Relativistic Quantum Mechanics (eBook)

Armin Wachter holds a Ph.D. in Physics from the John von Neumann Institute for Computing (NIC) / research Centre of Jülich, Germany. His research interests include theoretical elementary particle physics, heavy quark physics, heavy meson spectroscopy, algorithms on parallel computers, and lattice gauge theory. His publications at Springer include: Repetitorium Theoretische Physik, ISBN 3-540-21457-7 Compendium of Theoretical Physics, ISBN 0-387-25799-3 Relativistische Quantenmechanik, ISBN 3-540-22922-1

Preface 6
Table of Contents 11
List of Exercises 14
1. Relativistic Description of Spin-0 Particles 16
1.1 Klein-Gordon Equation 19
1.1.1 Canonical and Lorentz-covariant Formulationsof the Klein-Gordon Equation 19
1.1.2 Hamilton Formulation of the Klein-Gordon Equation 24
1.1.3 Interpretation of Negative Solutions, Antiparticles 27
Exercises 33
1.2 Symmetry Transformations 36
1.2.1 Active and Passive Transformations 36
1.2.2 Lorentz Transformations 38
1.2.3 Discrete Transformations 39
Exercises 44
1.3 One-Particle Interpretation of the Klein-Gordon Theory 45
1.3.1 Generalized Scalar Product 45
1.3.2 One-particle Operatorsand Feshbach-Villars Representation 48
1.3.3 Validity Range of the One-particle Concept 54
1.3.4 Klein Paradox 57
Exercises 61
1.4 Nonrelativistic Approximation of the Klein-Gordon Theory 66
1.4.1 Nonrelativistic Limit 66
1.4.2 Relativistic Corrections 68
Exercises 73
1.5 Simple One-Particle Systems 76
1.5.1 Potential Well 77
1.5.2 Radial Klein-Gordon Equation 81
1.5.3 Free Particle and Spherically Symmetric Potential Well 83
1.5.4 Coulomb Potential 88
1.5.5 Oscillator-Coulomb Potential 92
Exercises 97
2. Relativistic Description of Spin-1/2 Particles 100
2.1 Dirac Equation 101
2.1.1 Canonical Formulation of the Dirac Equation 101
2.1.2 Dirac Equation in Lorentz-Covariant Form 108
2.1.3 Properties of -Matrices and Covariant Bilinear Forms 112
2.1.4 Spin Operator 115
2.1.5 Projection Operators 118
2.1.6 Interpretation of Negative Solutions, Antiparticlesand Hole Theory 121
Exercises 128
2.2 Symmetry Transformations 136
2.2.1 Proper Lorentz Transformations 136
2.2.2 Spin of Dirac Solutions 141
2.2.3 Discrete Transformations 142
Exercises 148
2.3 One-Particle Interpretation of the Dirac Theory 152
2.3.1 One-Particle Operatorsand Feshbach-Villars Representation 152
2.3.2 Validity Range of the One-Particle Concept 156
2.3.3 Klein Paradox 158
Exercises 160
2.4 Nonrelativistic Approximation of the Dirac Theory 166
2.4.1 Nonrelativistic Limit 166
2.4.2 Relativistic Corrections 168
Exercises 173
2.5 Simple One-Particle Systems 175
2.5.1 Potential Well 175
2.5.2 Radial Form of the Dirac Equation 178
2.5.3 Free Particle and Centrally Symmetric Potential Well 181
2.5.4 Coulomb Potential 184
Exercises 190
3. Relativistic Scattering Theory 192
3.1 Review: Nonrelativistic Scattering Theory 193
3.1.1 Solution of the General Schrödinger Equation 194
3.1.2 Propagator Decomposition by Schrödinger Solutions 198
3.1.3 Scattering Formalism 200
3.1.4 Coulomb Scattering 208
Exercises 211
3.2 Scattering of Spin-1/2 Particles 217
3.2.1 Solution of the General Dirac Equation 218
3.2.2 Fourier Decomposition of the Free Fermion Propagator 221
3.2.3 Scattering Formalism 225
3.2.4 Trace Evaluations with -Matrices 230
Exercises 235
3.3 Spin-1/2 Scattering Processes 237
3.3.1 Coulomb Scattering of Electrons 239
3.3.2 Electron-Proton Scattering (I) 247
3.3.3 Electron-Proton Scattering (II) 259
3.3.4 Preliminary Feynman Rules in Momentum Space 267
3.3.5 Electron-Electron Scattering 270
3.3.6 Electron-Positron Scattering 276
3.3.7 Compton Scattering against Electrons 281
3.3.8 Electron-Positron Annihilation 289
3.3.9 Conclusion: Feynman Diagrams in Momentum Space 294
Exercises 298
3.4 Higher Order Corrections 307
3.4.1 Vacuum Polarization 310
3.4.2 Self-Energy 316
3.4.3 Vortex Correction 321
3.4.4 Physical Consequences 325
Exercises 332
3.5 Scattering of Spin-0 Particles 334
3.5.1 Solution of the General Klein-Gordon Equation 334
3.5.2 Scattering Formalism 336
3.5.3 Coulomb Scattering of Pions 339
3.5.4 Pion-Pion Scattering 342
3.5.5 Pion Production via Electrons 346
3.5.6 Compton Scattering against Pions 351
3.5.7 Conclusion: Enhanced Feynman Rulesin Momentum Space 356
Exercises 358
A. Appendix 364
A.1 Theory of Special Relativity 364
A.2 Bessel Functions, Spherical Bessel Functions 370
A.3 Legendre Functions, Legendre Polynomials,Spherical Harmonics 372
A.4 Dirac Matrices and Bispinors 374
Index 377