Quantum Mechanics and Quantum Information: A Guide through the Quantum World

A thorough definition of the basic concepts of quantum mechanics and quantum information and their interrelations. Alongside a thorough definition of the basic concepts and their interrelations, backed by numerous examples, this textbook features a rare discussion of the quantum information theory. It also deals with other important topics hardly found in the literature, including the Robertson-Schrodinger-relation, angle and angular momentum uncertainties, interaction-free measurements, and the limitations of the no-cloning theorem. With its interpretations of quantum mechanics and its discussions of quantum computing, this book is poised to become the standard textbook for advanced undergraduate and beginning graduate quantum mechanics courses and as an essential reference for physics students and physics professionals.

Table of Contents:
Preface XIII Abbreviations and Notations XIX 1 The Failure of Classical Physics 1 1.1 Blackbody Radiation 1 1.2 Heat Capacity 4 1.3 The Photoelectric Effect 9 1.4 Atoms and Their Spectra 12 1.5 The Double-Slit Experiment 14 Problem 19 References 19 2 The First Steps into the Unknown 21 2.1 The BBR and Planck's Formula 21 2.2 Einstein's Light Quanta and BBR 24 2.2.1 Discussion 27 2.3 PEE Revisited 30 2.4 The Third Breakthrough: de Broglie Waves 31 2.4.1 Exercise 33 Problems 35 References 35 3 Embryonic Quantum Mechanics: Basic Features 37 3.1 A Glimpse of the New Realm 37 3.2 Quantum-Mechanical Superposition of States 39 3.3 What Is Waving There (the Meaning of the C-Function)? 42 3.4 Observables and Their Operators 47 3.5 Quantum-Mechanical Indeterminacy 49 3.6 Indeterminacy and the World 53 3.7 Quantum Entanglement and Nonlocality 58 3.8 Quantum-Mechanical Phase Space 62 3.9 Determinism and Causality in Quantum World 63 3.9.1 Discussion 63 Problems 66 References 66 4 Playing with the Amplitudes 69 4.1 Composition of Amplitudes 69 4.2 Double Slit Revised I 74 4.3 Double Slit Revised II 77 4.4 Neutron Scattering in Crystals 78 4.5 Bosonic and Fermionic States 81 4.6 Path Integrals 89 Problems 93 References 93 5 Basic Features and Mathematical Structure of QM 95 5.1 Observables: the Domain of Classical and Quantum Mechanics 95 5.2 Quantum-Mechanical Operators 97 5.3 Algebra of Operators 100 5.4 Eigenvalues and Eigenstates 102 5.5 Orthogonality of Eigenstates 107 5.6 The Robertson-SchrEUROodinger Relation 110 5.7 The Wave Function and Measurements (Discussion) 112 Problems 116 References 117 6 Representations and the Hilbert Space 119 6.1 Various Faces of a State Function 119 6.2 Unitary Transformations 121 6.3 Operators in the Matrix Form 125 6.4 The Hilbert Space 129 6.5 Operations in the Hilbert Space 135 6.6 Nonorthogonal States 142 Problems 147 References 148 7 Angular Momentum 149 7.1 Orbital and Spin Angular Momenta 149 7.2 The Eigenstates and Eigenvalues of ^L 151 7.3 Operator ^L and Its Commutation Properties 154 7.4 Spin as an Intrinsic Angular Momentum 164 7.5 Angular Momentum of a Compound System 183 7.6 Spherical Harmonics 188 Problems 196 References 197 8 The SchrEUROodinger Equation 199 8.1 The SchrEUROodinger Equation 199 8.2 State Function and the Continuity Equation 200 8.3 Separation of Temporal and Spatial Variables: Stationary States 203 8.4 The Helmholtz Equation and Dispersion Equation for a Free Particle 205 8.5 Separation of Spatial Variables and the Radial SchrEUROodinger Equation 207 8.6 Superposition of Degenerate States 209 8.7 Phase Velocity and Group Velocity 212 8.8 de Broglie's Waves Revised 218 8.9 The SchrEUROodinger Equation in an Arbitrary Basis 222 Problems 226 References 226 9 Applications to Simple Systems: One Dimension 227 9.1 A Quasi-Free Particle 227 9.2 Potential Threshold 232 9.3 Tunneling through a Potential Barrier 