1、Chapter 3Relativistic Quantum MechanicsIntroductionNon-relativistic quantum mechanics relativistic quantum mechanicsSchrdinger equation Klein-Gordon equation S integerDirac equation S half integerSpin is automatically contained in Dirac equation3.1 Klein Gordon equationLorentz transormation time,spa
2、ce are of the same weightK G equation3.1 Klein Gordon equation3.1 Klein Gordon equation3.1 Klein Gordon equation3.1 Klein Gordon equationDiscussionNegative energy instable3.1 Klein Gordon equationNegative probability22*22*2221()ctt 3.1 Klein Gordon equation3.1 Klein Gordon equationNon-relativistic l
3、imit:K-G eq Sch eq3.1 Klein Gordon equation3.1 Klein Gordon equation22*2*()()2iimcimcmctt 3.1 Klein Gordon equationWith electromagnetic field3.1 Klein Gordon equation3.1 Klein Gordon equationCovariant form3.1 Klein Gordon equation3.1 Klein Gordon equation3.1 Klein Gordon equation3.2 Dirac equationHo
4、w to overcome the negative probability difficulty3.2 Dirac equation3.2 Dirac equation3.2 Dirac equation3.2 Dirac equationThe condition for and 1)They must follow the relation2)Operator H must be Hermitian3)Lorentz invariance22224Ec pm c3.2 Dirac equation3.2 Dirac equation3.2 Dirac equation3.2 Dirac
5、equation4 anti-commute matrices and 44 matrices3.2 Dirac equation3.2 Dirac equationConservation law of the probability flux3.2 Dirac equation321kkiii cmctx 321kkiii cmctx 321kkiii cmctx 3.2 Dirac equation31()()kkiii ctx ,kkjc 3.3 solutions of the free particle3.3 solutions of the free particle3.3 so
6、lutions of the free particle3.3 solutions of the free particle3.3 solutions of the free particle3.3 solutions of the free particle3.3 solutions of the free particle3.3 solutions of the free particle3.3 solutions of the free particleDiscussionPositive energy state(=+1)Negative energy state(=-1)Eigens
7、tates of momentum p2224,1pEc pm c 2224,1pEc pm c 3.3 solutions of the free particleOrbital angular momentum is not conserved3.3 solutions of the free particle1,()0 xxxxxyyzzxxxxyyzxzyzzyxdLL HdticLpppicLpLpLpicppcp()dLcpdt3.3 solutions of the free particleSpin angular momentum0(,)0iiiix y z 00 Or3.3
8、 solutions of the free particle,0,0ii,2,2,2xyzxzxzxyiii3.3 solutions of the free particle,2()2()xxxyyzzzyyzxHcpcppicppicp ,()2Hi cp 3.3 solutions of the free particle3.3 solutions of the free particleHelicity operator3.3 solutions of the free particle3.3 solutions of the free particleIf ,we find0,0,
9、ppEigenvalues:/23.3 solutions of the free particleEigenstates:3.3 solutions of the free particle3.3 solutions of the free particleDirac hole theory Dirac seaHole:(+Ep0,+m0,+e0)(positron)1932,Anderson discovered positron from cosmic ray using cloud chamber3.4 Dirac equation in the central force field
10、Equation in non-relativistic limit3.4 Dirac equation in the central force field3.4 Dirac equation in the central force field22cpmcEV 3.4 Dirac equation in the central force fieldIn non-relativistic approximation22EVmc22cpxmcEV 3.4 Dirac equation in the central force field21122EVpmcmc 22221()()()()24
11、1()()4Eppppmm cp VpVEm c 3.4 Dirac equation in the central force field3.4 Dirac equation in the central force field3.4 Dirac equation in the central force fieldNoting:up to the orderNormalization condition must be ensured3.4 Dirac equation in the central force field3.4 Dirac equation in the central
12、force field2222222(1)(1)88spm cm c3.4 Dirac equation in the central force field222222222222(1)81()()(1)2448sspEm cpE ppVp Vpmm cm cm c 3.4 Dirac equation in the central force field3.4 Dirac equation in the central force fieldBy using3.4 Dirac equation in the central force fieldRelativistic correctio
