Chapter1 QuantumMechanicsinHilbert Spaces 1.1 Theessentialresultsinquantummechanicsaregiventhroughpurelyalgebraicrelations. TheAbstractHilbertSpace Specicresultscanbederived,e.g.,forvectors2`2andmatricesbeinglinearmaps; generalityandthenconsideraspecicrepresentationofthebasisvectorsofthehilbert however,thoseresultsareessentiallyindependentofthespecicrepresentationofthe operators.forthespecicresultsonlyalgebraicrelationsbetweenoperatorsandabstract propertiesofthehilbertspaceenter.thispointofviewallowstoconsiderproblemsinfull forwhichadditionandmultiplicationwithcomplexnumbersisdened I.TheabstractHilbertspace`2isgivenbyasetofelementsH=(j spaceandtheoperators(e.g.,matrices,dierentialoperators). j i+j'i=j +'i2h i;j'i;ji;), aj i=ja (1.1) togetherwithascalarproduct (1.2) Withrespectto(1.1)and(1.2),Hisalinearvectorspace,i.e., h'j i2c: (1.3) j i+j'i=j'i+j 2 i

(j i+j'i)+jxi=j j j i+j? i+j0i=j i=0 i+(j'i+jxi) i Thelasttworelationsstatetheexistenceofa0-vectorandtheexistenceofanegative vectorwithrespecttoj i. (1.4) a(bj 1j i)=(ab)j i=j i a(j (a+b)j i+j'i)=aj i=aj i+aj'i i+bj i i II.Withrespecttothescalarproduct,Hisaunitaryvectorspace (1.5) and h j i0 (1.6) h j i=0)j j'i i=0 (1.7) h'j h'ja 1+ 2i=h'j i=ah'j 1i+h'j i Becauseof(1.6)aNormcanbedened 2i (1.8) wherethespeciccharacteristicsofthenormdependonthevectorspace. k k=qh j i; (1.9) Onehas k'+ jh'j ijk'kk kk'k+k k and k (1.10) ha'j i=ah'j 3 i: (1.11)

Furtherpostulatesare: III.Hiscomplete. IV.Hisseparable. previousvectorj vectorj countable.letfj AHilbertspacebeingseparablemeansthatthereexistsasetofvectorsdenseinHand fj'1i;j'niginwhichtheoriginalsequenceiscontained.thesetoffj'nigisvia kifromthissequence,whichcanberepresentedaslinearcombinationofthe 1i;;j kigbeasequenceofvectorsinh.ifwetakeoutevery constructiondenseinh.onecanassumethatfj'nigisasetoforthogonalvectors(if not,usee.g.,gram-schmidtorthogonalization). k?1i,thenweobtainasetoflinearindependentvectors Iffj'nigisdenseinH,wecanexpandeveryarbitraryvectoraccordingtothisbasis h'mj'ni=mn: (1.12) andthus j i=1xn=1j'nian (1.13) fromwhichfollowthateachvectorcanberepresentedas h'mj i=1xn=1h'mj'nian=1xn=1mnan=am (1.14) Therelation(1.15)isonlyvalidiffj'nigisacompleteset,andwehavethecompleteness j i=1xn=1j'nih'nj i: Writingthecompletenessrelationas k k2=1xn=1jh'nj ij2: (1.16) allowstorepresenth j i=h j1j i=1xn=1h j'nih'nj i (1.17) 1=1Xn=1j'nih'nj; 4 (1.18)

