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The Pauli Exclusion Principle the Pauli Exclusion Principle Origin, Verifications, and Applications THE PAULI EXCLUSION PRINCIPLE THE PAULI EXCLUSION PRINCIPLE ORIGIN, VERIFICATIONS, AND APPLICATIONS Ilya G. Kaplan Materials Research Institute, National Autonomous University of Mexico, Mexico This edition first published 2017 © 2017 John Wiley & Sons, Ltd. Registered Office John Wiley & Sons, Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom For details of our global editorial offices, for customer services and for information about how to apply for permission to reuse the copyright material in this book please see our website at www.wiley.com. The right of the author to be identified as the author of this work has been asserted in accordance with the Copyright, Designs and Patents Act 1988. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by the UK Copyright, Designs and Patents Act 1988, without the prior permission of the publisher. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. Designations used by companies to distinguish their products are often claimed as trademarks. All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners. The publisher is not associated with any product or vendor mentioned in this book. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. It is sold on the understanding that the publisher is not engaged in rendering professional services and neither the publisher nor the author shall be liable for damages arising herefrom. If professional advice or other expert assistance is required, the services of a competent professional should be sought Library of Congress Cataloging-in-Publication Data ⌒ Names: Kaplan, I. G. (Il ia Grigor evich) Title: The Pauli exclusion principle : origin, verifications and applications / Ilya Kaplan. Description: Chichester, West Sussex : John Wiley & Sons, Inc., 2017. | Includes bibliographical references and index. Identifiers: LCCN 2016018231 (print) | LCCN 2016020644 (ebook) | ISBN 9781118795323 (cloth) | ISBN 9781118795293 (epdf) | ISBN 9781118795248 (epub) Subjects: LCSH: Pauli exclusion principle. | Quantum theory. Classification: LCC QC174.17.P3 K37 2017 (print) | LCC QC174.17.P3 (ebook) | DDC 530.12–dc23 LC record available at https://lccn.loc.gov/2016018231 A catalogue record for this book is available from the British Library. Cover image: David Wells/EyeEm/Gettyimages Set in 10/12pt Times by SPi Global, Pondicherry, India 10987654321 To my wife Larisa Contents Preface xi 1 Historical Survey 1 1.1 Discovery of the Pauli Exclusion Principle and Early Developments 1 1.2 Further Developments and Still Existing Problems 11 References 21 2 Construction of Functions with a Definite Permutation Symmetry 25 2.1 Identical Particles in Quantum Mechanics and Indistinguishability Principle 25 2.2 Construction of Permutation-Symmetric Functions Using the Young Operators 29 2.3 The Total Wave Functions as a Product of Spatial and Spin Wave Functions 36 2.3.1 Two-Particle System 36 2.3.2 General Case of N-Particle System 41 References 49 3 Can the Pauli Exclusion Principle Be Proved? 50 3.1 Critical Analysis of the Existing Proofs of the Pauli Exclusion Principle 50 3.2 Some Contradictions with the Concept of Particle Identity and their Independence in the Case of the Multidimensional Permutation Representations 56 References 62 viii Contents 4 Classification of the Pauli-Allowed States in Atoms and Molecules 64 4.1 Electrons in a Central Field 64 4.1.1 Equivalent Electrons: L–S Coupling 64 4.1.2 Additional Quantum Numbers: The Seniority Number 71 4.1.3 Equivalent Electrons: j–j Coupling 72 4.2 The Connection between Molecular Terms and Nuclear Spin 74 4.2.1 Classification of Molecular Terms and the Total Nuclear Spin 74 4.2.2 The Determination of the Nuclear Statistical Weights of Spatial States 79 4.3 Determination of Electronic Molecular Multiplets 82 4.3.1 Valence Bond Method 82 4.3.2 Degenerate Orbitals and One Valence Electron on Each Atom 87 4.3.3 Several Electrons Specified on One of the Atoms 91 4.3.4 Diatomic Molecule with Identical Atoms 93 4.3.5 General Case I 98 4.3.6 General Case II 100 References 104 5 Parastatistics, Fractional Statistics, and Statistics of Quasiparticles of Different Kind 106 5.1 Short Account of Parastatistics 106 5.2 Statistics of Quasiparticles in a Periodical Lattice 109 5.2.1 Holes as Collective States 109 5.2.2 Statistics and Some Properties of Holon Gas 111 5.2.3 Statistics of Hole Pairs 117 5.3 Statistics of Cooper’s Pairs 121 5.4 Fractional Statistics 124 5.4.1 Eigenvalues of Angular Momentum in