QUANTUM MECHANICS: A MODERN DEVELOPMENT (2ND EDITION)

Main subject category: • Quantum MechanicsAlthough there are many textbooks that deal with the formal apparatus of quantum mechanics (QM) and its application to standard problems, none take into account the developments in the foundations of the subject which have taken place in the last few decades. There are specialized treatises on various aspects of the foundations of QM, but none that integrate those topics with the standard material. This book aims to remove that unfortunate dichotomy, which has divorced the practical aspects of the subject from the interpretation and broader implications of the theory.In this edition a new chapter on quantum information is added. As the topic is still in a state of rapid development, a comprehensive treatment is not feasible. The emphasis is on the fundamental principles and some key applications, including quantum cryptography, teleportation of states, and quantum computing. The impact of quantum information theory on the foundations of quantum mechanics is discussed. In addition, there are minor revisions to several chapters.The book is intended primarily as a graduate level textbook, but it will also be of interest to physicists and philosophers who study the foundations of QM. Parts of it can be used by senior undergraduates too. Contents Preface Introduction: The Phenomena of Quantum Mechanics Chapter 1 Mathematical Prerequisites 1.1 Linear Vector Space Dirac’s bra and ket notation 1.2 Linear Operators 1.3 Self-Adjoint Operators Properties of complete orthonormal sets The spectral theorem Commuting sets of operators 1.4 Hilbert Space and Rigged Hilbert Space 1.5 Probability Theory Interpretations of probability Probability and frequency Estimating a probability Further reading for Chapter 1 Vectors and operators Probability Problems Chapter 2 The Formulation of Quantum Mechanics 2.1 Basic Theoretical Concepts Mechanical aspect Statistical aspect 2.2 Conditions on Operators 2.3 General States and Pure States Pure states 2.4 Probability Distributions Further reading for Chapter 2 Problems Chapter 3 Kinematics and Dynamics 3.1 Transformations of States and Observables 3.2 The Symmetries of Space–Time 3.3 Generators of the Galilei Group Evaluation of commutators Multiples of identity 3.4 Identification of Operators with Dynamical Variables Case (i): A free particle with no internal degrees of freedom Case (ii): A free particle with spin Case (iii): A particle interacting with external fields Conventional notation adopted 3.5 Composite Systems 3.6 [[Quantizing a Classical System]] 3.7 Equations of Motion 3.8 Symmetries and Conservation Laws Further reading for Chapter 3 Problems Chapter 4 Coordinate Representation and Applications 4.1 Coordinate Representation 4.2 The Wave Equation and Its Interpretation 4.3 Galilei Transformation of Schrodinger’s Equation 4.4 Probability Flux 4.5 Conditions on Wave Functions 4.6 Energy Eigenfunctions for Free Particles 4.7 Tunneling 4.8 Path Integrals Classical limit of the path integral Imaginary time and statistical mechanics Discussion of the path integral method Further reading for Chapter 4 Problems Chapter 5 Momentum Representation and Applications 5.1 Momentum Representation 5.2 Momentum Distribution in an Atom 5.3 Bloch’s Theorem 5.4 Diffraction Scattering: Theory Diffraction by a periodic array (a) Position probability density (b) Momentum probability distribution Double slit diffraction 5.5 Diffraction Scattering: Experiment 5.6 Motion in a Uniform Force Field Further reading for Chapter 5 Problems Chapter 6 The Harmonic Oscillator 6.1 Algebraic Solution 6.2 Solution in Coordinate Representation 6.3 Solution in H Representation Problems Chapter 7 Angular Momemtum 7.1 Eigenvalues and Matrix Elements 7.2 Explicit Form of the Angular Momentum Operators Case (i): A one-component state function Case (ii): A multicomponent state function 7.3 Orbital Angular Momentum 7.4 Spin 7.5 Finite Rotations Active and passive rotations Rotation of angular momentum eigenvectors Relation to spherical harmonics 7.6 Rotation Through 2π Superselection versus ordinary symmetry 7.7 Addition of Angular Momenta Clebsch