scalarproduct whichiscalledthespectralrepresentationofthe1-operator.therepresentationofthe throughthecomponentsh'nj hxj iandh'njxiofthevectorsj i=1xn=1hxj'nih'nj i iandjxiwithrespectto (1.19) theorthonormalsystemfj'nigobtainedin(1.19)through"insertion"ofthe1-operator ofvectorsinorthonormalbasissystemsinaveryeconomicalway. asgivenin(1.18).thechosenbra-(h'j)andket-(j'i)notationallowstherepresentation 1.2 Arelation LinearOperatorsinH iscalledlinearoperatorinhifaj i=ja i=j 0i (1.20) assumethatda(therangeofa)isdenseinh. IngeneralAdoesnothavetobedenedonallvectors2H.Inthefollowingwewill A(aj i+bj'i)=aaj i+baj i: (1.21) 1.2.1 AdjointoperatorAy: DAisdenseinH;Ayisuniquelydened. Ay,ingeneraldierentfromA,iscalledthetoAadjointoperatorif(1.22)isfullled.If h'jaj i=hay'j i If 1.2.2 Hermitianoperators h'jaj i=ha'j 5 i (1.23)

Comparing(1.22)and(1.23)showsthatforahermitianoperatorAthereexistsalways thenaiscalledhermitian. anadjointoperatorwith atleastontherangedaofa.itcouldbethataydenedvia(1.22)existsonalarger rangedayda.ifthisisnotthecase,i.e.,day=da,thenaiscalledself-adjoint.it Ay=A (1.24) itisself-adjoint.sinceeigenvectors(oreigenfunctions)characterizequantummechanical shouldbementionedthatahermitianoperatoronlyhasacompletesetofeigenvectorsif systems,therequirementthatoperatorsareself-adjoint(notonlyhermitian),iftheyare supportedtocharacterizephysicalobservables,isquiterelevant. Foraself-adjointoperatorweobviouslyhave PropertiesofHermitianOperators:TheexpectationvalueofanobservableAinthe (Ay)y=A: (1.25) statej i(withk k=1)isgivenby Inorderfor(1.26)tobereal,onehastorequireAtobehermitian: hai =h h jaj j i i=h jaj i: (1.26) fromwhichfollowsthatforhermitianoperatorathequantityh h jaj i=ha j i=h ja i If jaj iisreal. thenj operatorsarereal. aiiscalledeigenvectortoawitheigenvaluea.theeigenvaluesofhermitian Aj ai=aj (1.27) orthogonal,i.e., Eigenvectorsj ai;j a0iofhermitianoperatorsatodierenteigenvaluesa6=a0are fora06=a. h a0j ai=0 (1.28) 6

Proof:From follows Aj ai=aj ai (1.29) and h a0jaj ai=ah a0j ai (1.30) andthus ha a0j ai=a0h a0j ai=ah a0j ai (1.31) Sinceaccordingtotheassumptiona06=a,i.e.,(a0?a)6=0,followsthath (a0?a)h a0j ai=0: a0j ai=0. (1.32) Self-adjointoperatorshaveaspecialroleinquantummechanics,sincetheyarerelatedto 1.2.3 Isometricandunitaryoperators physicalobservables.formanytheoreticalconsiderationsoneneedsinadditionso-called isometricoperators,whicharedenedas Becauseof h'j i=h'j i: (1.33) theyobviouslyfulllh'jyj i=h'j i=h'j1j i; (1.34) forinnitedimensionalvectorspacesthisisingeneralnotthecase.ifanoperatoru Innitedimensionalvectorspaces,therelation(1.35)wouldimplyy=1.However, y=1: fullls itiscalledunitary.anotherdenitionforanoperatortobeunitarycanbewrittenas UyU=UUy=1 (1.36) Uy=U?1: 7 (1.37)

