teoria de grupos symmetry, structure spectroscopy group theory外语英文电子书

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1、Chemistry239Symmetry,StructureandSpectroscopy:GroupTheoryanditsApplicationsPeterR.TaylorSanDiegoSupercomputerCenterandDepartmentofChemistryandBiochemistryUniversityofCalifornia,SanDiegotaylor@sdsc.eduhttp://www.sdsc.edu/~taylorSanDiegoSupercomputerCenterChemistry239Abs

2、tractGroupTheory•Considerasetofobjects{G}andaproductruledenoted◦thatallowsustocombinethem.•DenotedF◦G,whereF,G∈{G}.•{G}canbeobjectssuchasnumbersorvariables,oroperators.SanDiegoSupercomputerCenterChemistry239Examples•Theintegersandanyofthebinaryoperationsofarithmetic:◦=

3、+: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•Translationsorrotationsofaphysicalobj

4、ectintwoorthreedimensions.Here◦denotessuccessivetransformations.cosθ−sinθ0cosφ−sinφ0sinθcosθ0◦sinφcosφ0001001•Thesecommute,unlikecosθ−sinθ0cosφ0−sinφsinθcosθ0◦010001sinφ0cosφSanDiegoSupercomputerCenterChemistry239Examples•Permutationsofo

5、bjects: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}.•NotethatthisdoesnotimplyF◦G=G◦F.•Suchasetandclosedproductrul

6、ecompriseagruppoid.•Forexample,theintegersareclosedunderaddition,multiplication,andsubtraction,butnotunderdivision.•ThesetofpermutationsofNobjectsisclosedwithrespecttosuccessivepermutations.•SuccessiverotationsandtranslationsinMdimensionsareclosed.SanDiegoSupercomputer

7、CenterChemistry239Imposingfurtherrules•Gruppoidsareclearlyverygeneralthings.•Fewusefulpropertiesareknownforgruppoids—wehavetorestrictourselvesfurther.•Imposerestrictionsonoursetandproductrule.SanDiegoSupercomputerCenterChemistry239Associativity•RequirethatifF,G,H∈{G},w

8、ehave(F◦G)◦H=F◦(G◦H).•Forexample,theadditionandmultiplicationofintegersisassociative,whereassubtractionisnot.•Success