236 9.4 Cold Emission 241 9.5 Potential Well 244 9.6 Quantum Oscillator 249 9.7 Oscillator in the E-Representation 254 9.8 The Origin of Energy Bands 257 9.9 Periodic Structures 260 Problems 269 References 271 10 Three-Dimensional Systems 273 10.1 A Particle in a 3D Box 273 10.2 A Free Particle in 3D (Spherical Coordinates) 274 10.2.1 Discussion 277 10.3 Some Properties of Solutions in Spherically Symmetric Potential 277 10.4 Spherical Potential Well 278 10.5 States in the Coulomb Field and a Hydrogen Atom 281 10.6 Atomic Currents 287 10.7 Periodic Table 290 Problems 293 References 294 11 Evolution of Quantum States 295 11.1 The Time Evolution Operator 295 11.2 Evolution of Operators 299 11.3 Spreading of a Gaussian Packet 301 11.4 The B-Factor and Evolution of an Arbitrary State 303 11.5 The Fraudulent Life of an "Illegal" Spike 306 11.6 Jinnee Out of the Box 311 11.7 Inadequacy of Nonrelativistic Approximation in Description of Evolving Discontinuous States 315 11.7.1 Discussion 316 11.8 Quasi-Stationary States 317 11.8.1 Discussion 323 11.9 3D Barrier and Quasi-Stationary States 324 11.10 The Theory of Particle Decay 327 11.11 Particle-Antiparticle Oscillations 331 11.11.1 Discussion 337 11.12 AWatched Pot Never Boils (Quantum Zeno Effect) 339 11.13 AWatched Pot Boils Faster 344 Problems 350 References 352 12 Quantum Ensembles 355 12.1 Pure Ensembles 355 12.2 Mixtures 356 12.3 The Density Operator 358 12.4 Time Evolution of the Density Operator 366 12.5 Composite Systems 368 Problems 376 References 376 13 Indeterminacy Revisited 377 13.1 Indeterminacy Under Scrutiny 377 13.2 The Heisenberg Inequality Revised 380 13.3 The Indeterminacy of Angular Momentum 382 13.4 The Robertson-SchrEUROodinger Relation Revised 384 13.5 The N-f Indeterminacy 388 13.6 Dispersed Indeterminacy 390 Problems 394 References 395 14 Quantum Mechanics and Classical Mechanics 397 14.1 Relationship between Quantum and Classical Mechanics 397 14.2 QM and Optics 400 14.3 The Quasi-Classical State Function 401 14.4 The WKB Approximation 404 14.5 The Bohr-Sommerfeld Quantization Rules 406 Problems 409 References 410 15 Two-State Systems 411 15.1 Double Potential Well 411 15.2 The Ammonium Molecule 415 15.3 Qubits Introduced 419 Problem 422 References 422 16 Charge in Magnetic Field 423 16.1 A Charged Particle in EM Field 423 16.2 The Continuity Equation in EM Field 425 16.3 Origin of the A-Momentum 427 16.4 Charge in Magnetic Field 429 16.5 Spin Precession 432 16.6 The Aharonov-Bohm Effect 437 16.6.1 Discussion 441 16.7 The Zeeman Effect 442 Problems 444 References 445 17 Perturbations 447 17.1 Stationary Perturbation Theory 447 17.1.1 Discussion 450 17.2 Asymptotic Perturbations 455 17.3 Perturbations and Degeneracy 457 17.4 Symmetry, Degeneracy, and Perturbations 460 17.5 The Stark Effect 462 17.6 Time-Dependent Perturbations 465 Problems 471 References 471 18 Light-Matter Interactions 473 18.1 Optical Transitions 473 18.2 Dipole Radiation 474 18.3 Selection Rules 477 18.3.1 Oscillator 478 18.3.2 Hydrogen-Like Atom 478 Problems 480 Reference 480 19 Scattering 481 19.1 QM Description of Scattering 481 19.2 Stationary Scattering 487 19.3 Scattering Matrix and the Optical Theorem 490 19.4 Diffraction Scattering 494 19.5 Resonant Scattering 498 19.6 The Born Approximation 501 Problems 504 References 505 20 Submissive Quantum Mechanics 507 20.1 The Inverse Problem 507 20.2 Playing with Quantum States 509 20.3 Playing with Evolution: Discussion 514 Problems 522 References 522 21 Quantum Statistics 525 21.1 Bosons and Fermions: The Exclusion Principle 525 21.1.1 Discussion 531 21.2 Planck and Einstein Again 540 21.3 BBR Again 542 21.4 Lasers and Masers 543 Problems 545 References 546 22 Second Quantization 547 22.1 Quantum Oscillator Revisited 547 22.2 Creation and Annihilation Operators: Bosons 548 22.3 Creation and Annihilation Operators: Fermions 552 Problems 555 References 555 23 Quantum Mechanics and Measurements 557 23.1 Collapse or Explosion? 