XIV Table of Contents 9.The Foldy-Wouthuysen Transformation and Relativistic Corrections..............................181 9.1 The Foldy-Wouthuysen Transformation......... .181 9.1.1 Description of the Problem 1只1 91 2 Transfo mation for Free Particles 1只0 0121m ction with the Elec etic Field 1只3 9.2 Relativist ra trom ions and the Lamb Shift.. 187 9.2 Relativistic Corrections 187 9.2.2 Estimate of the Lamb Shift,,...,,.,...,,,,,,,,,.,.189 Problems ...................................................193 10.Physical Interpretation of the Solutions to the Dirac Equation....................195 l0.1 Wave Packets and“Zitterbewegung”. 10.1.1 Superposition of Positive Energy States.............196 10.1.2 The General Wave Packet.. .197 10.1.3 General Solution of the Free Dirac Equation in the Heisenberg Re tatic 200 10.1.4 Potential Steps and the Klein Paradox 202 10.2 The Hole Theory...............20 Problems..........207 11.Symmetries and Further Properties of the Dirac Equation.....................................209 "11.1 Active and Passive Transformations, Transformations of Vectors..............................209 11.2 Invariance and Conservation Laws........................212 11.2.1 The General Transformation. .212 11.2.2 Rotations 212 11.2.3 Translations )12 11.2.4 Spatial Relection (Parity Transformation) 912 11.3 Charge Conjugatic 214 11.4 Time Reversal (Motion Reversal) 21 11.4.1 Reversal of Motion in Classical Physics..............218 11.4.2 Time Reversal in Quantum Mechanics..... 221 11.4.3 Time-Reversal Invariance of the Dirac Equation...229 *11.4.4 Racah Time Reflection....... 235 *11.5 Helicity 226 11.6 Zero-Mass Fermions (Neutrinos)..................... )20 Problem 244 Bibliography for Part II …245XIV Table of Contents 9. The Foldy–Wouthuysen Transformation and Relativistic Corrections .............................. 181 9.1 The Foldy–Wouthuysen Transformation . . . . . . . . . . . . . . . . . . . 181 9.1.1 Description of the Problem . . . . . . . . . . . . . . . . . . . . . . . . 181 9.1.2 Transformation for Free Particles . . . . . . . . . . . . . . . . . . . 182 9.1.3 Interaction with the Electromagnetic Field . . . . . . . . . . 183 9.2 Relativistic Corrections and the Lamb Shift . . . . . . . . . . . . . . . . 187 9.2.1 Relativistic Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 9.2.2 Estimate of the Lamb Shift . . . . . . . . . . . . . . . . . . . . . . . . 189 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 10. Physical Interpretation of the Solutions to the Dirac Equation .................... 195 10.1 Wave Packets and “Zitterbewegung” . . . . . . . . . . . . . . . . . . . . . . 195 10.1.1 Superposition of Positive Energy States . . . . . . . . . . . . . 196 10.1.2 The General Wave Packet . . . . . . . . . . . . . . . . . . . . . . . . . 197 ∗10.1.3 General Solution of the Free Dirac Equation in the Heisenberg Representation . . . . . . . . . . . . . . . . . . . 200 ∗10.1.4 Potential Steps and the Klein Paradox . . . . . . . . . . . . . . 202 10.2 The Hole Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 11. Symmetries and Further Properties of the Dirac Equation..................................... 209 ∗11.1 Active and Passive Transformations, Transformations of Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 11.2 Invariance and Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . 212 11.2.1 The General Transformation . . . . . . . . . . . . . . . . . . . . . . . 212 11.2.2 Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 11.2.3 Translations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 11.2.4 Spatial Reflection (Parity Transformation) . . . . . . . . . . . 213 11.3 Charge Conjugation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 11.4 Time Reversal (Motion Reversal) . . . . . . . . . . . . . . . . . . . . . . . . . 217 11.4.1 Reversal of Motion in Classical Physics . . . . . . . . . . . . . . 218 11.4.2 Time Reversal in Quantum Mechanics . . . . . . . . . . . . . . 221 11.4.3 Time-Reversal Invariance of the Dirac Equation . . . . . . 229 ∗11.4.4 Racah Time Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 ∗11.5 Helicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 ∗11.6 Zero-Mass Fermions (Neutrinos) . . . . . . . . . . . . . . . . . . . . . . . . . . 239 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 Bibliography for Part II ...................................... 245