photons 284 125 Coherent States 287 126 References 294 127 Problems 294 Complement to Chapter 12: 12A Solving the Schrödinger Equation Directly 296 13 • WAVE MECHANICS IN THREE DIMENSIONS 301 131 The Schrödinger Equation in Three Dimensions 301 132 Central Potentials 304 133 Orbital Angular Momentum 310 134 The Hydrogen Atom 314 135 Multielectron Atoms 323 136 References 325 137 Problems 326 Complements to Chapter 13: 13A Quantum Dots 329 13B Series Solution to the Radial Equation 334 14 • TIME-INDEPENDENT PERTURBATION THEORY 337 141 Nondegenerate Theory 337 142 Degenerate Theory 344 143 Fine Structure of Hydrogen 348 144 Hyperfi ne Structure of Hydrogen 356 145 The Zeeman Effect 357 146 References 359 147 Problems 359 15 • TIME-DEPENDENT PERTURBATION THEORY 363 151 Time Evolution of the State 363 152 Sinusoidal Perturbations 367 153 Atoms and Fields 369 154 The Photoelectric Effect 375 155 References 377 156 Problems 378 Complement to Chapter 15: 15A Einstein’s A and B Coeffi cients 380 16 • QUANTUM FIELDS 383 161 The Schrödinger and Heisenberg Pictures of Quantum Mechanics 383 162 The Field Hamiltonian 385 163 Field Operators 387 164 Field States 390 165 Fully Quantum Mechanical Atom-Field Interactions 394 166 Quantum Theory of Photoelectric Detection 400 167 Beam Splitters 402 168 References 406 169 Problems 407 Complement to Chapter 16: 16A Second-Order Coherence and the Grangier Experiment 410 17 • QUANTUM INFORMATION 415 171 Qubits and Ebits 415 172 Quantum Cryptography 416 173 The No-Cloning Theorem 419 174 Quantum Teleportation 421 175 Quantum Computing 423 176 References 431 177 Problems 431 LABORATORIES 433 Getting Started 433 Before Lab 433 Important Laboratory Safety Tips 434 LAB 1 • SPONTANEOUS PARAMETRIC DOWNCONVERSION 435 Lab Ticket 435 L11 Introduction 435 L12 Aligning the Crystal 438 L13 Aligning Detector A 440 L14 Aligning Detector B 443 L15 Angular Correlations – Momentum Conservation 444 L16 Polarization 445 L17 Timing 446 L18 References 447 LAB 2 • ‘‘PROOF’’ OF THE EXISTENCE OF PHOTONS 449 Lab Ticket 449 L21 Introduction 449 L22 Theory 451 L23 Aligning the Irises and the Beam Splitter 456 L24 Aligning the B’ Detector 458 L25 Measuring g (2) (0) for a Single-Photon State 460 L26 Two-Detector Measurement of g (2) (0) 461 L27 References 462 LAB 3 • SINGLE-PHOTON INTERFERENCE 463 Lab Ticket 463 L31 Introduction 463 L32 Aligning the Polarization Interferometer 465 L33 Equalizing the Path Lengths 468 L34 The Polarization Interferometer 470 L35 Single-Photon Interference and the Quantum Eraser 471 L36 “Experiment 6” 472 L37 Particles and Waves 473 L38 References 474 LAB 4 • QUANTUM STATE MEASUREMENT 475 Lab Ticket 475 L41 Introduction 475 L42 Alignment 478 L43 Measurement of Linear Polarization States 479 L44 Measurement of Circular and Elliptical Polarization States 480 L45 References 481 LAB 5 • TESTING LOCAL REALISM 483 Lab Ticket 483 L51 Introduction 483 L52 Theory 485 L53 Alignment 486 L54 Creating the Bell State 487 L55 Exploring Quantum Correlations--Entangled States and Mixed States 490 L56 Testing the CHSH Inequality 491 L57 Measuring H 492 L58 Optimizing Your Results 493 L59 Last Experiment 495 L510 References 495