C reative
Particle
Higgs
CPH Theory is based on Generalized light velocity from energy into mass.
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Advanced Quantum Mechanics
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Advanced Quantum Mechanics Full Story PDF 1.58 MB
Advanced Quantum Mechanics Peter S. Riseborough March 14, 2007 Contents 1 Introduction 4 2 Quantum Mechanics of a Single Photon 5 2.1 Rotations and Intrinsic Spin . . . . . . . . . . . . . . . . . . . . . 6 2.2 Massless Particles with Spin Zero . . . . . . . . . . . . . . . . . . 10 2.3 Massless Particles with Spin One . . . . . . . . . . . . . . . . . . 11 3 Maxwell’s Equations 13 3.1 Vector and Scalar Potentials . . . . . . . . . . . . . . . . . . . . . 14 3.2 Gauge Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4 Relativistic Formulation of Electrodynamics 18 4.1 Lorentz Scalars and Vectors . . . . . . . . . . . . . . . . . . . . . 19 4.2 Covariant and Contravariant Derivatives . . . . . . . . . . . . . . 20 4.3 Lorentz Transformations . . . . . . . . . . . . . . . . . . . . . . . 23 4.4 Invariant Form of Maxwell’s Equations . . . . . . . . . . . . . . . 24 5 The Simplest Classical Field Theory 27 5.1 The Continuum Limit . . . . . . . . . . . . . . . . . . . . . . . . 32 5.2 Normal Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 5.3 Rules of Canonical Quantization . . . . . . . . . . . . . . . . . . 36 5.4 The joys of the number operator! . . . . . . . . . . . . . . . . . . 38 5.5 The Classical Limit . . . . . . . . . . . . . . . . . . . . . . . . . . 38 6 Classical Field Theory 39 6.1 The Hamiltonian Formulation . . . . . . . . . . . . . . . . . . . . 41 6.2 Symmetry and Conservation Laws . . . . . . . . . . . . . . . . . 42 6.2.1 Conservation Laws . . . . . . . . . . . . . . . . . . . . . . 42 6.2.2 Noether Charges . . . . . . . . . . . . . . . . . . . . . . . 44 6.2.3 Noether’s Theorem . . . . . . . . . . . . . . . . . . . . . . 45 6.3 The Energy-Momentum Tensor . . . . . . . . . . . . . . . . . . . 46 1 7 The Electromagnetic Lagrangian 49 7.1 Conservation Laws for Electromagnetic Fields . . . . . . . . . . . 53 7.2 Massive Spin-One Particles . . . . . . . . . . . . . . . . . . . . . 58 8 Symmetry Breaking and Mass Generation 59 8.1 Symmetry Breaking and Goldstone Bosons . . . . . . . . . . . . 59 8.2 The Kibble-Higgs Mechanism . . . . . . . . . . . . . . . . . . . . 61 9 Quantization of the Electromagnetic Field 63 9.1 The Lagrangian and Hamiltonian Density . . . . . . . . . . . . . 65 9.2 Quantizing the Normal Modes . . . . . . . . . . . . . . . . . . . . 67 9.2.1 The Energy of the Field . . . . . . . . . . . . . . . . . . . 69 9.2.2 The Electromagnetic Field . . . . . . . . . . . . . . . . . 70 9.2.3 The Momentum of the Field . . . . . . . . . . . . . . . . 72 9.2.4 The Angular Momentum of the Field . . . . . . . . . . . . 74 9.3 Uncertainty Relations . . . . . . . . . . . . . . . . . . . . . . . . 79 9.4 Coherent States . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 9.4.1 The Phase-Number Uncertainty Relation . . . . . . . . . 86 9.4.2 Argand Representation of Coherent States . . . . . . . . . 87 10 Non-Relativistic Quantum Electrodynamics 88 10.1 Emission and Absorption of Photons . . . . . . . . . . . . . . . . 