557 23.2 "SchrEUROodinger's Cat" and Classical Limits of QM 563 23.3 Von Neumann's Measurement Scheme 571 23.3.1 Discussion 575 23.4 Quantum Information and Measurements 578 23.5 Interaction-Free Measurements: Quantum Seeing in the Dark 586 23.6 QM and the Time Arrow 593 Problems 595 References 596 24 Quantum Nonlocality 599 24.1 Entangled Superpositions I 599 24.2 Entangled Superpositions II 601 24.2.1 Discussion 604 24.3 Quantum Teleportation 604 24.4 The "No-Cloning" Theorem 607 24.5 Hidden Variables and Bell's Theorem 613 24.6 Bell-State Measurements 619 24.7 QM and the Failure of FTL Proposals 627 24.8 Do Lasers Violate the No-Cloning Theorem? 628 24.9 Imperfect Cloning 636 24.10 The FLASH Proposal and Quantum Compounds 643 Problems 649 References 649 25 Quantum Measurements and POVMs 651 25.1 Projection Operator and Its Properties 651 25.2 Projective Measurements 655 25.3 POVMs 658 25.4 POVM as a Generalized Measurement 664 25.5 POVM Examples 666 25.6 Discrimination of Two Pure States 670 25.7 Neumark's Theorem 681 25.8 How to Implement a Given POVM 686 25.9 Comparison of States and Mixtures 695 25.10 Generalized Measurements 697 Problems 700 References 701 26 Quantum Information 703 26.1 Deterministic Information and Shannon Entropy 703 26.2 von Neumann Entropy 709 26.3 Conditional Probability and Bayes's Theorem 711 26.4 KL Divergence 716 26.5 Mutual Information 717 26.6 Renyi Entropy 719 26.7 Joint and Conditional Renyi Entropy 721 26.8 Universal Hashing 726 26.9 The Holevo Bound 731 26.10 Entropy of Entanglement 733 Problems 734 References 735 27 Quantum Gates 737 27.1 Truth Tables 737 27.2 Quantum Logic Gates 741 27.3 Shor's Algorithm 746 Problems 751 References 752 28 Quantum Key Distribution 753 28.1 Quantum Key Distribution (QKD) with EPR 753 28.2 BB84 Protocol 758 28.3 QKD as Communication Over a Channel 766 28.4 Postprocessing of the Key 769 28.5 B92 Protocol 776 28.6 Experimental Implementation of QKD Schemes 779 28.7 Advanced Eavesdropping Strategies 788 Problems 793 References 793 Appendix A: Classical Oscillator 795 Reference 799 Appendix B: Delta Function 801 Reference 807 Appendix C: Representation of Observables by Operators 809 Appendix D: Elements of Matrix Algebra 813 Appendix E: Eigenfunctions and Eigenvalues of the Orbital Angular Momentum Operator 817 Appendix F: Hermite Polynomials 821 Appendix G: Solutions of the Radial SchrEUROodinger Equation in Free Space 825 Appendix H: Bound States in the Coulomb Field 827 Reference 829 Index 831

About the Author :
Moses Fayngold is Special Lecturer at the Physics Department of the New Jersey Institute of Technology, where he is also involved in the research activities of the Multi-Disciplinary Optical Science and Engineering Program. He has been lecturing for some 20 years on special relativity, both to non-specialized audiences in public planetariums as well as to undergraduate and graduate students. Vadim Fayngold holds two bachelor degrees - one in Physics and one in Computer Science. While being a research assistant at the Department of Computer Engineering (Polytechnic University, New York), he worked on computer simulation of complex processes in fluid dynamics. He won a scholarship in the REU (Research Experience for Undergraduates) program sponsored by the NSF National Science Foundation,) and had worked in the area of Plasma Physics at the University of California (Irvine). He acquired an extensive experience in computer simulation while participating in the MECO (Muon-Electron Conversion) project. Vadim Fayngold came to the idea of writing this book while working on computer animations of various relativistic and quantum-mechanical phenomena.