1.3 Ifonehas MatrixRepresentationofLinearOperators usestherepresentationofthe1-operator(1.18)andmultipliesfromtheleftwithh'mj, oneobtains Aj i=j 0i (1.38) Hereh'nj iarethefouriercomponentsoftheexpansion(1.15)ofj Xnh'mjAj'nih'nj i=h'mj 0i: iwithrespectto (1.39) acompleteorthonormalsystemfj'nig.ifonedenes then(1.39)canbewrittenash'mjaj'ni=amn XnAmnan=a0m (1.40) abstracthilbertspaceassignseachvectoracolumnvectoroffouriercoecients: whichistheformofalinearmaprepresentedbymatrices.introducingabasisinthe (1.41) j i?! 0 B@ h'1j. h'nj ica 1 0 B@ a a2. 1 1 CA (1.42) andanoperatorcorrespondstothematrixrepresentationofalinearmap Hereallrulesderivedfromlinearalgebracanbeapplied. A?![h'mjAj'ni]=[Amn]: (1.43) Theeigenvectorsj'niofaself-adjointoperatorA 1.4 "A"-Representation Aj'ni=anj'ni 8 (1.44)

fordistincteigenvaluesandoingeneralnotformacompleteorthonormalsystem.ifthey doformacompleteorthonormalsystemfj vectorsforrepresentingotheroperators. Inthisparticularcaseonecanassigntoanarbitrarymap nig,thenonecanusethosevectorsasbasis therepresentation Xnh'mjBj'nih'nj Bj i=jb i=j 0i (1.45) i=h'mjb i=h'mj 0i: tation: Itiscalled"A"-Representation.Specically,theoperatorAisdiagonalinthisrepresen- (1.46) Thismeans:Alinearoperatorisdiagonalinitsownrepresentation.Ifoneusesthe eigenvectorsofthehamiltonianh,thenthisrepresentationiscalledh-orenergyrepresentation. Amn=h'mjAj'ni=mnan: (1.47) 1.5 Physicalobservablesareassignedtoself-adjointoperators,i.e.,onehasforpositionxand QuantumMechanicsinAbstractHilbertSpaces momentump x(t)?!x(t) p(t)?!p(t) whichobeythecommutationrelation (1.48) Allobservablesdependingonpandxcorrespondtoself-adjointoperators [P;X]=hi1: (1.49) e.g.,thehamiltonianoftheharmonicoscillatorisgivenas H=P2 A=A(P;X); (1.50) 2m+m2!2X2: 9 (1.51)

AquantummechanicalstateischaracterizedbyaHilbertspacevectorj tationvaluesofoperatorsinsuchastatearegivenas(providedk hai =h jaj i: k=1) i.theexpec- ThetimedependenceofanoperatorAisgiventhroughtheHamiltonianH(p;x)via (1.52) A=ih[H;A]: _ (1.53) Themeanvalue(expectationvalue)ofanobservableinagivenstatej 1.6 Root-Mean-SquareDeviation hai =h jaj i iisdenedas (withk themeanvalue(a?h k=1).theroot-meansquaredeviationfromthisexpectationvalueisgivenby jaj i2),i.e.,throughthenon-negativeexpression (1.54) (1.55)follows Itssquareroot(A) (A)2 iscalledroot-mean-squaredeviationorstandarddeviation.from =h j(a?h jaj i)2j i0: (1.55) andthus (A)2 =h ja2j i?2h jah jaj ij i+h jaj i2 (A)2 (1.56) Withthisdenition,oneprovesanessentialtheoreminquantummechanics =h ja2j i?h jaj i2: (1.57) zeroifj totheeigenvalue. Theroot-mean-squaredeviationofanobservableAinthestatej iiseigenvectorofa.theexpectationvalueinthisstatecorresponds iisexactly Letj aibeeigenvector,i.e., then Aj ai=aj ai; (1.58) h ajaj ai=a; 10 (1.59)

