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Moses Fayngold, Vadim Fayngold

Quantum Mechanics and Quantum Information

PHYSICS TEXTBOOK

Quantum

Mechanics

and Quantum

Information

ISBN 978-3-527-40647-0

Fayngold Fayngold

www.wiley-vch.de

Moses Fayngold graduated from the State University of Samarkand and got his PhD at the Nuclear Research Institute of Academy of Science in Uzbekistan (former USSR). He has combined teaching and research in colleges of USSR and USA, most recently as a Senior University Lectur-er at the Physics Department of the New Jersey Institute of Technology. He has lectured on Quantum Mech-anics and Special Relativity to both undergraduate and graduate students. His research interests and areas of activity include Special and General Relativity, Quantum Mechanics, Optics and optical imaging, particle scattering and propagation in peri-odic structures.

Vadim Fayngold holds two degrees M.S. in Physics and B.S. in Computer Science. While working as a research assistant at the Department of Com-puter Engineering (Polytechnic Uni-versity, New York), he focused on computer simulation of complex processes in fluid dynamics. The combined expertise he developed there has spurred his interest in the Quantum Information theory. Vadim came to the idea of writing this book while working on computer animations of various relativistic and quantum-mechanical phenomena.

Alongside a thorough definition of basic concepts and their interrelations, backed by numerous examples, this textbook features a rare discussion of quantum mechanics and informa-tion theory combined in one text. It deals with important topics hardly found in regular textbooks, including the Robertson-Schrdinger relation, incompatibility between angle and angular momentum, dispersed indeterminacy, interaction-free mea-surements, submissive quantum mechanics, and many others. With its in-depth discussion of key concepts complete with problems and exercises, this book is poised to become the standard textbook for advanced undergraduate and begin-ning graduate quantum mechanics courses and an essential reference for physics students and physics professionals.

From the contents:

Embryonic Quantum Mechanics: Basic Features Playing with the Amplitudes Representations and the Hilbert Space Angular Momentum Evolution of Quantum States Indeterminacy Revisited Submissive Quantum Mechanics Quantum Statistics Second Quantization Quantum Mechanics and Measurements Quantum Non-Locality Quantum Measurements and POVMs Quantum Information Quantum Gates Quantum Key Distribution

Moses Fayngold and Vadim

Fayngold

Quantum Mechanics andQuantum Information

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Moses Fayngold and Vadim Fayngold

Quantum Mechanics and QuantumInformation

A Guide through the Quantum World

The Authors

Moses FayngoldNJITDept. of Physi csNewark, NJ 07102-1982fayngold@mai laps.org

Vadim Fayngoldvadim.resear ch@gmail.com

All books published by Wiley-VCH are carefullyproduced . Neverthel ess, authors, editors, andpublisher do not warrant the information containedin these books, inclu ding this book, to be free oferrors. Rea ders are advised to keep in mind thatstatemen ts, data, illust rations, procedural details orother items may inadver tently be inaccurate .

Library of Congress Car d No.: applied for

British Library Catalogu ing-in-Publica tion Dat aA catalogu e record for this book is av ailable from theBritish Library.

Bibliog raphic inform ation publ ished by the DeutscheNation albibliothekThe D eutsche Na tionalbibliothe k lists thispublicatio n in the Deutsche Nationalbibli ografi e;detailed bibliographic data are available on theInternet at < http:// dnb.d-nb.d e> .

# 2013 Wiley-VCH Verlag GmbH & Co. KGaA,Boschstr. 12, 69469 Weinheim, Germany

All rights reserved (including those of translation intoother languages). No part of this book may bereproduced in any form by photoprinting,microfilm, or any other means nor transmitted ortranslated into a machine language without writtenpermission from the publishers. Registered names,trademarks, etc. used in this book, even when notspecifically marked as such, are not to be consideredunprotected by law.

