Table Of Content

Undergraduate Lecture Notes in Physics Gabor Kunstatter Saurya Das A First Course on Symmetry, Special Relativity and Quantum Mechanics The Foundations of Physics Undergraduate Lecture Notes in Physics Series Editors Neil Ashby, University of Colorado, Boulder, CO, USA WilliamBrantley,DepartmentofPhysics,FurmanUniversity,Greenville,SC,USA MatthewDeady,PhysicsProgram,BardCollege,Annandale-on-Hudson,NY,USA Michael Fowler, Department of Physics, University of Virginia, Charlottesville, VA, USA Morten Hjorth-Jensen, Department of Physics, University of Oslo, Oslo, Norway Michael Inglis, Department of Physical Sciences, SUNY Suffolk County Community College, Selden, NY, USA Undergraduate Lecture Notes in Physics (ULNP) publishes authoritative texts covering topics throughout pure and applied physics. Each title in the series is suitable as a basis for undergraduate instruction, typically containing practice problems,workedexamples,chaptersummaries,andsuggestionsforfurtherreading. ULNP titles must provide at least one of the following: (cid:129) An exceptionally clear and concise treatment of a standard undergraduate subject. (cid:129) A solid undergraduate-level introduction to a graduate, advanced, or non-standard subject. (cid:129) A novel perspective or an unusual approach to teaching a subject. ULNPespeciallyencouragesnew,original,andidiosyncraticapproachestophysics teaching at the undergraduate level. The purpose of ULNP is to provide intriguing, absorbing books that will continue to be the reader’s preferred reference throughout their academic career. More information about this series at http://www.springer.com/series/8917 Gabor Kunstatter Saurya Das (cid:129) A First Course on Symmetry, Special Relativity and Quantum Mechanics The Foundations of Physics 123 Gabor Kunstatter Saurya Das University of Winnipeg University of Lethbridge Winnipeg, MB,Canada Lethbridge, AB,Canada ISSN 2192-4791 ISSN 2192-4805 (electronic) Undergraduate Lecture Notesin Physics ISBN978-3-030-55419-4 ISBN978-3-030-55420-0 (eBook) https://doi.org/10.1007/978-3-030-55420-0 ©SpringerNatureSwitzerlandAG2020 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained hereinorforanyerrorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregard tojurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface Special relativity and quantum mechanics provide the beautiful but sometimes counter-intuitive underpinnings of twenty-first century physics. A less known crucialingredientisthenotionofsymmetry.Justassymmetryisintimatelylinkedto humans’perceptionofbeauty,italsoplaysakeyroleindeterminingthebeautyand desirability of physical theories. More important than the aesthetic significance of symmetryisitsremarkabeutility.ThroughthepowerfultheoremprovenbyEmmy Noether in1915,symmetryleadstoconservationlaws thatgive usconfidencethat some things do remain constant despite the rapid and often dizzying changes occurring in the world. More practically, symmetry provides invaluable tools for decodingcomplicatedstructuresandsimplifyingproblemsthatwouldotherwisebe intractable. The purpose of this text is to provide undergraduate students with a compre- hensiveandmathematicallyrigorousintroductiontospecialrelativityandquantum mechanics.Anovelaspectofthepresentationisthatsymmetryisgivenitsrightful prominence as an integral part of the foundation of physics. Students are given a conceptual understanding of symmetry and the important role it plays in physics. They are also provided with mathematical tools that allow for quantitative applications. The primary target audience consists of second-year physics majors who have completed an introductory calculus course and a first-year physics course that includes Newtonian mechanics and electrostatics. Some knowledge of linear algebra is useful but not essential. The book is intended to provide material for a self-contained two-semester course on the foundations of physics. It contains clearly marked supplementary sections that introduce more advanced topics such as variational mechanics and a proof of Noether’s theorem, four vectors and tensors and relativistic quantum mechanics. We have included many pedagogical descriptions of relevant topics of general interest.Theseincludetheroleofsymmetryinthediscoveryofspecialandgeneral relativity, the connection between symmetry and conservation laws, and the geo- metrical nature of Einstein’s theory of general relativity. Other specific subjects of v vi Preface current interest are gravitational waves, cosmology, quantum computers, Bell’s theorem,entanglement,andtheassociated“spooky”actionatadistanceinquantum mechanics. There are many worked Examples and Exercises for the student interspersed throughoutthetext.TheExamplesaresolvedwithinthetext,whilesolutionstothe Exercisesareprovidedinaseparatesolutionsmanual.Afirstreadingofthetextcan be accomplished with only a cursory examination of the Examples and Exercises, but a complete understanding of the material requires the student to work through them carefully. Lethbridge, AB, Canada Saurya Das Victoria, BC, Canada Gabor Kunstatter Acknowledgements The authors thank Pasquale Bosso, Kevin Brown, Ramin Daghigh, Esmat Elhami, Valerio Faraoni, Mary Kunstatter, Vesna Milosevic-Zdjelar, Elias