Wave Equations in Higher Dimensions PDF
Preview Wave Equations in Higher Dimensions
Wave Equations in Higher Dimensions Shi-Hai Dong Wave Equations in Higher Dimensions Shi-HaiDong EscuelaSuperiordeFísicayMatemáticas UnidadProfesionalAdolfoLópezMateos InstitutoPolitécnicoNacional Edificio9 MexicoDF07738 Mexico [email protected] ISBN978-94-007-1916-3 e-ISBN978-94-007-1917-0 DOI10.1007/978-94-007-1917-0 SpringerDordrechtHeidelbergLondonNewYork LibraryofCongressControlNumber:2011933232 ©SpringerScience+BusinessMediaB.V.2011 Nopartofthisworkmaybereproduced,storedinaretrievalsystem,ortransmittedinanyformorby anymeans,electronic,mechanical,photocopying,microfilming,recordingorotherwise,withoutwritten permissionfromthePublisher,withtheexceptionofanymaterialsuppliedspecificallyforthepurpose ofbeingenteredandexecutedonacomputersystem,forexclusiveusebythepurchaserofthework. Coverdesign:VTeXUAB,Lithuania Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Thisbookisdedicatedtomy wifeGuo-HuaSun and lovelychildrenBoDongand JazminYueDongSun. Preface This work will introduce the wave equations in higher dimensions at an advanced leveladdressingstudentsofphysics,mathematicsandchemistry.Theaimistoput the mathematical and physical concepts and techniques like the wave equations, group theory, generalized hypervirial theorem, the Levinson theorem, exact and proper quantizationrules related tothe higherdimensionsat thereader’s disposal. For this purpose, we attempt to provide a comprehensive description of the wave equationsincludingthenon-relativisticSchrödingerequation,relativisticDiracand Klein-Gordonequationsinhigherdimensionsandtheirwideapplicationsinquan- tum mechanics which complements the traditional coverage found in the existing quantummechanicstextbooks.Relatedtothisfieldarethequantummechanicsand group theory. In fact, the author’s driving force has been his desire to provide a comprehensivereviewvolumethatincludessomenewandsignificantresultsabout thewaveequationsinhigherdimensionsdrawnfromtheteachingandresearchex- perienceoftheauthorsincetheliteratureisinundatedwithscatteredarticlesinthis fieldandtopavethereader’swayintothisterritoryasrapidlyaspossible.Wehave madetheefforttopresenttheclearandunderstandablederivationsandincludethe necessarymathematicalstepssothattheintelligentanddiligentreaderisabletofol- lowthetextwithrelativeease,inparticular,whenmathematicallydifficultmaterial ispresented.TheauthoralsoembracesenthusiasticallythepotentialoftheLaTeX typesettinglanguagetoenrichthepresentationoftheformulasastomakethelog- icalpatternbehindthemathematicsmoretransparent.Inaddition,anysuggestions and criticism to improve the text are most welcome. It should be pointed out that the maineffort to follow the textand master the material is left to the reader even thoughthisbookmakesanefforttoservethereaderasmuchaswaspossibleforthe author. ThisbookstartsoutinChap.1withacomprehensivereviewforthewaveequa- tionsinhigherdimensionsandbuildsonthistointroduceinChap.2thefundamen- tal theory about the SO(N) group to be used in the successive Chaps. 3–5 includ- ing the non-relativistic Schrödinger equation, relativistic Dirac and Klein-Gordon equations. As important applications in non-relativistic quantum mechanics, from Chap.6toChap.12,weshallapplythetheoriesproposedinPartIItostudysome vii viii Preface importantquantumsystemssuchastheharmonicoscillator,Coulombpotential,the Levinson theorem, generalized hypervirial theorem, exact and proper quantization rules and Langer modification, the Schrödinger equation with position-dependent massandothers.WeshallillustratetwoimportantapplicationsinrelativisticDirac and Klein-Gordon equations with the Coulomb potential in Chaps. 13 and 14. As crucial generalized applications of Dirac equation in higher dimensions, we shall studytheLevinsontheorem,generalizedhypervirialtheoremandKaluza-Kleinthe- oryinChaps.15–17.SomeconclusionsandoutlooksaregiveninChap.18.Some usefulreferencematerialssuchasgrouptheory,grouprepresentations,fundamental properties of Lie groups and Lie algebras, the angular momentum theory and the confluenthypergeometricfunctionsaresketchedinAppendicesA–E. This book is in a stage of continuing development, various chapters, e.g., on thequantumgravity,ontheKaluza-Kleintheory,onthesupersymmetryandstring theory,onthehighdimensionalbranewillbeaddedtotheextentthattherespective topics expand. At the present stage, however, the work presented for such topics shouldbecompleteenoughtoservethereader. This book shall give the theoretical physicists and researchers a fresh outlook andnewwaysofhandlingsomeimportantandinterestingquantumsystemsinsev- eral branches of physics. This book can be used by graduate students and young researchers in physics, especially theoretical and mathematical physics. It is also suitableforsomegraduatestudentsintheoreticalchemistry. Mexicocity,Mexico Shi-HaiDong Acknowledgements First,IwouldliketothankProf.Zhong-QiMaforencouragementandcontinuous support in my research. Prof. Ma was always there to give me some positive and helpfulsuggestions. Second, I shall give special thanks to Profs. E. Ley-Koo, S.A. Cruz and L.H. FabilaCastillo,whohaveofferedmeunselfishhelpinmylife.Inparticular,Iwould like to thank Prof. Ley-Koo for many positive and invaluable suggestions on this topicin2004.ThemildandbracingclimateinMexicocityhaskeptmeinagood spirit. Third,IamveryindebtedtomycollaboratorsProfs.Child,Lin,Lemus,Frank, Lozada-Cassou, Popov, Vezerra, Gu, Sun, Tang, Lara, Qiang, Chen, Yu, García, Peña, Wei and others. I also would like to thank Prof. Alwyn van der Merwe for greatlysupportingthisbookproposal.Inparticular,Ireallyappreciatethesupport fromprojects20100297-SIP-IPNand20110491-SIP-IPN,Mexico. Fourth,IthankmyprevioussupervisorProf.FengPanforgettingmeinterested ingrouptheoryandLiealgebras. Last,butnotleast,Iwouldliketothankmyfamily:mywife,Guo-HuaSun,for givingmecontinuousencouragementandfordevotingherselftothewholefamily except for her own’s study and job; my lovely son and daughter, Bo Dong and JazminYueDongSunforgivingmeencouragement;myparents,Ji-TangDongand Gui-RongWang,for givingmelife,for unconditionalsupportandencouragement topursuemyinterests,evenwhentheinterestswentbeyondboundariesoflanguage and geography; my older brother and sister S.S. Dong and X.F. Dong for looking afterourparentsmeticulouslyinChina. Shi-HaiDong ix
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