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, space 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 probability3.1 Klein Gordon equation3.1 Klein Gordon equationNon-relativistic limit: K-G eq Sch eq3.1 Klein Gordon equation3.1 Klein Gordon equation3.1 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 equationHow 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 invariance3.2 Dirac equation3.2 Dirac equation3.2 Dirac equation3.2 Dirac equation4 anti-commute matrices and 44 matrices3.2 Dirac equation3.2 Dirac equationConservation law of the probability flux3.2 Dirac equation3.2 Dirac equation3.3 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 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)Eigenstates of momentum p3.3 solutions of the free particleOrbital angular momentum is not conserved3.3 solutions of the free particle3.3 solutions of the free particleSpin angular momentumOr3.3 solutions of the free particle3.3 solutions of the free particle3.3 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 findEigenvalues:3.3 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 fieldEquation in non-relativistic limit3.4 Dirac equation in the central force field3.4 Dirac equation in the central force field3.4 Dirac equation in the central force fieldIn non-relativistic approximation3.4 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 fieldNoting: up to the orderNormalization condition must be ensured3.4 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 field3.4 Dirac equation in the central force fieldBy using3.4 Dirac equation in the central force fieldRelativistic correction 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 field3.4 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 field3.4 Dirac equation in the central force field3.4 Dirac equation in the central force field3.4 Dirac equation in the central force fieldRadial equations3.4 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 fieldWe take3.5 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 field3.5 Solution of the Dirac equation in the Coulomb field3.5 Solution of the Dirac equation in the Coulomb field3.5 Solution of the Dirac equation in the Coulomb field3.5 Solution of the Dirac equation in the Coulomb fieldn=0,1,2, , n=1,2,33.5 Solution of the Dirac equation in the Coulomb field3.5 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 equation 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 potential3.6 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 statesBound statesCannot confine quarksconfinement3.6 Klein paradoxThe physics of Klein paradox3.6 Klein paradox3.6 Klein paradox3.6 Klein paradox3.6 Klein paradox3.6 Klein paradox3.6 Klein paradox3.6 Klein paradox3.6 Klein paradox3.6 Klein paradox3.6 Klein paradox3.6 Klein paradox3.6 Klein paradox3.6 Klein paradox3.6 Klein paradoxIf p=mc3.6 Klein paradoxThe explanation of Klein paradox3.6 Klein paradox3.6 Klein paradox3.6 Klein paradox3.6 Klein paradox3.6 Klein paradox3.6 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 bag 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