1、University of Electronic Science and Technology of China 2005-3-1 Prof.Zhang XiaoxiaReviewUniversity of Electronic Science and Technology of China 2005-3-1 Prof.Zhang XiaoxiaChapter 4 Dirac Notation and Operator TechniquesThe Operation Rules of OperatorThe Momentum Operators and The Angular Momentum
2、 OperatorsThe Eigenvalues and Eigenfunctions of Hermition Operation The Relation between Operation and Dynamical Quantities Dirac Notation15234backUniversity of Electronic Science and Technology of China 2005-3-1 Prof.Zhang XiaoxiaThe Definition of Operator The Character of OperatorThe Operation Rul
3、es of OperatorbackUniversity of Electronic Science and Technology of China 2005-3-1 Prof.Zhang Xiaoxiawhen an operator acts on a wave function it produces a new wave function u=voperator acts on a wave function u,it produces a new wavefunction v，is operator.1）du/dx=v，d/dx is an operator.2）x u=v，x is
4、 also an operator.An operator must acts on a wave functionThe Definition of OperatorbackUniversity of Electronic Science and Technology of China 2005-3-1 Prof.Zhang Xiaoxialinearity Operator(c11+c22)=c11+c22Equal OperatorIf =，So =for arbitrary functionsbackThe Character of OperatorPlus Operator(+)=+
5、=+=for arbitrary functionsUniversity of Electronic Science and Technology of China 2005-3-1 Prof.Zhang XiaoxiaMultiply Operator()=()=for arbitrary functionsUsually and do not commuteis The commutation relations:do not commute xxxExampleoperatorspi(1)()xxxxpx ii x xxxxxpp xixpp xi（）xxxxiixixp )()2(ba
6、ckIn fact the momentum and positionoperators do not commute is at the heart of quantum mechanics.the canonical commutation relationUniversity of Electronic Science and Technology of China 2005-3-1 Prof.Zhang XiaoxiaThe commutation relationsbetween momentum and position operators yyzzypp yizpp zi0000
7、00000 zxxzyzzyxyyxyyxxzzxxzzyyppppppppppppzppzzppzyppyyppyxppxxppxzyxppppixppx,0 These commutation relations are at the heart of quantum mechanics.backUniversity of Electronic Science and Technology of China 2005-3-1 Prof.Zhang Xiaoxiacommutation bracketcommutation bracket,-1),=-,2),+=,+,3),=,+,4),+
8、,+,=0The last equation is Jacobi identical equation.ipx,backThe commutation relationsbetween momentum and position operatorsUniversity of Electronic Science and Technology of China 2005-3-1 Prof.Zhang Xiaoxiadefinition of*dUdU transpose operators：xxExample：*xdxProof:When|x|,When|x|,0 00)(*xxdxxxxx 0
9、)(xxpp 、are arbitrary are arbitrary functions functions *xdx xdx*|*xdx*Transpose operatorsbackUniversity of Electronic Science and Technology of China 2005-3-1 Prof.Zhang XiaoxiaThe hermitian conjugate operators *)(*OdOd *)(*OdOdTranspose operators*OO Also writeThe hermitian conjugate operators+:()+
10、=+(.)+=.+*)(*Od *Od *OdbackUniversity of Electronic Science and Technology of China 2005-3-1 Prof.Zhang XiaoxiaThe hermitian operatorsThe hermitian operators:*()*dOdOOOorcharacter+=,+=(+)+=+=(+)()+=+=only ,=0,()+=backUniversity of Electronic Science and Technology of China 2005-3-1 Prof.Zhang Xiaoxi
11、aThe Momentum Operators The momentum operator is hermitian The eigenvalue equation of momentum The universe NormalizableThe Angular Momentum Operators The Angular Momentum Operators The eigenvalue equation of Angular momentum The commutation relations of Angular momentumbackThe Momentum Operators an