C. Vagenas, DwightVincent,MarkWaltonandJonathanZiprickforinvaluablediscussionsand feedback. SD thanks Paritosh Kumar Das, Jayasri Das, Sangeeta Barua, Aalok Banerjee and Brishti Das for their unwavering support and encouragement. GK is indebted to the many undergraduates at the University of Winnipeg who had the dubious honour of being test subjects for much of the material in this textbook. In addition, GK is deeply grateful toMary Kunstatter for blessing him with her love,supportandencouragementduringthepreparationofthistextandalsoduring the previous 45 years spent learning to appreciate the symmetry and beauty of Nature. Last but by no means least, GK thanks David and Shauna Kunstatter for their love and support, and for being a continual source of inspiration. vii Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 The Goal of Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 The Connection Between Physics and Mathematics. . . . . . . . . 2 1.3 Paradigm Shifts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 The Correspondence Principle . . . . . . . . . . . . . . . . . . . . . . . . 6 2 Symmetry and Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1 Learning Outcomes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 What Is Symmetry?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 Role of Symmetry in Physics. . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3.1 Symmetry as a Guiding Principle . . . . . . . . . . . . . . . 10 2.3.2 Symmetry and Conserved Quantities . . . . . . . . . . . . . 11 2.3.3 Symmetry as a Tool for Simplifying Problems. . . . . . 11 2.4 Symmetries Were Made to Be Broken . . . . . . . . . . . . . . . . . . 12 2.4.1 Spacetime Symmetries . . . . . . . . . . . . . . . . . . . . . . . 12 2.4.2 Parity Violation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.4.3 Spontaneously Broken Symmetries . . . . . . . . . . . . . . 16 2.4.4 Variational Calculations: Lifeguards and Light Rays. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3 Formal Aspects of Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.1 Learning Outcomes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.2 Symmetries as Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.2.1 Definition of a Symmetry Operation . . . . . . . . . . . . . 23 3.2.2 Rules Obeyed by Symmetry Operations. . . . . . . . . . . 25 3.2.3 Multiplication Tables . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2.4 Symmetry and Group Theory . . . . . . . . . . . . . . . . . . 28 ix x Contents 3.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.3.1 The Identity Operation . . . . . . . . . . . . . . . . . . . . . . . 28 3.3.2 Permutations of Two Identical Objects. . . . . . . . . . . . 29 3.3.3 Permutations of Three Identical Objects. . . . . . . . . . . 29 3.3.4 Rotations of Regular Polygons . . . . . . . . . . . . . . . . . 30 3.4 Continuous Versus Discrete Symmetries. . . . . . . . . . . . . . . . . 31 3.5 Noether’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.6 Supplementary: Variational Mechanics and the Proof of Noether’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.6.1 Variational Mechanics: Principle of Least Action . . . . 34 3.6.2 Euler-Lagrange Equations . . . . . . . . . . . . . . . . . . . . . 38 3.6.3 Proof of Noether’s Theorem . . . . . . . . . . . . . . . . . . . 40 4 Symmetries and Linear Transformations . . . . . . . . . . . . . . . . . . . . 45 4.1 Learning Outcomes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.2 Review of Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.2.1 Coordinate Free Definitions. . . . . . . . . . . . . . . . . . . . 45 4.2.2 Cartesian Coordinates . . . . . . . . . . . . . . . . . . . . . . . . 50 4.2.3 Vector Operations in Component Form . . . . . . . . . . . 50 4.2.4 Position Vector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.2.5 Velocity and Acceleration: Differentiation of Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.3 Linear Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.3.1 Definition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.3.2 Translations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.3.3 Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.3.4 Reflections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.4 Linear Transformations and Matrices . . . . . . . . . . . . . . . . . . . 60 4.4.1 General Discussion. . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.4.2 Identity Transformation and Inverse. . . . . . . . . . . . . . 62 4.4.3 Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.4.4 Reflections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.4.5 Matrix Representation of Permutations of Three Objects. . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.5 Pythagoras and Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 5 Special Relativity I: The Basics. . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 5.1 Learning Outcomes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 5.2 Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 5.2.1 Frames of Reference. . . . . . . . . . . . . . . . . . . . . . . . . 69 5.2.2 Spacetime Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . 70 5.2.3 Newtonian Relativity and Galilean Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73