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1、arXiv:quant-ph/0410061v414Feb200912QUANTUMMECHANICS?HitoshiKitada?December28,2003?c1998–2003byHitoshiKitada,AllRightsReserved?GraduateSchoolofMathematicalSciences,UniversityofTokyo,Komaba,Meguro,Tokyo153-8914,Japan,e-mail:kitada@kims.ms.u-tokyo.ac.jp,webpag
2、e:http://kims.ms.u-tokyo.ac.jp/iiPrefaceIconsiderinthisbookaformulationofQuantumMechanics,whichisoftenabbreviatedasQM.UsuallyQMisformulatedbasedonthenotionoftimeandspace,bothofwhicharethoughtapriorigivenquantitiesornotions.However,whenwetrytode?nethenotiono
3、fvelocityormomentum,weencounteradi?cultyaswewillseeinchapter1.Theproblemisthatifthenotionoftimeisgivenapriori,thevelocityisde?nitelydeterminedwhengivenaposition,whichcontradictstheuncertaintyprincipleofHeisenberg.WethensetthebasisofQMonthenotionofpositionan
4、dmomentumoperatorsasinchapter2.Timeofalocalsystemthenisde?nedapproximatelyasaratio
5、x
6、/
7、v
8、betweenthespacecoordinatexandthevelocityv,where
9、x
10、,etc.denotestheabsolutevalueorlengthofavectorx.InthisformulationofQM,wecankeeptheuncertaintyprinciple,andtimeisaquanti
11、tythatdoesnothaveprecisevaluesunliketheusuallysupposednotionoftimehas.Thefeatureoflocaltimeisthatitisatimepropertoeachlocalsystem,whichisde?nedasa?nitesetofquantummechanicalparticles.Wenowhaveanin?nitenumberoflocaltimesthatareuniqueandpropertoeachlocalsyste
12、m.Basedonthenotionoflocaltime,themotioninsidealocalsystemisdescribedbytheusualSchr¨odingerequation.WeinvestigatesuchmotioninagivenlocalsysteminpartII.Thisisausualquantummechanics.Aftersomeexcursionoftheinvestigationoflocalmotion,weconsiderinpartIIItherelati
13、verelationormotionbetweenplurallocalsystems.Weregardeachlocalsystem’scenterofmassasaclassicalparticle.Thenastherelativecoordinateinsidealocalsystemisindependentofitscenterofmass,wecansetanarbitraryruleontherelationamongthosecentersofmassoflocalsystems.Weado
14、pttheprinciplesofgeneralrelativityastherulesthatgoverntherelationsofplurallocalsystems.Bythereasonthatthecenterofmassandtheinnercoordinateareindependent,wecancombinequantummechanicsandgeneralrelativity