andthuswith(1.58):(a)2 Proofofinversedirection:Iftheroot-mean-squaredeviationofAiszeroforastatej a=h aj(a?a)2j ai=0: (1.60) then 0=(A)2 j(a?h jaj i)2j i i, =h(a?h =k(a?h jaj i) i) j(a?h k2 jaj i) i andthus (A?h jaj i)j i=0: (1.62) (1.61) Thismeansthatj valueofainthestatej Ifonepreparesanensembleofstatesexperimentallyinsuchawaythattheexpectation iiseigenvectorofawitheigenvalueh i. jaj i,i.e.,tothemean valueofastateisgivenwithzerodeviation(i.e.,sharp),whichmeansthatthisensemble BoundStates:Inquantummechanicsboundstateshavediscretevalues.Ifwedene Aj hasthesamevaluean,thentheensembleischaracterizedbytheeigenvectorj ni=anj ni. nivia boundstatej then,accordingtotheabovetheorem,onehas nivia Hj (H) ni=enj n=0; ni; (1.63) i.e.,boundstatesareobtainedbysolvingtheeigenvalueequationforthehamiltonianh. (1.64) LetAandBbehermitianoperators.WithSchwartz'inequalityfollows 1.7 UncertaintyPrinciple jh j(a?a)(b?b)j ij=jh(a?a) k(a?a) =A B kk(b?b) j(b?b) ik 11 (1.65)

HereAh jimh jaj j(a?a)(b?b)j i.theleft-handsideof(1.65)canbeestimatedfrombelowvia Onealsohas ijjh j(a?a)(b?b)j ij: (1.66) jimh j(a?a)(b?b)j j(a?a)(b?b)j ij j(a?a)(b?b)?(b?b)(a?a)j i?h(a?a)(b?b) ijj ij From(1.65),(1.66)and(1.67)followsthat =12jh jab?baj ij (1.67) Thus,ifAandBdonotcommute,theuncertaintyrelationgivesanestimateforthe 12jh root-mean-squaredeviationofaandb.thisisonlytrueifj j[a;b]j ij(a) (B): iisnotaneigenstateof (1.68) AorB.Ifj Becauseof(1.49)onehasforXandPoperators vanish,andtheequationwouldbemeaningless. iwouldbeaneigenstateofeitheroperator,thenbothsidesof(1.68)would h2=12h jhi1j ij= (P) 12jh (X) j[p;x]j ; ij i.e,independentof theproductofthedeviationsislimitedfrombelowas (1.69) AnimmediateconsequenceofthisrelationisthatneitherXnorPvanish.Thus, h2(p) accordingtothetheorem,noeigenvectorsexistforeitherpnorx. (X): (1.70) normalizeable.thismeanstheyarenotvectorsinahilbertspace. Remark:ItispossibletointroduceeigenvectorstoPandX;however,thosearenot 12

Chapter2 SymmetriesI 2.1 Accordingto(1.53)thetimedependenceofanoperatorAisgiventhroughtheHamiltonianH(P;X)via TheobservableAiscalledconstantofmotionofthesystem,if (i)aiscompatiblewithh,i.e. A=@A _ @t=ih[h;a] (2.1) ConstantsofMotion (ii) [H;A]=0 @A @t=0 (2.3) (2.2) Condition(ii)statesthatAdoesnothaveanyexplicittimedependence.(Moreontime dependencelater.) Ingeneral,symmetriesorinvariancepropertiesleadtoconservationlaws.Therearetwo distinctkindsofsymmetries: onlyslightlyfromunity Letaninnitesimalunitarytransformationdependonarealparameter"andvary discretesymmetriesandcontinuoussymmetries. ^U"(^G)=1+i"^G; 13 (2.4)

where^giscalledthegeneratoroftheinnitesimaltransformation. ^U"isunitaryonlyif"isrealand^GHermitian ^U"^Uy"=(1+i"^G)(1?i"^G)y onlyif =1+i"(^G^Gy)+O("2) ^G=^Gy: (2.6) (2.5) Applyonstatevector TransformationoftheoperatorA: ^U"j i=(1+i"^g)j i=j i+i"^g)j i=j i+j i (2.7) ThenA0=Aonlyif[^G;A]=0. A0=^U"A^Uy"=(1+i"(^G)A((1+i"(^G)y 'A+i"[^G;A] (2.8) Finiteunitarytransformationscanbebuiltfrominnitesimaltransformationsby successiveapplication Theoperator^Uisunitaryifisrealand^GHermitian.Then ^U(^G)=lim N!1NYk=1(1+iN^G)=lim N!1(1+i^G)N=ei^G: (2.9) ForthetransformationofanoperatorAoneobtains ei^gy=e?i^g=ei^g?1: ei^gae?i^g=a+i[^g;a]+(i)2 (2.10) onlyif[^g;a]=0. 2![^G;[^G;A]]+ (2.11) Considerthefollowingexamples: 1.TimeTranslation 14

