The Pauli exclusion principle: origin, verifications and applications PDF
Preview 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 Thiseditionfirstpublished2017 ©2017JohnWiley&Sons,Ltd. RegisteredOffice JohnWiley&Sons,Ltd,TheAtrium,SouthernGate,Chichester,WestSussex,PO198SQ, UnitedKingdom Fordetailsofourglobaleditorialoffices,forcustomerservicesandforinformationabouthowtoapply forpermissiontoreusethecopyrightmaterialinthisbookpleaseseeourwebsiteatwww.wiley.com. Therightoftheauthortobeidentifiedastheauthorofthisworkhasbeenassertedinaccordancewiththe Copyright,DesignsandPatentsAct1988. Allrightsreserved.Nopartofthispublicationmaybereproduced,storedinaretrievalsystem,or transmitted,inanyformorbyanymeans,electronic,mechanical,photocopying,recordingorotherwise, exceptaspermittedbytheUKCopyright,DesignsandPatentsAct1988,withoutthepriorpermissionof thepublisher. 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Identifiers:LCCN2016018231(print)|LCCN2016020644(ebook)|ISBN9781118795323(cloth)| ISBN9781118795293(epdf)|ISBN9781118795248(epub) Subjects:LCSH:Pauliexclusionprinciple.|Quantumtheory. Classification:LCCQC174.17.P3K372017(print)|LCCQC174.17.P3(ebook)|DDC530.12–dc23 LCrecordavailableathttps://lccn.loc.gov/2016018231 AcataloguerecordforthisbookisavailablefromtheBritishLibrary. Coverimage:DavidWells/EyeEm/Gettyimages Setin10/12ptTimesbySPiGlobal,Pondicherry,India 10 9 8 7 6 5 4 3 2 1 Tomy wife Larisa Contents Preface xi 1 Historical Survey 1 1.1 Discovery of the Pauli Exclusion Principle andEarly Developments 1 1.2 Further Developments andStill Existing Problems 11 References 21 2 Construction of Functions with aDefinite 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 thePauliExclusion 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 AdditionalQuantum Numbers: TheSeniority 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 andOne Valence Electronon Each Atom 87 4.3.3 SeveralElectrons Specified onOne 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 ShortAccountof 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 Subgroupsand Cosets 139 A.1.5 Conjugate Elements. Classes 140 Contents ix A.2 Representationof 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 Decompositionof aReducible 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 BasisFunctions 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 anAntisymmetric Function from the Basis Functions for Two Conjugate Representations 175 B.2.5 Young Operators 176 B.2.6 The Construction of BasisFunctions for the Standard Representationfrom aProduct of N Orthogonal Functions 178 References 181 Appendix C: The Interconnection between Linear Groups and PermutationGroups 182 C.1 ContinuousGroups 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 TwoIrreducible Representations 194 x Contents C.2.4 Reductionof the DirectProduct of kIrreducible Representations. 3n−j Symbols 197 C.3 Tensor Representations 201 C.3.1 Construction of aTensorRepresentation 201 C.3.2 Reductionof aTensor Representation into Reducible Components 202 C.3.3 Littlewood’s Theorem 207 C.3.4 The Reduction of U R 209 2j+1 3 λ C.4 Tables of the Reductionof the Representations U 2j+1 to the Group R 214 3 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 ThisbookisdevotedtothePauliexclusionprinciple,whichisafundamentalprin- cipleofquantummechanicsandhasbeennaturallykeptinmindinallitsnumerous applicationsinphysics,chemistry,molecularbiology,andeveninastronomy,see Chapter1.Nevertheless,tothebestofmyknowledge,itisthefirstscientific(not philosophical)bookdevotedtothePauliexclusionprinciple.AlthoughWolfgang Pauliformulatedhisprinciplemorethan90yearsago,itsrigoroustheoreticalfoun- dationsarestillabsent.Inthehistoricalsurvey(Chapter1)andinotherchaptersof the book, I discuss in detail still existing unsolved problems connected with the Pauli exclusion principle and for some of them suggest possible solutions. FromthebeginningofmyscientificactivityIhavebeeninterestedintheissuesof symmetryinquantummechanicsandinthemathematicaldescriptionofit,thatis,in thegrouptheoryand,particularly,inthepermutationgrouptheory.Iwasimpressed