Quantum Mechanics for Pedestrians 2: Applications and Extensions (Undergraduate Lecture Notes in Physics)

Description

This book, the second in a two-volume set, provides an introduction to the basics of (mainly) non-relativistic quantum mechanics. While the first volume addresses the basic principles, this second volume discusses applications and extensions to more complex problems. In addition to topics dealt with in traditional quantum mechanics texts, such as symmetries or many-body problems, it also treats issues of current interest such as entanglement, Bell’s inequality, decoherence and various aspects of quantum information in detail. Furthermore, questions concerning the basis of quantum mechanics and epistemological issues which are relevant e.g. to the realism debate are discussed explicitly. A chapter on the interpretations of quantum mechanics rounds out the book. Readers are introduced to the requisite mathematical tools step by step. In the appendix, the most relevant mathematics is compiled in compact form, and more advanced topics such as the Lenz vector, Hardy’s experiment and Shor’s algorithm are treated in more detail. As an essential aid to learning and teaching, 130 exercises are included, most of them with solutions. This revised second edition is expanded by an introduction into some ideas and problems of relativistic quantum mechanics. In this second volume, an overview of quantum field theory is given and basic conceptions of quantum electrodynamics are treated in some detail. Originally written as a course for students of science education, the book addresses all those science students and others who are looking for a reasonably simple, fresh and modern introduction to the field. Preface to the Second Edition, Volume 2 Preface to the First Edition, Volume 2 Contents Contents of Volume 1 Introduction Overview of Volume 2 Part II Applications and Extensions 15 One-Dimensional Piecewise-Constant Potentials 15.1 General Remarks 15.2 Potential Steps 15.2.1 Potential Step, EV0 15.3 Finite Potential Well 15.3.1 Potential Well, E0 15.4 Potential Barrier, Tunnel Effect 15.5 From the Finite to the Infinite Potential Well 15.6 Wave Packets 15.7 Exercises 16 Angular Momentum 16.1 Orbital Angular Momentum Operator 16.2 Generalized Angular Momentum, Spectrum 16.3 Matrix Representation of Angular Momentum Operators 16.4 Orbital Angular Momentum: Spatial Representation of the Eigenfunctions 16.5 Addition of Angular Momenta 16.6 Exercises 17 The Hydrogen Atom 17.1 Central Potential 17.2 The Hydrogen Atom 17.3 Complete System of Commuting Observables 17.4 On Modelling 17.5 Exercises 18 The Harmonic Oscillator 18.1 Algebraic Approach 18.1.1 Creation and Annihilation Operators 18.1.2 Properties of the Occupation-Number Operator 18.1.3 Derivation of the Spectrum 18.1.4 Spectrum of the Harmonic Oscillator 18.2 Analytic Approach (Position Representation) 18.3 Exercises 19 Perturbation Theory 19.1 Stationary Perturbation Theory, Nondegenerate 19.1.1 Calculation of the First-Order Energy Correction 19.1.2 Calculation of the First-Order State Correction 19.2 Stationary Perturbation Theory, Degenerate 19.3 Hydrogen: Fine Structure 19.3.1 Relativistic Corrections to the Hamiltonian 19.3.2 Results of Perturbation Theory 19.3.3 Comparison with the Results of the Dirac Equation 19.4 Hydrogen: Lamb Shift and Hyperfine Structure 19.5 Exercises 20 Entanglement, EPR, Bell 20.1 Product Space 20.2 Entangled States 20.2.1 Definition 20.2.2 Single Measurements on Entangled States 20.2.3 Schrödinger's Cat 20.2.4 A Misunderstanding 20.3 The EPR Paradox 20.4 Bell's Inequality 20.4.1 Derivation of Bell's Inequality 20.4.2 EPR Photon Pairs 20.4.3 EPR and Bell 20.5 Conclusions 20.6 Exercises 21 Symmetries and Conservation Laws 21.1 Continuous Symmetry Transformations 21.1.1 General: Symmetries and Conservation Laws 21.1.2 Time Translation 21.1.3 Spatial Translation 21.1.4 