Set ^G1hH andapplythistoastate ^Ut=1+ihtH "=t 1+ihtHj (2.12) onehas wheretheexplicitformoftheschrodingerequation,hj (t)i=j(t)i?t@@tj (t)ij (t)i=ih@@tj (t?t)i; (t)iwasused.thus (2.13) wherehisthegeneratoroftimetranslations. ^Utj (t)i=1+ihthj(t)i=j(t?t)i; (2.14) Considerforsimplicationone-dimensionalspacetranslationsinx-direction.Then 2.SpaceTranslation Applicationonastatevectoryields^U"=1+ih"^Px ^G1h^Px 1+ih"^Pxj (2.15) (x)i=j j (x)i+ih"^pxj (x+"i; (x)i+"@ @xj(x)i (x)i wherethecoordinatespacerepresentation^px=?ih@ gets @xwasused.fortheoperatorxone (2.16) X0=^U"(^Px)X^U?1 "(^Px)=1+ih"^PxX1?ih"^Px X+ih"[^Px;X] Fornitetranslationsonehas =X+": (2.17) ^Ua(P) (r)=eihap 15(r)= (r+a); (2.18)

transformationexp(i^g).thenthehamiltoniantransformsas whereexp(ihap)isthegeneratorofnitetranslations. Ingeneral,symmetriesandconservationlawsarecloselyrelated.Considertheunitary H0=ei^GHe?i^G=H+i[^G;H]+(i)2 doesnotexplicitydependonthetime,i.e.^gisaconstantofmotion.thatis Onlyif[^G;H]=0followsH0=H.Wehaveshownbefore,thatif[^G;H]=0,then^G 2![^G;[^G;H]]+ (2.19) ddt^g=1 ih[[^g;h]+@^g @t=0: (2.20) Therstexampleofadiscretesymmetryisinversion. 2.2 Inversion Denition:Inversionisdenedastransformation whichmeansforcartesiancoordinates ~x!~x0=?~x; (2.21) intoleft-handedonesandviceversa. Thegeometricalinterpretationisthatinversionturnsright-handedcoordinatesystems (x;y;z)!(x0;y0;z0)=(?x;?y;?z) (2.22) Studyingthesymmetrypropertiesofasystemunderinversionmeansconsideringthe behaviorofthehamiltonianunderthetransformation ConsiderthekineticenergyoperatorP2 2m=?h2 H(~X;~P2)!H(?~X0;~P2)H0(~X0;~P2): 2m,whichisgivenas (2.23) 2mr2=?h 2m(@2 @x2+@2 @y2+@2 writesthehamiltonianas Obviously,thekineticenergyoperatorisinvariantif(x;y;z)!(?x;?y;?z).Ifone @z2): (2.24) H(~X;~P2)=P2 162m+V(~X) (2.25)

