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Quantum Mechanics: Genesis and Achievements Alexander Komech Quantum Mechanics: Genesis and Achievements AlexanderKomech FacultyofMathematics UniversityofVienna Vienna,Austria Resume Theintentofthisbookisaconciseintroductiontothefieldofnonrelativisticand relativisticquantummechanics.AhistoricalaccountoftheHeisenbergandtheSchrödinger theoriesisgiven,andtraditionalapplicationstothehydrogenatomareoutlined.Alldetails ofcalculationsaregivenandsupplementedwithrelatedmaterialsfromtheMaxwellElectro- dynamicsandSpecialRelativity. Thebookcanbeusedasasourceforatwo-semesterlecturecourseonNonrelativisticQuan- tumMechanics,aswellasforaone-semestercourseoneithertheRelativisticQuantumMe- chanics,ClassicalElectrodynamics,ortheClassicalFieldTheory. ISBN978-94-007-5541-3 ISBN978-94-007-5542-0(eBook) DOI10.1007/978-94-007-5542-0 SpringerDordrechtHeidelbergNewYorkLondon LibraryofCongressControlNumber:2012951292 ©SpringerScience+BusinessMediaDordrecht2013 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. 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Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) To thememoryof Vera andAlexeyKomech, EstherBraginskaya, andMarkVishik Preface This book is based on a three-semester course of lectures delivered by the author attheFacultyofMathematicsoftheViennaUniversity.Forpreliminaryversionsof themanuscript,see[113,114]. The main emphasis of this book is on nonrelativistic and relativistic quantum mechanics with standard applications to the hydrogen atom. Our main intention is to present the quantum mechanics in a comprehensive manner, accessible for amathematician.Theexpositionisformalized(wheneverpossible)onthebasisof coupledMaxwell–SchrödingerandMaxwell–Diracequations.Thisintentionagrees with Hilbert’s 6th Problem (“Axiomatize Theoretical Physics”), and Heisenberg’s nonlinearprogramme[83,84]. Our exposition starts with a chronological analysis of crucial empirical obser- vations and their theoretical systematization, explaining inter alia the motivation behindtheHeisenbergandSchrödingerequations.Theintroductionofquantumob- servables stems from the agreement with corresponding classical observables for short wavelength solutions, Hamilton–Jacobi’s theorem being taken into account. Also, the relation between the quantum observables and the Noether symmetry theory is discussed. The Lagrangian formalism is used as a fundamental unifying principletolaythebasisforintroductionofthecoupledMaxwell–Schrödingerand Maxwell–Diracequations. Ofcourse,theequationsandobservablescouldbe(andformallyshouldbe)ac- cepted as axioms. On the other hand, it is crucially important to know the experi- mentalandmathematicalfactsbehindquantumformalismtoembeditinthewhole ofphysics. Moreover, the modern form of quantum theory seems to be far from its com- pleteness,likegeometryinpre-Euclideanera.Itisthereforeparticularlyimportant to understand the degree of confidence to individual constituents of the quantum formalism.Thisiswhywepaysomuchattentiontotheoriginandmotivationofthe formalism. The hydrogen spectrum and the atom radiation are calculated with all detail. Parallelsbetweenquantumandclassicaldescriptionaretracedeverywheretomoti- vate the introductionof quantum differential cross section,magneticmoment,etc. vii viii Preface The scattering problems are solved by application of the perturbation procedure to the coupled Maxwell–Schrödinger equations. We point out some deficiency in theperturbationprocedure,whichshouldbefixedwithanonperturbativeapproach, howeverthiscorrectionisstillanopenproblem. The introduction of the electron spin is discussed in detail from experimental and theoretical point of view. We calculate the Landé formula for the gyromag- neticratioviathespin-orbitalinteractionofRussell–Saunders,whichexplainsthe Einstein–deHaasexperimentandtheanomalousZeemaneffect.Furtherweprove therelativisticcovarianceoftheDiracequation,obtainthecorrespondingintrinsic spinmomentum,andthecorrespondingnonrelativisticapproximations.Finally,we calculatethehydrogenspectrumviatheDiracequation. Wemakeexplicitinvokedassumptionsandapproximations,anddiscussaplausi- bletreatmentofsomelogicalleapsinthetheory.However,wedidnottrytoestablish newrigorousresults.Generally,ourexpositionisnotmathematicallyrigorous.For example, we do not distinguish between Hermitian symmetric and the selfadjoint operators,eventhoughthespectralresolutionisusedrepeatedly. Inappendices(Chaps.12and13),weexplainrelateddetailsfromClassicalElec- trodynamicsandSpecialRelativity,GeometricalOptics,theHamilton–Jacobitheo- rem,anupdatedversionoftheNoethertheoryofcurrents,andthelimitingamplitude principle. InChap.14,wecollectclassicalcalculationslyinginthebaseofthe‘oldquan- tummechanics’. FurtherReading Ourmaingoalistogiveaconciseexplanationofmathematical principles of Quantum Mechanics. More technical details and a systematic com- parison with experimental data can be found in [7, 11, 12, 20, 23, 34, 63, 75, 81, 130, 131, 145, 160, 171, 179, 191]. The books [46, 79, 89, 143] and [93, 185], respectively,explainbasicconceptsofQuantumMechanicsandClassicalElectro- dynamics. A suitable introduction to the mathematical theory of the Schrödinger equationiscontainedin[10]. Wedevelopthemethodsofquantummechanicsforthehydrogenatomwhichcan be extended to other one-electron atoms (lithium, sodium, potassium atoms, etc), anddonottouchmulti-electronproblemsofquantumchemistry[34,41,178,187]. WealsodonottouchtheStabilityofMatter[31,134],theQuantumElectrodynam- icsandQuantumFieldTheory[13,14,33,64,77,85,137,138,158,159,163,189, 195,196]. Acknowledgements Over a period of years the author has been pleased to have manystimulatingdiscussionswithA.ShnirelmanandH.Spohn.Theauthorisgrate- fultoE.Kopylovaforreadingandcheckingthemanuscript,andtoC.Adamforhis helpinwritingpreliminaryversionsoftheExercises(Sects.14.1–14.6).Theauthor is indebted to the Faculty of Mathematics of Vienna University, and the Institute fortheInformationTransmissionProblemsoftheRussianAcademyofSciencesfor providingcongenialfacilitiesforthework. Preface ix The author is also grateful to the Max Planck Institute for Mathematical Sci- ences(Leipzig),theMünchenTechnicalUniversity,andtheTeamONDES/POEMS (INRIA,Rocquencourt)fortheirhospitality. TheworkwassupportedinpartbytheDepartmentofMechanicsandMathemat- icsoftheMoscowStateUniversity,bytheAlexandervonHumboldtFoundation,by the Austrian Science Fund (FWF) (projects Nos. P19138-N13 and P22198-N13), andbygrantsfromtheDeutscheForschungsgemeinschaftandtheRussianFundfor BasicResearch. Leipzig–Moscow–München–Rocquencourt–Vienna A.Komech