Print ISBN: 978-3-527-40647-0

Cover Design Adam Design, Weinheim, Germany

Typesetting Thomson Digital, Noida, India

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Contents

Preface XIIIAbbreviations and Notations XIX

1 The Failure of Classical Physics 11.1 Blackbody Radiation 11.2 Heat Capacity 41.3 The Photoelectric Effect 91.4 Atoms and Their Spectra 121.5 The Double-Slit Experiment 14

Problem 19References 19

2 The First Steps into the Unknown 212.1 The BBR and Plancks Formula 212.2 Einsteins Light Quanta and BBR 242.2.1 Discussion 272.3 PEE Revisited 302.4 The Third Breakthrough: de Broglie Waves 312.4.1 Exercise 33

Problems 35References 35

3 Embryonic Quantum Mechanics: Basic Features 373.1 A Glimpse of the New Realm 373.2 Quantum-Mechanical Superposition of States 393.3 What Is Waving There (the Meaning of the C-Function)? 423.4 Observables and Their Operators 473.5 Quantum-Mechanical Indeterminacy 493.6 Indeterminacy and the World 533.7 Quantum Entanglement and Nonlocality 583.8 Quantum-Mechanical Phase Space 623.9 Determinism and Causality in Quantum World 63

jV

3.9.1 Discussion 63Problems 66References 66

4 Playing with the Amplitudes 694.1 Composition of Amplitudes 694.2 Double Slit Revised I 744.3 Double Slit Revised II 774.4 Neutron Scattering in Crystals 784.5 Bosonic and Fermionic States 814.6 Path Integrals 89

Problems 93References 93

5 Basic Features and Mathematical Structure of QM 955.1 Observables: the Domain of Classical and Quantum

Mechanics 955.2 Quantum-Mechanical Operators 975.3 Algebra of Operators 1005.4 Eigenvalues and Eigenstates 1025.5 Orthogonality of Eigenstates 1075.6 The RobertsonSchrodinger Relation 1105.7 The Wave Function and Measurements (Discussion) 112

Problems 116References 117

6 Representations and the Hilbert Space 1196.1 Various Faces of a State Function 1196.2 Unitary Transformations 1216.3 Operators in the Matrix Form 1256.4 The Hilbert Space 1296.5 Operations in the Hilbert Space 1356.6 Nonorthogonal States 142

Problems 147References 148

7 Angular Momentum 1497.1 Orbital and Spin Angular Momenta 1497.2 The Eigenstates and Eigenvalues of L 1517.3 Operator L and Its Commutation Properties 1547.4 Spin as an Intrinsic Angular Momentum 1647.5 Angular Momentum of a Compound System 1837.6 Spherical Harmonics 188

Problems 196References 197

VIj Contents

8 The Schrodinger Equation 1998.1 The Schrodinger Equation 1998.2 State Function and the Continuity Equation 2008.3 Separation of Temporal and Spatial Variables: Stationary

States 2038.4 The Helmholtz Equation and Dispersion Equation for a Free

Particle 2058.5 Separation of Spatial Variables and the Radial Schrodinger

Equation 2078.6 Superposition of Degenerate States 2098.7 Phase Velocity and Group Velocity 2128.8 de Broglies Waves Revised 2188.9 The Schrodinger Equation in an Arbitrary Basis 222

Problems 226References 226

9 Applications to Simple Systems: One Dimension 2279.1 A Quasi-Free Particle 2279.2 Potential Threshold 2329.3 Tunneling through a Potential Barrier 2369.4 Cold Emission 2419.5 Potential Well 2449.6 Quantum Oscillator 2499.7 Oscillator in the E-Representation 2549.8 The Origin of Energy Bands 2579.9 Periodic Structures 260

Problems 269References 271

10 Three-Dimensional Systems 27310.1 A Particle in a 3D Box 27310.2 A Free Particle in 3D (Spherical Coordinates) 27410.2.1 Discussion 27710.3 Some Properties of Solutions in Spherically Symmetric

Potential 27710.4 Spherical Potential Well 27810.5 States in the Coulomb Field and a Hydrogen Atom 28110.6 Atomic Currents 28710.7 Periodic Table 290

Problems 293References 294

11 Evolution of Quantum States 29511.1 The Time Evolution Operator 29511.2 Evolution of Operators 299

Contents jVII

11.3 Spreading of a Gaussian Packet 30111.4 The B-Factor and Evolution of an Arbitrary State 30311.5 The Fraudulent Life of an Illegal Spike 30611.6 Jinnee Out of the Box 31111.7 Inadequacy of Nonrelativistic Approximation in Description

of Evolving Discontinuous States 31511.7.1 Discussion 31611.8 Quasi-Stationary S