Gordan coefficients Recursion relations 3 j symbols 7.8 Irreducible Tensor Operators Matrix elements of tensor operators Products of tensors 7.9 Rotational Motion of a Rigid Body Further reading for Chapter 7 Problems Chapter 8 State Preparation and Determination 8.1 State Preparation No-cloning theorem 8.2 State Determination Spin state s= 12 Spin state s=1 Orbital state of a spinless particle 8.3 States of Composite Systems Classification of states 8.4 Indeterminacy Relations Operational significance Further reading for Chapter 8 Problems Chapter 9 Measurement and the Interpretation of States 9.1 An Example of Spin Measurement 9.2 A General Theorem of Measurement Theory 9.3 The Interpretation of a State Vector The measurement theorem for general states 9.4 Which Wave Function? 9.5 Spin Recombination Experiment 9.6 Joint and Conditional Probabilities Filtering-type measurements Application to spin measurements Further reading for Chapter 9 Problems Chapter 10 Formation of Bound States 10.1 Spherical Potential Well Square well potential 10.2 The Hydrogen Atom Solution in spherical coordinates Solution in parabolic coordinates 10.3 Estimates from Indeterminacy Relations 10.4 Some Unusual Bound States 10.5 Stationary State Perturbation Theory Brillouin-Wigner perturbation theory 10.6 Variational Method Upper and lower bounds on eigenvalues Problems Chapter 11 Charged Particle in a Magnetic Field 11.1 Classical Theory 11.2 Quantum Theory Heisenberg equation of motion Coordinate representation Gauge transformations Probability current density 11.3 Motion in a Uniform Static Magnetic Field Energy levels Solution in rectangular coordinates Orbit center coordinates Degeneracy of energy levels Orbit radius and angular momentum 11.4 The Aharonov–Bohm Effect Bound state Aharonov-Bohm effect 11.5 The Zeeman Effect Further reading for Chapter 11 Problems Chapter 12 Time-Dependent Phenomena 12.1 Spin Dynamics Spin precession Spin resonance 12.2 Exponential and Nonexponential Decay The exponential decay law The decay probability in quantum mechanics The “watched pot” paradox 12.3 Energy–Time Indeterminacy Relations 12.4 Quantum Beats 12.5 Time-Dependent Perturbation Theory Harmonic perturbation Harmonic perturbation of long duration 12.6 Atomic Radiation The gauge problem The electric dipole approximation Induced emission and absorption Spontaneous emission 12.7 Adiabatic Approximation The Berry phase Further reading for Chapter 12 Problems Chapter 13 Discrete Symmetries 13.1 Space Inversion 13.2 Parity Nonconservation 13.3 Time Reversal Properties of antilinear operators Time reversal of the Schrodinger equation Time reversal and spin Time reversal squared Further reading for Chapter 13 Problems Chapter 14 The Classical Limit 14.1 Ehrenfest’s Theorem and Beyond Ehrenfest’s theorem Corrections to Ehrenfest’s theorem 14.2 The Hamilton–Jacobi Equation and the Quantum Potential 14.3 Quantal Trajectories 14.4 The Large Quantum Number Limit Further reading for Chapter 14 Problems Chapter 15 Quantum Mechanics in Phase Space 15.1 Why Phase Space Distributions? 15.2 The Wigner Representation Time dependence of the Wigner function 15.3 The Husimi Distribution Indeterminacy relation for the Husimi distribution Further reading for Chapter 15 Problems Chapter 16 Scattering 16.1 Cross Section Laboratory and center-of-mass frames The quantum state function in scattering 16.2 Scattering by a Spherical Potential Phase shifts Calculation of phase shifts 16.3 General Scattering Theory Scattering cross sections Scattering amplitude theorem 16.4 Born Approximation and DWBA 16.5 Scattering Operators Outgoing waves and the limit E + iε Properties of the scattering states Unitarity of the S matrix Symmetries of the S matrix 16.6 Scattering Resonances Decay of a resonant state Virtual bound states 16.7 Diverse Topics General behavior of phase shifts Validity of the Born approximation Multiple scattering The inverse scattering problem Further reading for Chapter 16 Problems Chapter 17 Identical Particles 17.1 Permutation Symmetry 17.2 Indistinguishability of Particles 17.3 The Symmetrization Postulate 17.4 Creation and Annihilation Operators Fermions Bosons