Theory and Experiment

Transcription

1 Theory and Experiment Mark Beck OXPORD UNIVERSITY PRESS

2 Contents Table of Symbols Preface xiii xix 1 MATHEMATICAL PRELIMINARIES Probability and Statistics LinearAlgebra References Problems 17 2 CLASSICAL DESCRIPTION OF POLARIZATION Polarization Birefringence Modifying the Polarization Jones Vectors and Jones Matrices Polarization Interferometer References Problems 40 Complement to Chapter 2: 2.A Coherence and Interference 42 3 QUANTUM STATES State Vectors Basis States Other States Probabilities Complex Probability Amplitudes Row and Column Vector Notation 57

3 3.7 Interference Problems 62 4 OPERATORS Operators The Adjoint Operator The Projection Operator The Matrix Representation of Operators Changing Bases Hermitian Operators References Problems 81 Complement to Chapter 4: 4.A Similarity Transformations 83 5 MEASUREMENT Measuring Polarization The Postulates of Quantum Mechanics Expectation Values Operators and Measurements Commutation and Indeterminacy Relations Complementarity References Problems 103 Complement to Chapter 5 5.A "Measuring" a Quantum State SPIN-1/2 HI 6.1 The Stern-Gerlach Experiment Spin States More Spin States Commutation Relations Particle Interference References Problems ANGULAR MOMENTUM AND ROTATION Commuting Observables Angular Momentum Operators Eigenvalues and Eigenstates Spin Rotation 134 viii CONTENTS

4 7.6 Spin of a Photon References Problems 139 Complements to Chapter 7 7.A Compatible Observables B Eigenvalues and Eigenstates of Angular Momentum 146 8» TWO-PARTICLE SYSTEMS AND ENTANGLEMENT Pairs of Photons Entangled States Mixed States Testing Local Realism References Problems 171 Complements to Chapter 8: 8.A The Density Operator B The Bell-Clauser-Horne Inequality C Two Spin-1/2 Particles TIME EVOLUTION AND THE SCHRODINGER EQUATION The Time-Evolution Operator The Schrodinger Equation Expectation Values Spin-1/2 Particle in a Magnetic Field Neutrino Oscillations References Problems 203 Complement to Chapter 9: 9.A Magnetic Resonance POSITION AND MOMENTUM Position Momentum The Momentum Basis Problems 231 Complement to Chapter 10: 10.A Useful Mathematics * WAVE MECHANICS AND THE SCHRODINGER EQUATION The Schrodinger Equation Revisited Constant Potential-the Free Particle Potential Step 247 CONTENTS Ix

5 11.4 Tunneling Infinite Square Well References Problems 265 Complement to Chapter 11: 11. A Free Particle Propagation THE HARMONIC OSCILLATOR Why Study the Harmonic Oscillator? Creation, Annihilation, and Number Operators Wave Functions Fock States and Photons Coherent States References Problems 294 Complement to Chapter 12: 12.A Solving the Schrodinger Equation Directly WAVE MECHANICS IN THREE DIMENSIONS The Schrodinger Equation in Three Dimensions Central Potentials Orbital Angular Momentum The Hydrogen Atom Multielectron Atoms References Problems 326 Complements to Chapter 13: 13.A Quantum Dots B Series Solution to the Radial Equation TIME-INDEPENDENT PERTURBATION THEORY Nondegenerate Theory Degenerate Theory Fine Structure of Hydrogen Hyperfine Structure of Hydrogen The Zeeman Effect References Problems TIME-DEPENDENT PERTURBATION THEORY Time Evolution of the State Sinusoidal Perturbations Atoms and Fields 369 x CONTENTS

6 15.4 The Photoelectric Effect References Problems 378 Complement to Chapter 15: 15.A Einstein's A and B Coefficients QUANTUM FIELDS The Schrodinger and Heisenberg Pictures of Quantum Mechanics The Field Hamiltonian Field Operators Field States Fully Quantum Mechanical Atom-Field Interactions Quantum Theory of Photoelectric Detection Beamsplitters References Problems 407 Complement to Chapter 16: 16.A Second-Order Coherence and the Grangier Experiment QUANTUM INFORMATION Qubits and Ebits Quantum Cryptography The No-Cloning Theorem Quantum Teleportation Quantum Computing References 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

7 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