Taylor--Symmetry， Structure &ampamp;amp; Spectroscopy Group Theory ( 加州圣地牙哥超级电脑中心课件 ).pdf

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1、Chemistry239Symmetry,StructureandSpectroscopy:GroupTheoryanditsApplicationsPeterR.TaylorSanDiegoSupercomputerCenterandDepartmentofChemistryandBiochemistryUniversityofCalifornia,SanDiegotaylor@sdsc.eduhttp://www.sdsc.edu/~taylorSanDiegoSupercomputerCenterChemis

2、try239AbstractGroupTheory•Considerasetofobjects{G}andaproductruledenoted◦thatallowsustocombinethem.•DenotedF◦G,whereF,G∈{G}.•{G}canbeobjectssuchasnumbersorvariables,oroperators.SanDiegoSupercomputerCenterChemistry239Examples•Theintegersandanyofthebinaryoperati

3、onsofarithmetic:◦=+:1+5=6(1)◦=−:1−5=−4=5−1(2)(12−3)−7=3=12−(3−7)=16(3)◦=÷:12÷3=4=3÷12(notevenaninteger)(4)•Notethatsofartherearenorequirementsthat◦shouldobeycertainrules,suchascommutativityorclosure.SanDiegoSupercomputerCenterChemistry239Examples•Translatio

4、nsorrotationsofaphysicalobjectintwoorthreedimensions.Here◦denotessuccessivetransformations.cosθ−sinθ0cosφ−sinφ0sinθcosθ0◦sinφcosφ0001001•Thesecommute,unlikecosθ−sinθ0cosφ0−sinφsinθcosθ0◦010001sinφ0cosφSanDiegoSupercomputerCenter

5、Chemistry239Examples•Permutationsofobjects:supposewehaveaset{ABC}andwehavepermutationsde£nedby,e.g.,(312){ABC}={CAB}•Then(312)◦(213){ABC}=(312){BAC}={CBA}SanDiegoSupercomputerCenterChemistry239Closure•RequirethatifF,G∈{G},thenF◦G∈{G}andG◦F∈{G}.•Notethatthisdoe

6、snotimplyF◦G=G◦F.•Suchasetandclosedproductrulecompriseagruppoid.•Forexample,theintegersareclosedunderaddition,multiplication,andsubtraction,butnotunderdivision.•ThesetofpermutationsofNobjectsisclosedwithrespecttosuccessivepermutations.•Successiverotationsandtr

7、anslationsinMdimensionsareclosed.SanDiegoSupercomputerCenterChemistry239Imposingfurtherrules•Gruppoidsareclearlyverygeneralthings.•Fewusefulpropertiesareknownforgruppoids—wehavetorestrictourselvesfurther.•Imposerestrictionsonoursetandproductrule.SanDiegoSuperc

8、omputerCenterChemistry239Associativity•RequirethatifF,G,H∈{G},wehave(F◦G)◦H=F◦(G◦H).•Forexample,theadditionandmultiplicationofintegersisassociative,whereassubtractionisnot.•Success