oneonlyneedstoconsiderthetransformationpropertiesofv(~x).anycentralpotential V(~X)V(jj~Xjj)isinvariantunderinversion.Apotentialoftheform where~aisanarbitraryvectorisinvariantunderinversionprovidedv1=v2.ifhis invariantunderinversion,i.e.h(~x;~p2)=h(?~x;~p2); V(~X)V1(j~x+~aj)+V2(j~x?~aj); (2.26) howdowavefunctionsh~xji(~x)behave? (2.27) Deneanoperator(parityoperator)Psuchthat Applying~x!?~x0gives P(~x)=0(~x)=(?~x): (~x)=(~x0)=0(~x0): (2.28) Combining(2.28)and(2.29)gives (~x)=0(~x0)=p(~x0)=p(?~x)=p2(~x): (2.30) (2.29) Since(2.30)holdsforanyji,theoperatorPhastofulll Thus LetbetheeigenvalueoftheoperatorP,thenP2musthavetheeigenvalue2=1. P2=1: (2.31) fromwhichfollows P(~x)=(?~x); P(~x)=(~x) (?~x)=(~x); (2.33) (2.32) >FromP2=1follows i.e.parityeigenstateswitheigenvalue+1(?1)areeven(odd)functionsof~x. Thespecicchoiceof(~x)H(~x) P(~x)=PH(~x;::) (~x)gives (~x)=h(?~x;::) P?1=P (?~x) (2.34) =PH(~x;::)P?1P 17(~x)=PH(~x;::)P?1 (?~x): (2.35)

Since (~x)isarbitrary,onehastheoperatorequation whichleadsto PH(~x;::)P?1=H(?~x;::); [P;H(~x;::)]=0 (2.37) (2.36) Thus,iftheHamiltonianisinvariantunderinversion,itcommuteswiththeparityoperator,andparityisaconstantofmotion. Considerboundstateswithdiscreteeigenvaluesasgivenin(1.64):IfHisinvariantunder inversiononeobtainsfromh(~x;::) H(~x;::) n(~x)=en n(?~x)=en n(~x)whenapplyingp ifh(~x;::)isinvariantunderinversion. Twodierentcasesarise: n(?~x): (2.38) Then constantfactor~: (a)theeigenvalueenisnon-degenerate. (~x)and(?~x)areessentiallythesamefunction,andcandieratmostbya Applying~x!?~xgives (~x)=~ (?~x)=~ (?~x)=~2 (~x): (~x): (2.39) theparityoperatorp,andthus Thus,~2=1and~1,whichshowsthat~isthepreviouslyintroducedeigenvalueof (2.40) oddfunctionsof~x,havingeithereven(=+1)orodd(=?1)parity. Thismeansthattheeigenfunctionstoanon-degenerateeigenvalueEareeitherevenor (?~x)=(~x): (2.41) (b)theeigenvalueenisdegenerate. If(2.38)togetherwith(1.64)holds,thenanylinearcombination isalsoeigenfunctionofh(~x;::)tothesameeigenvalueen.onecanusethisfreedomto chooselinearcombinationswhichareevenoroddparitystates,i.e. a (~x)+b(?~x) (2.42) (~x) (?~x) (2.43) 18

2.3 TheHamiltonianoftheclassicalharmonicoscillatorisaquadraticfunctionofposition Ladder-OperatorsandtheU(1)Symmetry andmomenta.inthesimplestcase(providesm=!1),itreads whichcanbedecomposedas Hclass=12(p2+x2); (2.44) where Hclass=12(x?ip)(X+ip)=aa; a:=1p2(x+ip): (2.46) (2.45) InthequantummechanicsonedenesacorrespondingoperatorA A:=1 DuetotheoperatorcharacterofA,thefactorizationoftheHamiltonianisslightlymore p2(x+ip): (2.47) complicated,andoneobtainsanadditiveterm12; Thesamefactorizationcanbeappliedtothedierentialequationfortheharmonicoscillator wherethedierentialoperatorcanbedecomposedas d2x(t) H=12(P2+Q2)=AyA+121: (2.48) d2 dt2 +!2x(t)=0; (2.49) dt2+!2= ddt?i!! ddt+i!!: Thusoneobtainssolutionsto(2.49)ifonesolvesoneofthefollowingrst-orderdierential (2.50) equations: ddt?i!!x(t)=0 or ddt+i!!x(t)=0: (2.51) 19