12、d The Angular Momentum OperatorsUniversity of Electronic Science and Technology of China 2005-3-1 Prof.Zhang XiaoxiaThe Momentum OperatorsThe momentum operator is hermitiandxidxpdxdx )(*The eigenvalue equation of momentum)()(rpripp )()()()()()(rprirprirpripzpzpypypxpx Proof：dxiidxd*)(|*dxidxd *)(dxp
13、x *)(When|x|,When|x|,0 0University of Electronic Science and Technology of China 2005-3-1 Prof.Zhang XiaoxiaI.Solutions)()()()(zyxrp zdzzdziydyydyixdxxdxippp)()()()()()(rpzpypxpppppiziyixizyxceecececzyxzyxr 321)()()()()()()()()()()()()(321zeczyecyxecxzziyyixxipzppyppxp )()2(|)()(32)(22*ppcdecdeecdrr
14、rpprprpppiii II.Normalizable coefficient)()(rpripp University of Electronic Science and Technology of China 2005-3-1 Prof.Zhang XiaoxiaxyzAAoLThe universe Normalizableperiodicity boundary condition22zpypLpizpypLpizyxzyxcece 12120,1,2,xip Lxxxxxenp LnpLnWe find：22,0,1,2,yzyzyznnppLLnnSimilarly：zyLrA,
15、2 zyLrA,2University of Electronic Science and Technology of China 2005-3-1 Prof.Zhang Xiaoxia222)()(zyxnnnprppLznLynLxnizyxicercer 1*322/2/22/2/LcdcdLLppLL rpVrpLnnniizyxee 12/31)(c=L-3/2，Wave functionsUniversity of Electronic Science and Technology of China 2005-3-1 Prof.Zhang XiaoxiaThe Angular Mo
16、mentum OperatorsThe Angular Momentum OperatorsprL riprLThe cartesian coordinates )()()(xyxyzzxzxyyzyzxyxipypxLxzipxpzLzyipzpyL22222222)()()()()()(xyzxyzxyzxyzzyxyxxzzypypxpxpzpzpyLLLL The square of Angular Momentum OperatorsThe classical angular momentum corresponding quantum operatorUniversity of E
17、lectronic Science and Technology of China 2005-3-1 Prof.Zhang Xiaoxia )3(/tan)2(/cos)1(cossinsincossin2222xyrzzyxrrzryrx 123,iiiiffrffxrxxxx xxx y z whererxrxxxryryyyrzrzzzo r cossinsincossinzrsyrxr rxzThe spherical coordinate system.r yThe spherical coordinate sin1sincos1coscos1rzryrx 0sincos1sinsi
18、n1zryrx University of Electronic Science and Technology of China 2005-3-1 Prof.Zhang Xiaoxia iLiLiLzyxsincotcoscoscotsin 0sin1cossincos1sincos1sinsinsinsin1coscos1cossin rrzrrryrrrxThe cartesian componentsof the angular momentumvector in terms of spherical coordinates yieldingsin1)(sinsin122222 LUni
19、versity of Electronic Science and Technology of China 2005-3-1 Prof.Zhang XiaoxiaThe eigenvalue equation()()()()izzzldLildce S olu tion：The eigenvalue equation:Lz)2()(2112|2202220 ccdcd)(02120mndeeinim )2(zizillcece1/2sin/2cos2 zzllilezi 220,1,2,zlmm so,2,1,0 mmlzmninimdee 2021Normalizable coefficie
20、ntUniversity of Electronic Science and Technology of China 2005-3-1 Prof.Zhang XiaoxiaThe eigenvalueThe eigenfunctions,2,1,021)(memlimmz *()*zzLdLd didLz )(*20Equal zero0zzill ，（）C*(2)(2)(0)(0)0(2(0)(0)(2)）or1)0()2(Periodicity boundary condition dii *)(|*2020 dii *)(|*2020 dLiz *)(|*2020University o
21、f Electronic Science and Technology of China 2005-3-1 Prof.Zhang XiaoxiaThe eigenvalue equation:L L2 22222222222(,)(,)11(sin)(,)(,)sinsin11(sin)(,)(,)sinsinLYYYYYY or：=（+1),where =0,1,2,.lmYYlmePNYmlmlmimmllmmlm ,3,2,1),()1(),(,2,1,0)(cos)1(),(*20*01sin),(),(ddYYlmlm|)!|(4)12(|)!|(mllmlNlm Solution:
22、YSolution:Yl l m m(,)University of Electronic Science and Technology of China 2005-3-1 Prof.Zhang XiaoxiaThe orthonormal condition：20*0sin),(),(mml lmllmddYYThe degeneration of the eigenvalue：m:0,1,2,3,.,(2 +1)lmYYlmePNYmlmlmimmllmmlm ,3,2,1),()1(),(,2,1,0)(cos)1(),(*University of Electronic Science