the Three- and Two-Dimensional Space 124 5.4.2 Anyons and Fractional Statistics 128 References 133 Appendix A: Necessary Basic Concepts and Theorems of Group Theory 135 A.1 Properties of Group Operations 135 A.1.1 Group Postulates 135 A.1.2 Examples of Groups 137 A.1.3 Isomorphism and Homomorphism 138 A.1.4 Subgroups and Cosets 139 A.1.5 Conjugate Elements. Classes 140 Contents ix A.2 Representation of Groups 141 A.2.1 Definition 141 A.2.2 Vector Spaces 142 A.2.3 Reducibility of Representations 145 A.2.4 Properties of Irreducible Representations 147 A.2.5 Characters 148 A.2.6 The Decomposition of a Reducible Representation 149 A.2.7 The Direct Product of Representations 151 A.2.8 Clebsch–Gordan Coefficients 154 A.2.9 The Regular Representation 156 A.2.10 The Construction of Basis Functions for Irreducible Representation 157 References 160 Appendix B: The Permutation Group 161 B.1 General Information 161 B.1.1 Operations with Permutation 161 B.1.2 Classes 164 B.1.3 Young Diagrams and Irreducible Representations 165 B.2 The Standard Young–Yamanouchi Orthogonal Representation 167 B.2.1 Young Tableaux 167 B.2.2 Explicit Determination of the Matrices of the Standard Representation 170 B.2.3 The Conjugate Representation 173 B.2.4 The Construction of an Antisymmetric Function from the Basis Functions for Two Conjugate Representations 175 B.2.5 Young Operators 176 B.2.6 The Construction of Basis Functions for the Standard Representation from a Product of N Orthogonal Functions 178 References 181 Appendix C: The Interconnection between Linear Groups and Permutation Groups 182 C.1 Continuous Groups 182 C.1.1 Definition 182 C.1.2 Examples of Linear Groups 185 C.1.3 Infinitesimal Operators 187 C.2 The Three-Dimensional Rotation Group 189 C.2.1 Rotation Operators and Angular Momentum Operators 189 C.2.2 Irreducible Representations 191 C.2.3 Reduction of the Direct Product of Two Irreducible Representations 194 x Contents C.2.4 Reduction of the Direct Product of k Irreducible Representations. 3n − j Symbols 197 C.3 Tensor Representations 201 C.3.1 Construction of a Tensor Representation 201 C.3.2 Reduction of a Tensor Representation into Reducible Components 202 C.3.3 Littlewood’s Theorem 207 C.3.4 The Reduction of U2j + 1 R3 209 λ C.4 Tables of the Reduction of the Representations U2j +1 to the Group R3 214 References 216 Appendix D: Irreducible Tensor Operators 217 D.1 Definition 217 D.2 The Wigner–Eckart Theorem 220 References 222 Appendix E: Second Quantization 223 References 227 Index 228 Preface This book is devoted to the Pauli exclusion principle, which is a fundamental prin- ciple of quantum mechanics and has been naturally kept in mind in all its numerous applications in physics, chemistry, molecular biology, and even in astronomy, see Chapter 1. Nevertheless, to the best of my knowledge, it is the first scientific (not philosophical) book devoted to the Pauli exclusion principle. Although Wolfgang Pauli formulated his principle more than 90 years ago, its rigorous theoretical foun- dations are still absent. In the historical survey (Chapter 1) and in other chapters of the book, I discuss in detail still existing unsolved problems connected with the Pauli exclusion principle and for some of them suggest possible solutions. From the beginning of my scientific activity I have been interested in the issues of symmetry in quantum mechanics and in the mathematical description of it, that is, in the group theory and, particularly, in the permutation group theory. I was impressed by the simplicity and clearness of Young’s mathematical language (about Young diagrams and Young tableaux, see Appendix B), especially if one takes into account Young’s individuality: most of Young’s papers on the permutation group were writ- ten when he was a country parish priest. For many years I was occupied with the following question: why, according to the Pauli exclusion principle, in our Nature only the antisymmetric or symmetric permutation symmetries for identical particles are realized, while the Schrödinger equation is satisfied by functions with any per- mutation symmetry. The possible answers on this question I discuss in Chapter 3. I was always impressed by the Pauli deep physical intuition, which several times struck physical community. In fact, the formulation by Pauli of his principle was to a great extent based on his intuition (it was done before the creation of modern quantum mechanics), and it gave rise to the discovery of such important quantum mechanical conception as spin (I describe this dramatic story in Chapter 1).
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