by the simplicity and clearness of Young’s mathematical language (about Young diagramsandYoungtableaux,seeAppendixB),especiallyifonetakesintoaccount Young’sindividuality:mostofYoung’spapersonthepermutationgroupwerewrit- ten when he was a countryparishpriest. For manyyears I was occupied withthe followingquestion:why,accordingtothePauliexclusionprinciple,inourNature onlytheantisymmetricorsymmetricpermutationsymmetriesforidenticalparticles arerealized,whiletheSchrödingerequationissatisfiedbyfunctionswithanyper- mutationsymmetry.ThepossibleanswersonthisquestionIdiscussinChapter3. IwasalwaysimpressedbythePaulideepphysicalintuition,whichseveraltimes struckphysicalcommunity.Infact,theformulationbyPauliofhisprinciplewasto a great extent based on his intuition (it was done before the creation of modern quantummechanics),anditgaverisetothediscoveryofsuchimportantquantum mechanical conception as spin (I describe this dramatic story in Chapter 1). Anotherwell-knownexampleisthePaulipredictionofneutrino.Paulimadethis predictionwithoutanyexperimentalandtheoreticalindicationsthatthischargeless xii Preface and,asthoughtatthattime,masslessparticlecanexist.Hetriedtosavetheenergy conservationlawintheβ-decay,becausehedidnotagreewithNielsBohrwhoat thattimewassurethattheenergyconservationlawisnotvalidformicroparticles. It turns out that Pauli was right. Theconstructionoffunctionswithagivenpermutationsymmetryisdiscussedin detailinChapter2.AllnecessarymathematicalapparatusisgiveninAppendicesA andB.IfthetotalwavefunctionofNidenticalparticlesystemisrepresentedasa productofthespatialandspinwavefunctionssymmetrizedaccordingtotheappro- priateirreduciblerepresentationsofthepermutationgroup,itautomaticallysatis- fiesthePauliexclusionprincipleanddescribesthestatewithadefinitevalueofthe total electronic or nuclear spin. Theapplicationofthepermutationgrouptheoryfortheconstructionofmolecu- lar wave functions makes possible elaborating effective and elegant methods for finding the Pauli-allowed states in atomic, molecular, and nuclear spectroscopy (Chapter4).Intheelaboratedmethodsthelineargroupsandtheirinterconnection withthepermutationgroupareapplied.Thenecessarymathematicalapparatusis representedinAppendixC.TheclassificationofthePauli-allowedstatesisrepre- sentedforalltypesofmany-atommoleculeswiththeexplicitformulaeforthechar- acters of reducible representations formed by agiven setof atomicstates.In last sectionsofthischapterthemethodsoffindingthePauli-allowedstatesforanarbi- trary many-particle system, containing subsystems characterized by their local symmetry,aredescribed.Thesemethodsdonotdependonthenumberofparticles in subsystems. Chapter 5 is devoted to exotic statistics: parastatistics and fractional statistics. Although the elementary particles obeying the parastatistics are not detected, Idemonstratethatthequasiparticles(collectiveexcitations)inaperiodicallattice areobeyingthemodifiedparafermistatistics;amongthemaretheholepairs,which are analogue of Cooper’s pairs in the high T superconductivity, and such well- c known quasiparticles as excitons and magnons. The fractional statistics is also realized in ourNature for excitationsin the fractional quantum Hall effect; these excitations can be considered as quasiparticles with fractional charge. However, the theoretical suggestions that the fractional statistics is realized in the high T c superconductivityhave not been confirmed byexperiment. Itriedtowritethebookforabroadaudiencefromacademicresearcherstogradu- atestudentsconnectedintheirworkorstudywithquantummechanics.Significant effortsweremadetopresentthebooksoasitwillbeself-sufficientforreaders,since allnecessaryapparatusofthegrouptheoryisdescribedintheappendices. IwouldliketoacknowledgeLucienPiela,LevPitaevsky,OlgaRodimova,Oleg Vasyutinsky, Vladimir Yurovsky,andSergeZagoulaevforusefuldiscussions of differentproblemsconnectedwiththetopicofthebook.Specialacknowledgment goes to Ulises Miranda andAlberto Lopez who helped me to correct the book. Mexico February 2016
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