23、 and Technology of China 2005-3-1 Prof.Zhang Xiaoxia,zyxzpxpzpzpy The commutation relations of Angular momentum,zxyzyxpxpzpzpyLL Proof,yzxzxyLLi LL Li LSimilarly,zxyzxzpxpzpzpxpzpy ,zyxyzzxzpxpzpzpzpxpypzpy zyxLiLL,yzzyzxxzppxzpxpzppzypzpy,yzxzppxzpzpy,yzyzxzxzppxzppzxpzpyppyz,yxpixpiy)()(xypypxi zL
24、i 123,1123,L LiLcalled Levi Civitax y z iswhere，orUniversity of Electronic Science and Technology of China 2005-3-1 Prof.Zhang XiaoxiaHome worksChapter 4P127:6,8,University of Electronic Science and Technology of China 2005-3-1 Prof.Zhang Xiaoxia Average value of the hermitian operators The eigenval
25、ue equation of the hermitian operators The orthogonality of the eigenfunctions for hermitian operators ExampleThe Eigenvalues and Eigenfunctions of Hermition OperationbackUniversity of Electronic Science and Technology of China 2005-3-1 Prof.Zhang XiaoxiaThI：Average value of the hermitian operators
26、must be real.Proof：FdF*)(Fd*Fd*F Average value of the hermitian operatorsUniversity of Electronic Science and Technology of China 2005-3-1 Prof.Zhang Xiaoxia dFFFFF222)(*)()(FFhermitian operatorsMust be realMust be realFF 22*FdF0|)(|)(222 dFFdFF FFd*)(2|Fd0 The eigenvalue equation of0)(2 F()0FFForco
27、nstantnnnFF ProofProof：The eigenvalue equation of the hermitian operatorshermitian operatorsUniversity of Electronic Science and Technology of China 2005-3-1 Prof.Zhang Xiaoxia nnFdF *ThII：The Eigenvalues of Hermition Operation are all to be realProof nnndF *nF.realisFsorealbemustFn，According theore
28、m I(I)All physically observable quantities are associated with hermitian operators.),(prFF ipprrr),(),(prFFprFF (II)When the state is an eigenstate of an operator,measurement of its associated observable must yield the eigenvalue belonging to that eigenstate.,2,1 nFFnnn University of Electronic Scie
29、nce and Technology of China 2005-3-1 Prof.Zhang XiaoxiaTH III：The eigenfunctions are orthonormal,if the eigenvalues are not degenerate.Proof:mmmnnnFFFF dnn*)*(mmmFF dFdFnmmnm*)(dFdFdFnmnnmnm*)(0*)(dFFnmnmFmFn，0*dnm1.The orthonormal condition for quantum set：mnnmnmnnddd *0*1*2.The orthonormal conditi
30、on for continue：)(*dThe orthogonality of the eigenfunctions for hermitian operatorsUniversity of Electronic Science and Technology of China 2005-3-1 Prof.Zhang XiaoxiaThe expectation values of dynamical quantitiesThe average of dynamical quantities（1 1）The eigenfunctions of dynamical quantities oper
31、ation form a complete set.（2 2）The expectation value and probability of t the dynamical quantities（3 3）The condition for dynamical quantities have certain value The Relation between Operation and Dynamical QuantitiesbackbackExampleUniversity of Electronic Science and Technology of China 2005-3-1 Pro
32、f.Zhang Xiaoxia(1)The eigenfunctions of dynamical quantities operation form a complete set.pdrpcrorpdrtpctrpp33)()()()(),(),(Example：The eigenfunctions of momentum operation form a complete set.The expectation values of dynamical quantities()()nnnxcxUniversity of Electronic Science and Technology of
33、 China 2005-3-1 Prof.Zhang Xiaoxia（2）The expectation value and probability of the dynamical quantities,2,1)()(nxxFnnn )()(xcxnnn dxxcxdxxxnnnmm)()()()(dxxxcnmnn)()(*mmnnncc dxxxcsonn)()(University of Electronic Science and Technology of China 2005-3-1 Prof.Zhang XiaoxiaThe condition for dynamical qu
