Quantum Mechanics II: Advanced Topics

Description

Quantum Mechanics II: Advanced Topics offers a comprehensive exploration of the state-of-the-art in various advanced topics of current research interest. A follow-up to the authors’ introductory book Quantum Mechanics I: The Fundamentals, this book expounds basic principles, theoretical treatment, case studies, worked-out examples and applications of advanced topics including quantum technologies. A thoroughly revised and updated this unique volume presents an in-depth and up-to-date progress on the growing topics including latest achievements on quantum technology. In the second edition six new chapters are included and the other ten chapters are extensively revised. Features Covers classical and quantum field theories, path integral formalism and supersymmetric quantum mechanics. Highlights coherent and squeezed states, Berry’s phase, Aharonov―Bohm effect and Wigner function. Explores salient features of quantum entanglement and quantum cryptography. Presents basic concepts of quantum computers and the features of no-cloning theorem and quantum cloning machines. Describes the theory and techniques of quantum tomography, quantum simulation and quantum error correction. Introduces other novel topics including quantum versions of theory of gravity, cosmology, Zeno effect, teleportation, games, chaos and steering. Outlines the quantum technologies of ghost imaging, detection of weak amplitudes and displacements, lithography, metrology, teleportation of optical images, sensors, batteries and internet. Contains several worked-out problems and exercises in each chapter. Quantum Mechanics II: Advanced Topics addresses various currently emerging exciting topics of quantum mechanics. It emphasizes the fundamentals behind the latest cutting-edge developments to help explain the motivation for deeper exploration. The book is a valuable resource for graduate students in physics and engineering wishing to pursue research in quantum mechanics. Cover Half Title Title Page Copyright Page Dedication Contents Preface About the Authors 1. Quantum Field Theory 1.1. Introduction 1.2. Why Quantum Field Theory? 1.3. What is a Field? 1.4. Classical Field Theory 1.5. Quantum Equations for Fields 1.6. Quantization of Nonrelativistic Wave Equation 1.7. Electromagnetic Field in Vacuum 1.8. Interaction of Charged Particles with Electromagnetic Field 1.9. Quantization of Klein–Gordon Equation 1.10. Quantization of Dirac Field 1.11. Gauge Field Theories 1.12. Concluding Remarks 1.13. Bibliography 1.14. Exercises 2. Path Integral Formulation 2.1. Introduction 2.2. Time Evolution of Wave Function and Propagator 2.3. Path Integral Representation of Propagator 2.4. Connection Between Propagator and Classical Action 2.5. Schrodinger Equation From Path Integral Formulation 2.6. Transition Amplitude of a Free Particle 2.7. Systems with Quadratic Lagrangian 2.8. Path Integral Version of Ehrenfest's Theorem 2.9. Concluding Remarks 2.10. Bibliography 2.11. Exercises 3. Supersymmetric Quantum Mechanics 3.1. Introduction 3.2. Supersymmetric Potentials 3.3. Relations Between the Eigenstates of Two Supersymmetric Hamiltonians 3.4. Hierarchy of Supersymmetric Hamiltonians 3.5. Applications 3.6. Generation of Complex Potentials with Real Eigenvalues 3.7. Concluding Remarks 3.8. Bibliography 3.9. Exercises 4. Coherent and Squeezed States 4.1. Introduction 4.2. The Uncertainty Product of Harmonic Oscillator 4.3. Coherent States: De nition, Uncertainty Product and Physical Meaning 4.4. Generation and Properties of Coherent States 4.5. Spin Coherent States 4.6. Coherent States of Position-Dependent Mass Systems 4.7. Squeezed States 4.8. Deformed Oscillators and Nonlinear Coherent States 4.9. Concluding Remarks 4.10. Bibliography 4.11. Exercises 5. Berry's Phase, Aharonov–Bohm and Sagnac Effects 5.1. Introduction 5.2. Derivation of Berry's Phase 5.3. Origin and Properties of Berry's Phase 5.4. Classical Analogue of Berry's Phase 5.5. Berry's Phase in Solid State Physics 5.6. Examples and E ects of Berry's Phase 5.7. Applications of Berry's Phase 5.8. Experimental Veri cation of Berry's Phase 5.9. Pancharatnam's Work 5.10. Cumulants Associated with Geometric Phases 5.11. The Aharonov–Bohm Effect 5.12. Sagnac Effect 5.13. Concluding Remarks 5.14. Bibliography 