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1、GroupTheoryProblemSet1October12,2001Note:ProblemsmarkedwithanasteriskareforRapidFeedback;problemsmarkedwithadoubleasteriskareoptional.1.Showthatthewaveequationforthepropagationofanimpulseatthespeedoflightc,1@2u@2u@2u@2u=++;c2@t2@x2@y2@z2iscovariantundertheLorentztransf
2、ormationµ¶0000vx=°(x¡vt);y=y;z=z;t=°t¡x;c2where°=(1¡v2=c2)¡1=2.2.¤TheSchrÄodingerequationforafreeparticleofmassmis@'¹h2@2'i¹h=¡:@t2m@x2Showthatthisequationisinvarianttotheglobalchangeofphaseofthewavefunction:'!'0=ei®';where®isanyrealnumber.Thisisanexampleofaninternalsy
3、mmetrytransformation,sinceitdoesnotinvolvethespace-timecoordinates.AccordingtoNoether'stheorem,thissymmetryimpliestheexistenceofaconservationR1law.Showthatthequantityj'(x;t)j2dxisindependentoftimeforsolutionsofthe¡1free-particleSchrÄodingerequation.3.¤Considerthefollow
4、ingsetsofelementsandcompositionlaws.Determinewhethertheyaregroupsand,ifnot,identifywhichgrouppropertyisviolated.(a)Therationalnumbers,excludingzero,undermultiplication.(b)Thenon-negativeintegersunderaddition.(c)Theevenintegersunderaddition.(d)Thenthrootsofunity,i.e.,e2
5、¼mi=n,form=0;1;:::;n¡1,undermultiplication.(e)Thesetofintegersunderordinarysubtraction.4.¤¤ThegeneralformoftheLiouvilleequationis·¸ddy£¤p(x)+q(x)+¸r(x)y=0dxdxwherep,qandrarereal-valuedfunctionsofxwithpandrtakingonlypositivevalues.Thequantity¸iscalledtheeigenvalueandthe
6、functiony,calledtheeigenfunction,isassumedtobede¯nedoveraninterval[a;b].Wetaketheboundaryconditionstobey(a)=y(b)=0buttheresultderivedbelowisalsovalidformoregeneralboundaryconditions.No-ticethattheLiouvilleequationscontainstheone-dimensionalSchrÄodingerequationasaspecia
7、lcase.Letu(x;¸)andv(x;¸)bethefundamentalsolutionsoftheLiouvilleequation,i.e.uandvaretwolinearly-independentsolutionsintermsofwhichallothersolutionsmaybeexpressed(foragivenvalue¸).ThenthereareconstantsAandBwhichallowanysolutionytobeexpressedasalinearcombinationofthisfun
8、damentalset:y(x;¸)=Au(x;¸)+Bv(x;¸)Theseconstantsaredeterminedbyrequiringy(x;¸)tosatisfytheboundaryconditions:y(a;¸)=A