34、antities have certain value=m,2,1)()(mnxxFnnn )()(xcxnnn m，|cm|2=1,|c1|2=|c2|2=.=0)()()(xxcxmnnn mnmncn01|2University of Electronic Science and Technology of China 2005-3-1 Prof.Zhang Xiaoxia dxxFxF)()(*dxxcFxcmmmnnn)()(dxxFxccmnmmnn)()(*dxxxccmnmmnmn)()(*nmmmnmncc *nnnc 2|iiixxxxxxxx 22112106110421
35、1064nnncF 2|dxxxdxxFxFccFnnnnn)()()()(|*22 dxxFxF)()(*The average of dynamical quantitiesUniversity of Electronic Science and Technology of China 2005-3-1 Prof.Zhang XiaoxiaExample 1Example 1：),(32),(312111 YY findfind：（1 1）Is Is the eigenstatethe eigenstate of L of L2 2 or not or not？（2 2）Is the ei
36、genstate of L Lz z or not？（3 3）The average of L2 ；（4 4）The expectation value and probability of L L2 2 and L and Lz z Solution:Solution:),(32),(31)1(211122 YYLL 212112)12(232)11(131YY 211122312YY University of Electronic Science and Technology of China 2005-3-1 Prof.Zhang Xiaoxia ),(32),(31)2(2111 Y
37、YLLzz21113231YY 21113231YY（3 3）The average of L L2 2I IdxxFxF)()(*dc *21 dYYYYc2111211123231*3231 dYYYYYYYYc11212111212111112*92*92*94*9122959491cc 53 cUniversity of Electronic Science and Technology of China 2005-3-1 Prof.Zhang Xiaoxia 21113231YYc dLL2*2 dYYLYY211122111251*251 dYYYY2121122111262*25
38、1 dYY221221122425122252624251 IIII 2111251YY nnncFuse2|222222526652251 L 21112111251323153YYYY 5451222262cL（4 4）12cLzUniversity of Electronic Science and Technology of China 2005-3-1 Prof.Zhang XiaoxiaExample 2Example 2：When t=0 When t=0，(x)=A sin(x)=A sin2 2kx+(1/2)coskx kx+(1/2)coskx Find the aver
39、age of kinetic energy and momentum.Find the average of kinetic energy and momentum.2224ikxikxikxikxAeeee Solution:)()()(21221ikxikxikxikxieeeeAx )()()()()()(543215432121xpixpixpixpixpiepcepcepcepcepcx kpkpkpkpp 54321220kpkpkpkpp 54321220University of Electronic Science and Technology of China 2005-3
40、-1 Prof.Zhang Xiaoxia 24)()(24)()(242)(54321ApcpcApcpcApc1|11)1()1(2216|)(|2222222251 AApcii kpkpkpkpp54321220 1 A 42)()(42)()(22)(54321pcpcpcpcpcUniversity of Electronic Science and Technology of China 2005-3-1 Prof.Zhang Xiaoxia1 1）the average of momentum kpkpkpkpp54321220 42)()(42)()(22)(54321pcp
41、cpcpcpc0)(4242)2(42242022|)(|22222251kkkkppcpiii2 2）the average of kinetic energy 85)(81)(81)2(81)2(81021)(42)(42)2(42)2(42022212|)(|2222222222222222251kkkkkkkkkppcTiii University of Electronic Science and Technology of China 2005-3-1 Prof.Zhang XiaoxiaHome worksChapter 4P128:11,12,University of Ele
42、ctronic Science and Technology of China 2005-3-1 Prof.Zhang XiaoxiaThe basic simplification of the Dirac notation is to introduce an abstract state that is an eigenvector of the position operator.That iswhere the state x is called a ket and its conjugateis called a bra.We require thatDirac Notationx
43、xxx|*|xx|()x xxx|1xdxxUniversity of Electronic Science and Technology of China 2005-3-1 Prof.Zhang Xiaoxia*(,)|()(,)()()|()|()1(,)(,)1mnmnmnmnwavefunctionx ttoperatorF riFux ux dxnormalizationQQeigenfunctionttx tx t dx*()()()|()()()()|1|1()()()qqnnnnnqqorthogonalux ux dxqqqqqquxuxxxQQqdqquxux dqxx*(,)(,)(,)|()|()(,)()()|xx tF x px ttFtF rprrFFFdxFF*|(,)(,)(,)|()|()mnmnmnFFdxFm F ndir tH rir titHttdt We can now define a wave function to be