Spatial Rotation 21.1.5 Special Galilean Transformation 21.2 Discrete Symmetry Transformations 21.2.1 Parity 21.2.2 Time Reversal 21.3 Exercises 22 The Density Operator 22.1 Pure States 22.2 Mixed States 22.3 Reduced Density Operator 22.3.1 Example 22.3.2 Comparison 22.3.3 General Formulation 22.4 Exercises 23 Identical Particles 23.1 Distinguishable Particles 23.2 Identical Particles 23.2.1 A Simple Example 23.2.2 The General Case 23.3 The Pauli Exclusion Principle 23.4 The Helium Atom 23.4.1 Spectrum Without V1,2 23.4.2 Spectrum with V1,2 (Perturbation Theory) 23.5 The Ritz Method 23.6 How Far does the Pauli Principle Reach? 23.6.1 Distinguishable Quantum Objects 23.6.2 Identical Quantum Objects 23.7 Exercises 24 Decoherence 24.1 A Simple Example 24.2 Decoherence 24.2.1 The Effect of the Environment I 24.2.2 Simplified Description 24.2.3 The Effect of the Environment II 24.2.4 Interim Review 24.2.5 Formal Treatment 24.3 Time Scales, Universality 24.4 Decoherence-Free Subspaces, Basis 24.5 Historical Side Note 24.6 Conclusions 24.7 Exercises 25 Scattering 25.1 Basic Idea; Scattering Cross Section 25.1.1 Classical Mechanics 25.1.2 Quantum Mechanics 25.2 The Partial-Wave Method 25.3 Integral Equations, Born Approximation 25.4 Exercises 26 Quantum Information 26.1 No-Cloning Theorem (Quantum Copier) 26.2 Quantum Cryptography 26.3 Quantum Teleportation 26.4 The Quantum Computer 26.4.1 Qubits, Registers (Basic Concepts) 26.4.2 Quantum Gates and Quantum Computers 26.4.3 The Basic Idea of the Quantum Computer 26.4.4 The Deutsch Algorithm 26.4.5 Grover's Search Algorithm 26.4.6 Shor's Algorithm 26.4.7 On The Construction of Real Quantum Computers 26.5 Exercises 27 Is Quantum Mechanics Complete? 27.1 The Kochen–Specker Theorem 27.1.1 Value Function 27.1.2 From the Value Function to Coloring 27.1.3 Coloring 27.1.4 Interim Review: The Kochen–Specker Theorem 27.2 GHZ States 27.3 Discussion and Outlook 27.4 Exercises 28 Interpretations of Quantum Mechanics 28.1 Preliminary Remarks 28.1.1 Problematic Issues 28.1.2 Difficulties in the Representation of Interpretations 28.2 Some Interpretations in Short Form 28.2.1 Copenhagen Interpretation(s) 28.2.2 Ensemble Interpretation 28.2.3 Bohm's Interpretation 28.2.4 Many-Worlds Interpretation 28.2.5 Consistent-Histories Interpretation 28.2.6 Collapse Theories 28.2.7 Other Interpretations 28.3 Conclusion A Abbreviations and Notations B Special Functions B.1 Spherical Harmonics B.2 Spherical Bessel Functions B.3 Eigenfunctions of the Hydrogen Atom B.4 Hermite Polynomials B.5 Waves C Tensor Product C.1 Direct Product C.2 Direct Sum of Vector Spaces C.3 Properties of the Tensor Product C.4 Examples C.4.1 General Examples C.4.2 Example with Reference to Chap.[Entang]20 D Wave Packets D.1 General Remarks D.1.1 One-Dimensional Wave Packet D.1.2 Example: Bell Curve D.1.3 Many-Dimensional Wave Packet D.2 Potential Step and Wave Packet D.3 Exercises E Laboratory System, Center-of-Mass System E.1 The Equivalent One-Body Problem E.2 Transformation Laboratory System Center-of-Mass System E.2.1 First Transformation, Then Transition to Quantum Mechanics E.2.2 First Transition to Quantum Mechanics, Then Transformation F Analytic Treatment of the Hydrogen Atom F.1 Nonrelativistic Case: Schrödinger equation F.2 Relativistic Case: Dirac equation F.2.1 From 4-Spinor to 2-Spinors F.2.2 Angular Part of the 2-Spinors F.2.3 From 2-Spinors to 4-Spinor F.2.4 Coupled Radial Equations, Solution F.3 Exercises and Solutions G The Lenz Vector G.1 In Classical Mechanics G.2 In Quantum Mechanics G.3 General Theorems on Vector Operators G.3.1 General Commutator Relations G.3.2 Vector Operators G.4 Exercises H Perturbative Calculation of the Hydrogen Atom H.1 Calculation of the Matrix Elements H.1.1 Matrix Elements of Wmp H.1.2 Matrix Elements of Wls H.1.3 Matrix Elements of WD H.2 Fine Structure Corrections I The Production of Entangled Photons I.1 Atomic Sources I.2 