13、n of kinetic energy3.4 Dirac equation in the central force fieldThomas termDarwin term3.4 Dirac equation in the central force field3.4 Dirac equation in the central force fieldQuantum number K3.4 Dirac equation in the central force field3.4 Dirac equation in the central force field,2K HHcpcLpcpcLpcp
14、 501003.4 Dirac equation in the central force field()pLipL3.4 Dirac equation in the central force field3.4 Dirac equation in the central force field3.4 Dirac equation in the central force field3.4 Dirac equation in the central force field22234LJL3.4 Dirac equation in the central force field3.4 Dirac
15、 equation in the central force fieldRadial equations3.4 Dirac equation in the central force field1122112210012121jjjmmmjjjllllmlmyYYll 3.4 Dirac equation in the central force field3.4 Dirac equation in the central force field3.4 Dirac equation in the central force fieldWe take12()();()()u rrg ru rrf
16、 r3.5 Solution of the Dirac equation in the Coulomb fieldMotivationDiscussion the Hydrogen atomFine structure3.5 Solution of the Dirac equation in the Coulomb field3.5 Solution of the Dirac equation in the Coulomb field11211121()0()0qqqqqqqqsqaabbsqbbaa 1020smmmsmmmuebuea3.5 Solution of the Dirac eq
17、uation in the Coulomb field0000()0()0sabsba3.5 Solution of the Dirac equation in the Coulomb field12nnab 3.5 Solution of the Dirac equation in the Coulomb field112121()()0nnsnaasnab 222()()mcEsnE3.5 Solution of the Dirac equation in the Coulomb fieldn=0,1,2,n=1,2,33.5 Solution of the Dirac equation
18、in the Coulomb field22222222222311()1()1()13()224ionEmcEmcZnZZmcZnnn3.5 Solution of the Dirac equation in the Coulomb fieldGround state 1S1/2(n=0,=-1,n=1,j=1/2)3.5 Solution of the Dirac equation in the Coulomb field3.5 Solution of the Dirac equation in the Coulomb field3.5 Solution of the Dirac equa
19、tion in the Coulomb fieldQuestionDirac eq+non-relativistic limit Sch eq Z137?No limit for ZUniform charged sphere?No limit for ZFine structure Enj En for Sch eq3.6 Klein paradoxAnother question for the non-relativistic limit of Dirac equationDoes positron existScalarlike potential and vectorlike pot
20、ential3.6 Klein paradox3.6 Klein paradox3.6 Klein paradoxAt the infinity,the wave function is not zero,which means that there is only the scattering state solution instead of bound state solution3.6 Klein paradoxDirac equation(non-relativistic limit)Sch eqV(r)=grV(r)=gr|roscillating|r0Scattering sta
21、tesBound statesCannot confine quarksconfinement3.6 Klein paradoxThe physics of Klein paradox3.6 Klein paradox3.6 Klein paradox313100kkkkkkEeVmcicxEeVmcicx0000eVVzeVz3.6 Klein paradox3.6 Klein paradox3.6 Klein paradox3.6 Klein paradox3.6 Klein paradox0220(2)()()()riiV cE cpuuFuV cppriuu F2202202()()(
22、)rriiV cu uuE cp uV cpp2rriiiipccuuu vuu uE3.6 Klein paradox3.6 Klein paradox3.6 Klein paradox3.6 Klein paradox3.6 Klein paradox3.6 Klein paradox2000000222002222222lim()lim()2()()VEVVVVVccccEEccEEEcccV mR Vpppmcppp3.6 Klein paradoxIf p=mc3.6 Klein paradoxThe explanation of Klein paradox3.6 Klein par
23、adox3.6 Klein paradox3.6 Klein paradox3.6 Klein paradox3.6 Klein paradox2*1122*122*220222zrefIzIIzp cja aeEmcp cjC CeEmcp cjb beVEmc 3.6 Klein paradox3.6 Klein paradox3.6 Klein paradox3.7 MIT bag modelMotivation:can we establish a model to confine quark scalarlike potential3.7 MIT bag model3.7 MIT b
24、ag model3.7 MIT bag modelIntroducing scalarlike potential3.7 MIT bag model3.7 MIT bag model3.7 MIT bag modelWhen gg we find a exponentially decaying solution3.7 MIT bag modelSolution of step function3.7 MIT bag model3.7 MIT bag model3.7 MIT bag model3.7 MIT bag model3.7 MIT bag model3.7 MIT bag model3.7 MIT bag model