Thesolutionsofthoseare Finally,thereisaspecicsymmetrypropertyin(2.44)ifoneconsidersthatinthe(x;p) x(t)=x0ei!t and x(t)=x0e?i!t: (2.52) changeifonecarriesoutarotationinthephasespacegivenby phasespacehclassdescribesthe"length"ofavectorinthatspace.thus,hclassdoesnot x0=xcos+psin ThisrotationleavesHclassinvariant,i.e.,Hclass(x0;p0)=Hclass(x;p).Forthequantities p0=?xsin+pcos: (2.53) afrom(2.45),thecorrespondingsymmetrytransformationisgivenby Theanalogoussymmetrycarriesoverintoquantummechanics,sinceN=AyAisinvariant a0=eia and a0=e?ia: (2.54) under Accordingtothegeneralsymmetryprincipleinquantummechanics,thissymmetryoperationmustbegeneratedbyaunitaryoperator.Thus,aunitaryoperatorU()must exist,whichcarriedaintoa0,a0=u()yau(): U()canbeexpressedasexponentialofaHamiltonianoperator (2.56) A0=eiA and Ay=e?iAy: (2.55) andonehastriviallyfor=0 U()=eiX() (2.57) Importantisthatthetransformationintroducedin(2.56)formsagroup,i.e.,carrying U(0)=1 and X(0)=0: (2.58) outtransformationsofthiskindgivesanotheroneofthesamekind, Especially,theinverseelementexists: U()U()=U(+): (2.59) U(?)=U?1(): 20 (2.60)

Thegeometricinterpretationofthisgroupisobviousifoneconsidersthe(x;p)formofthe d=1;u(1). thecomplexformof(2.55),theunderlyinggroupistheunitarygroupwithdimension transformationinphasespace.theoperatoru()describesarotationinthe(x;p)plane andthegroupiscalledo(2)orthogonalgroupintwodimensions.ifoneconsiders Oneobtainsimportantinsightintothequantumtheoryifoneexplicitlyconstructsthe valuesoftheparameter,i.e.,so-calledinnitesimaltransformations. Becauseof(2.58),onecanexpand operatoru().thisprocedurefollowsthegeneralrulethatoneconsidersrstsmall Applyingtheseto(2.56)gives X()=Y+ and ()=1+iY+ (2.61) Thus A0=(1?iY)A(1+iY)A+i(AY?YA): (2.62) Thismeansthattheinnitesimaltransformationisdeterminedbythecommutator[A;Y]. Thus,Yiscalledtheinnitesimalgeneratorofthegroup(hereU(1)).Uptonowwe A0=A+i[A;Y]+::: (2.63) havenotusedanyspecicpropertiesofthegroupu(1).thiscomesintoplaywhenwe expand(2.55)inpowersofa0=eiaa+ia+::: Comparing(2.64)with(2.63)leadsto[A;Y]=A (2.64) whichisanequationfory,whichcanbesolvedbyusing (2.65) Thesolutionis(uptoanadditivec-number) [N;A]=?A: (2.66) Thus,thenumberoperatorNistheinnitesimalgeneratorofthegroupU(1). Y=N: (2.67) 21

Thus,onehasalinearapproximationX()N.Becauseof(2.59)X()depends linearlyon,sothatx()=?nisexact.therefore, isanexactrepresentation. U()=e?iN (2.68) Theresultsderivedfortheharmonicoscillatorhavenumerousapplicationsinmodern physics,sincetransformationsoftheform(2.56)andoperatorswiththepropertiesof charge,baryonnumber,strangeness,etc.inallcases,thequantitiesare"quantized," Nappearinmanyareasoftheoreticalphysics.Examplesareparticlenumber,electric understandingquantumnumbers. i.e.,havediscretevalues,whichare,ingeneral,integers.thus,wehaveaparadigmfor 22