Representation of operators Wick’s theorem Further reading for Chapter 17 Problems Chapter 18 Many-Fermion Systems 18.1 Exchange The Fermi sea The exchange interaction 18.2 The Hartree–Fock Method 18.3 Dynamic Correlations Two-electron atoms Electrons in a metal 18.4 Fundamental Consequences for Theory 18.5 BCS Pairing Theory BCS ground state Bogoliubov transformation Energy minimization Elementary excitations Simple model Further reading for Chapter 18 Problems Chapter 19 Quantum Mechanics of the Electromagnetic Field 19.1 Normal Modes of the Field 19.2 Electric and Magnetic Field Operators Complex basis functions 19.3 Zero-Point Energy and the Casimir Force 19.4 States of the EM Field Photon number eigenstates Coherent states — Mathematical relations Coherent states — Physical properties 19.5 Spontaneous Emission Enhancement and inhibition of spontaneous radiation 19.6 Photon Detectors Detection of n photons Semiclassical theory 19.7 Correlation Functions First order correlations: Interference Second order correlations Quantum beats 19.8 Coherence First order coherence Coherence and monochromaticity Coherent states versus Pure states Classical theory Photon bunching and antibunching The single-photon state 19.9 Optical Homodyne Tomography — Determining the Quantum State of the Field Further reading for Chapter 19 Problems Chapter 20 Bell’s Theorem and Its Consequences 20.1 The Argument of Einstein, Podolsky, and Rosen 20.2 Spin Correlations 20.3 Bell’s Inequality 20.4 A Stronger Proof of Bell’s Theorem 20.5 Polarization Correlations Positronium decay J = 0→1→0 cascade Experimental tests 20.6 Bell’s Theorem Without Probabilities The KS theorem Bell’s theorem — new proof 20.7 Implications of Bell’s Theorem Is the contradiction due to some hypothesis other than locality? Are the experiments conclusive? Is quantum mechanics incompatible with relativity? Further reading for Chapter 20 Problems Chapter 21 Quantum Information 21.1 Quantum States as Carriers of Information Bits and Qubits Classical vs Quantum Information 21.2 Some Quantum Information Theorems 21.3 Quantum Transmission of Information Relevant Kinds of Probability Information Transmission – Examples 21.4 Cryptography Quantum Cryptography 21.5 Entanglement Definition of Entanglement Identifying Entanglement Physical Meaning of Entanglement 21.6 Teleportation of Quantum States 21.7 Quantum Information from Independent Pairs 21.8 Measurable “in Principle” Local Observables Nonlocal Observables of Two Objects 21.9 Quantum Computing Quantum Operations (Quantum Gates) Some Impossible Quantum Operations Advantages and Applications of a Quantum Computer 21.10 Quantum Information and Quantum Foundations Interpretations of Quantum States Summary Further reading for Chapter 21 Problems Appendix A Schur’s Lemma Appendix B Irreducibility of Q and P Appendix C Proof of Wick’s Theorem Appendix D Solutions to Selected Problems Bibliography Index "Although there are many textbooks that deal with the formal apparatus of quantum mechanics (QM) and its application to standard problems, none take into account the developments in the foundations of the subject which have taken place in the last few decades. There are specialized treatises on various aspects of the foundations of QM, but none that integrate those topics with the standard material. This book aims to remove that unfortunate dichotomy, which has divorced the practical aspects of the subject from the interpretation and broader implications of the theory. In this edition a new chapter on quantum information is added. As the topic is still in a state of rapid development, a comprehensive treatment is not feasible. The emphasis is on the fundamental principles and some key applications, including quantum cryptography, teleportation of states, and quantum computing. The impact of quantum information theory on the foundations of quantum mechanics is discussed. In addition, there are minor revisions to several chapters. The book is intended primarily as a graduate level textbook, but it will also be of interest to physicists and philosophers who study the foundations of QM. Parts of it can be used by senior undergraduates too."--Back cover