Parametric Fluorescence I.3 Semiconductor Sources I.4 Concluding Remarks J The Hardy Experiment J.1 The Experiment J.2 Calculation of the Probabilities K Set-Theoretical Derivation of the Bell Inequality L The Special Galilei Transformation L.1 Special Galilei Transformation L.1.1 Abstract Notation L.1.2 Position Representation L.1.3 Several Quantum Objects L.2 The Special Galilei Transformation and Kinetic Energy L.2.1 One-Dimensional Case L.2.2 Three-Dimensional Case L.3 Exercises M Kramers' Theorem N Coulomb Energy and Exchange Energy in the Helium Atom O The Scattering of Identical Particles P The Hadamard Transformation P.1 The MZI and the Hadamard Transformation P.2 The Beam Splitter and the Hadamard Transformation P.3 The Hadamard Transformation and Quantum Information Q From the Interferometer to the Computer R The Grover Algorithm, Algebraically S Shor Algorithm S.1 Classical Part S.2 Quantum-Mechanical Part S.3 Supplement on Modular Arithmetic S.4 Exercises T The Gleason Theorem U What is Real? Some Quotations V Remarks on Some Interpretations of Quantum Mechanics V.1 Bohmian Interpretation V.1.1 Sketch of the Formalism V.1.2 Example: Free Motion V.1.3 Conclusions V.2 The Many-Worlds Interpretation V.3 Consistent Histories V.3.1 Definitions V.3.2 A Simple Example V.3.3 Conclusions V.4 Ghirardi-Rimini-Weber W Elements of Quantum Field Theory W.1 Foreword W.2 Quantizing a Field - A Toy Example W.2.1 The Classical Case W.2.2 Quantization W.2.3 Creation and Annihilation Operators, Hamiltonian W.2.4 Generalization W.2.5 Exercises and Solutions W.3 Quantization of Free Fields, Introduction W.4 Quantization of Free Fields, Klein–Gordon W.4.1 Lagrangian, Conjugated Momentum, Poisson Brackets, Hamiltonian W.4.2 Canonical Quantization W.4.3 Exercises and Solutions W.5 Quantization of Free Fields, Dirac W.5.1 No Classical Spinor Field W.5.2 Lagrangian, Conjugated Momentum, Hamiltonian W.5.3 The Free Solutions W.5.4 Energy W.5.5 Interpretation of br( p) and dr( p) , Commutation Relations, Pauli Principle, Number Operator W.5.6 Again Infinities W.5.7 Anticommutators for Field Operators W.5.8 Conclusion W.5.9 Exercises and Solutions W.6 Quantization of Free Fields, Photons W.6.1 Determination of mathcalH W.6.2 Determination of H W.6.3 Exercises and Solutions W.7 Operator Ordering W.7.1 Normal Order W.7.2 Time Order W.7.3 Time Ordering and Normal Ordering W.7.4 Exercises and Solutions W.8 Interacting Fields, Quantum Electrodynamics W.8.1 Lagrangian W.8.2 Conjugated Momentum, Hamiltonian W.8.3 Interaction Picture, Time Evolution Operator W.8.4 S-Operator W.8.5 Approximating S W.9 S-Matrix, First Order W.9.1 Preliminary Note: Virtual Particles W.9.2 Field Operators W.9.3 Eight Elementary Processes of mathcalHI W.9.4 Two Worked Out Examples W.9.5 External Fields W.9.6 Feynman Diagrams W.9.7 First Feynman Rules W.9.8 Exercises and Solutions W.10 Contraction, Propagator, Wick's Theorem W.10.1 Contraction W.10.2 Propagators W.10.3 Wick's Theorem W.10.4 Exercises and Solutions W.11 S-Matrix, 2. Order, General W.11.1 Applying Wick's Theorem W.11.2 Physical Interpretation W.12 S-Matrix, 2. Order, 4 Lepton Scattering W.12.1 Bhabha Scattering, e+e-rightarrowe+e- W.12.2 Møller Scattering, e-e-rightarrowe-e- W.12.3 Scattering Cross Section and Feynman Amplitude W.12.4 Exercises and Solutions W.13 High Precision and Infinities W.13.1 Feynman Rules, Diagrams, Amplitudes for QED W.13.2 Extraordinary Precision W.13.3 Problematic Loops, Infinities, Renormalization W.13.4 Conclusion X Exercises and Solutions X.1 Exercises, Chap.15 X.2 Exercises, Chap.16 X.3 Exercises, Chap.17 X.4 Exercises, Chap.18 X.5 Exercises, Chap.19 X.6 Exercises, Chap.20 X.7 Exercises, Chap.[chap21]21 X.8 Exercises, Chap.22 X.9 Exercises, Chap.23 X.10 Exercises, Chap.24 X.11 Exercises, Chap.25 X.12 Exercises, Chap.26 X.13 Exercises, Chap.27 Further Reading Index of Volume 1 Lake Index Index of Volume 2 Index