5.15. Exercises 6. Phase Space Picture and Canonical Transformations 6.1. Introduction 6.2. Squeeze and Rotation in Phase Space 6.3. Linear Canonical Transformations 6.4. Wigner Function 6.5. Time Evolution of the Wigner Function 6.6. Applications 6.7. Advantages of the Wigner Function 6.8. Concluding Remarks 6.9. Bibliography 6.10. Exercises 7. Quantum Entanglement 7.1. Introduction 7.2. States in Classical Mechanics 7.3. Quantum Entangled States 7.4. Mixed States 7.5. Bipartite Systems 7.6. Separability Criteria 7.7. Multipartite Entanglement 7.8. Quantifying Entanglement 7.9. Applications of Entanglement 7.10. Concluding Remarks 7.11. Bibliography 7.12. Exercises 8. Quantum Decoherence 8.1. Introduction 8.2. Decoherence and Interference Damping 8.3. Interaction of a Detector on the Double-Slit Experiment 8.4. Decoherence Due to Phase Randomization 8.5. Position Decoherence Due to Environmental Scattering 8.6. Master Equations 8.7. Decoherence Models 8.8. Decoherence Experiments 8.9. The Role of Decoherence in the Interpretation of Quantum Mechanics 8.10. Concluding Remarks 8.11. Bibliography 8.12. Exercises 9. Quantum Computers 9.1. Introduction 9.2. What is a Quantum Computer? 9.3. Why is a Quantum Computer? 9.4. Fundamental Properties 9.5. Quantum Algorithms 9.6. Testing Quantum Computers Using Grover's Algorithm 9.7. Features of Quantum Computation 9.8. Quantum Computation Through NMR 9.9. Why is Making a Quantum Computer Extremely Diffcult? 9.10. Concluding Remarks 9.11. Bibliography 9.12. Exercises 10. Quantum Cryptography 10.1. Introduction 10.2. Standard Cryptosystems 10.3. Quantum Cryptography–Basic Principle 10.4. Types of Quantum Cryptography 10.5. Multiparty Quantum Secret Sharing 10.6. Applications of Quantum Cryptography 10.7. Implementation and Limitations 10.8. Fiber-Optical Quantum Key Distribution 10.9. Quantum Cheque Scheme 10.10. Concluding Remarks 10.11. Bibliography 10.12. Exercises 11. No-Cloning Theorem and Quantum Cloning Machines 11.1. Introduction 11.2. Proof of No-Cloning Theorem 11.3. No-Broadcasting Theorem 11.4. No-Cloning and No-Superluminar Signalling 11.5. Quantum Cloning Machines 11.6. Quantum Telecloning 11.7. Other No-Go Theorems 11.8. Concluding Remarks 11.9. Bibliography 11.10. Exercises 12. Quantum Tomography 12.1. Introduction 12.2. Pauli Problem 12.3. Recovery of Density Matrix from Wigner Function 12.4. Optical Homodyne Tomography 12.5. Qubit Quantum Tomography 12.6. Experimental Measure of Polarization of a Photonic Qubit 12.7. Multiqubit Tomography 12.8. Quantum Process Tomography 12.9. Conclusion 12.10. Bibliography 12.11. Exercises 13. Quantum Simulation 13.1. Introduction 13.2. Limitations of Classical Computers in Simulating Quantum Systems 13.3. Quantum Simulators 13.4. Analog Quantum Simulators 13.5. Digital Quantum Simulators 13.6. Theory of Quantum Simulation of the Schrodinger Equation 13.7. Quantum Simulators Using Quantum Computers 13.8. Quantum Circuits 13.9. Quantum Circuits for Final Measurements 13.10. Concluding Remarks 13.11. Bibliography 13.12. Exercises 14. Quantum Error Correction 14.1. Introduction 14.2. Sources of Errors in Quantum Information Processing 14.3. Di culties of Using Classical Error Correction Techniques to QEC 14.4. Digitization of Quantum Errors 14.5. QEC Mechanisms Using Quantum Redundancy 14.6. QEC with Stabilizer Codes 14.7. The Surface Code 14.8. Practical Issues in the Implementation of QEC Codes 14.9. Concluding Remarks 14.10. Bibliography 14.11. Exercises 15. Some Other Advanced Topics 15.1. Introduction 15.2. Quantum Theory of Gravity 15.3. Quantum Cosmology 15.4. Quantum Zeno Effect 15.5. Quantum Teleportation 15.6. Quantum Games 15.7. Quantum Pseudo-Telepathy Games 15.8. Quantum Steering 15.9. Quantum Diffusion 15.10. Quantum Chaos 15.11. Concluding Remarks 15.12. Bibliography 15.13. Exercises 16. Quantum Technologies 16.1. Introduction 16.2. Quantum Entangled Photons 16.3. Ghost Imaging 16.4. Detection of Weak Amplitude Object 16.5. Entangled Two-Photon Microscopy 16.6. Detection of Small Displacements 16.7. Quantum Lithography 16.8. Quantum Metrology 16.9. Quantum Teleportation of Optical Images 16.10. Quantum Sensors 16.11. Quantum Batteries 16.12. Quantum Internet 16.13. Concluding Remarks 16.14. Bibliography 16.15. Exercises Solutions to Selected Exercises Index