Table of Contents XIII Part II.Relativistic Wave Equations 5.Relativistic Wave Equations and their Derivation......................................115 5.1 Introduction...... .115 5.2 The Klein-Gordon Equation. .116 5.2.1 Derivation by Means of the Correspondence Principle.116 5.2.2 The Continuity Equation 110 100 5.3 Dirac E uation.. 120 5.3. Derivation of the Dirac Equation ................. 120 5.3.2 The Continuity Equation..........................122 5.3.3 Properties of the Dirac Matrices.. 123 5.3.4 The Dirac Equation in Covariant Form..............123 5.3.5 Nonrelativistic Limit and Coupling to the electromagnetic field 195 Problems .130 6.Lorentz Transformations and Covariance of the Dirac Equation ,.131 6.1 Lorentz Transformations 121 6.2 entz Covariance of the Dirac Equation 135 0 Lorentz Co e and Transformation of Spinors.... 135 6.2.2 Determination of the Representation S(A)..........13 6.2.3 Further Properties of S...........................142 6.2.4 Transformation of Bilinear Forms................... 144 6.2.5 Properties of the y Matrices.... 6.3 Solutions of the Dirac Equation for Free Particles. 146 6.3.1 Spinors with Finite Momentum 146 6.3.2 Orthogonality Relations and Density 1A0 6.3.3 Projection Operators 151 Problems................................. 152 7.Orbital Angular Mom 155 ve and Acti orma 15 Rotations and Angular Momentum...................... Problems.................................................. 159 The Coulomb Potential... 8.1 Klein-Gordon Equation with Electromagnetic Field... 8.1.1 Coupling to the Electromagnetic Field.. 8.1.2 Klein-Gordon Equation in a Coulomb Field.........162 8.2 Dirac Equation for the Coulomb Potential.................168 Problems.. ..179 Table of Contents XIII Part II. Relativistic Wave Equations 5. Relativistic Wave Equations and their Derivation ...................................... 115 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.2 The Klein–Gordon Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 5.2.1 Derivation by Means of the Correspondence Principle . 116 5.2.2 The Continuity Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 119 5.2.3 Free Solutions of the Klein–Gordon Equation . . . . . . . . 120 5.3 Dirac Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 5.3.1 Derivation of the Dirac Equation . . . . . . . . . . . . . . . . . . . 120 5.3.2 The Continuity Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 122 5.3.3 Properties of the Dirac Matrices . . . . . . . . . . . . . . . . . . . . 123 5.3.4 The Dirac Equation in Covariant Form . . . . . . . . . . . . . . 123 5.3.5 Nonrelativistic Limit and Coupling to the Electromagnetic Field . . . . . . . . . . . . . . . . . . . . . . . 125 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 6. Lorentz Transformations and Covariance of the Dirac Equation .................... 131 6.1 Lorentz Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 6.2 Lorentz Covariance of the Dirac Equation . . . . . . . . . . . . . . . . . 135 6.2.1 Lorentz Covariance and Transformation of Spinors . . . . 135 6.2.2 Determination of the Representation S(Λ) . . . . . . . . . . 136 6.2.3 Further Properties of S . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 6.2.4 Transformation of Bilinear Forms . . . . . . . . . . . . . . . . . . . 144 6.2.5 Properties of the γ Matrices . . . . . . . . . . . . . . . . . . . . . . . 145 6.3 Solutions of the Dirac Equation for Free Particles . . . . . . . . . . . 146 6.3.1 Spinors with Finite Momentum . . . . . . . . . . . . . . . . . . . . 146 6.3.2 Orthogonality Relations and Density . . . . . . . . . . . . . . . . 149 6.3.3 Projection Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 7. Orbital Angular Momentum and Spin .................... 155 7.1 Passive and Active Transformations . . . . . . . . . . . . . . . . . . . . . . . 155 7.2 Rotations and Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . 156 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 8. The Coulomb Potential ................................... 161 8.1 Klein–Gordon Equation with Electromagnetic Field . . . . . . . . . 161 8.1.1 Coupling to the Electromagnetic Field . . . . . . . . . . . . . . 161 8.1.2 Klein–Gordon Equation in a Coulomb Field . . . . . . . . . 162 8.2 Dirac Equation for the Coulomb Potential . . . . . . . . . . . . . . . . . 168 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179