91 10.1.1 The Emission of Radiation . . . . . . . . . . . . . . . . . 91 10.1.2 The Dipole Approximation . . . . . . . . . . . . . . . . . 93 10.1.3 Electric Dipole Radiation Selection Rules . . . . . . . . . 97 10.1.4 Angular Distribution of Dipole Radiation . . . . . . . . . 106 10.1.5 The Decay Rate from Dipole Transitions. . . . . . . . . . 112 10.1.6 The 2p ! 1s Electric Dipole Transition Rate. . . . . . . . 114 10.1.7 Electric Quadrupole and Magnetic Dipole Transitions. . . 117 10.1.8 The 3d ! 1s Electric Quadrupole Transition Rate . . . . 121 10.1.9 Two-photon decay of the 2s state of Hydrogen. . . . . . . 124 10.1.10 The Absorption of Radiation . . . . . . . . . . . . . . . . 131 10.1.11 The Photoelectric Effect . . . . . . . . . . . . . . . . . . . 136 10.1.12Impossibility of absorption of photons by free-electrons. . 139 10.2 Scattering of Light . . . . . . . . . . . . . . . . . . . . . . . . . . 141 10.2.1 Rayleigh Scattering . . . . . . . . . . . . . . . . . . . . . 145 10.2.2 Thompson Scattering . . . . . . . . . . . . . . . . . . . . 148 10.2.3 Raman Scattering . . . . . . . . . . . . . . . . . . . . . . 152 10.2.4 Radiation Damping and Resonance Fluorescence . . . . . 153 10.2.5 Natural Line-Widths . . . . . . . . . . . . . . . . . . . . . 156 10.3 Renormalization . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 10.3.1 The Casimir Effect . . . . . . . . . . . . . . . . . . . . . . 159 10.3.2 The Lamb Shift . . . . . . . . . . . . . . . . . . . . . . . . 164 10.3.3 The Self-Energy of a Free Electron . . . . . . . . . . . . . 166 10.3.4 The Self-Energy of a Bound Electron . . . . . . . . . . . . 169 10.3.5 Brehmstrahlung . . . . . . . . . . . . . . . . . . . . . . . 173
11 The Dirac Equation 179 11.1 Conservation of Probability . . . . . . . . . . . . . . . . . . . . . 183 11.2 Covariant Form of the Dirac Equation . . . . . . . . . . . . . . . 185 11.3 The Field Free Solution . . . . . . . . . . . . . . . . . . . . . . . 187 11.4 Coupling to Fields . . . . . . . . . . . . . . . . . . . . . . . . . . 193 11.4.1 Mott Scattering . . . . . . . . . . . . . . . . . . . . . . . . 194 11.4.2 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . 197 11.4.3 The Gordon Decomposition . . . . . . . . . . . . . . . . . 198 11.5 Lorentz Covariance of the Dirac Equation . . . . . . . . . . . . . 202 11.6 The Non-Relativistic Limit . . . . . . . . . . . . . . . . . . . . . 202 11.7 Conservation of Angular Momentum . . . . . . . . . . . . . . . . 205 11.8 Conservation of Parity . . . . . . . . . . . . . . . . . . . . . . . . 207 11.9 The Radial Dirac Equation . . . . . . . . . . . . . . . . . . . . . 210 11.9.1 The Hydrogen Atom . . . . . . . . . . . . . . . . . . . . . 216 11.9.2 A Particle in a Spherical Square Well . . . . . . . . . . . 224 11.9.3 The MIT Bag Model . . . . . . . . . . . . . . . . . . . . . 229 11.9.4 The Temple Meson Model . . . . . . . . . . . . . . . . . . 232 11.10An Electron in a Uniform Magnetic Field . . . . . . . . . . . . . 234 11.11Symmetry Considerations . . . . . . . . . . . . . . . . . . . . . . 236 11.12The Limit of Zero Mass . . . . . . . . . . . . . . . . . . . . . . . 236 11.13Classical Dirac Field Theory . . . . . . . . . . . . . . . . . . . . . 239 11.13.1 Chiral Gauge Symmetry . . . . . . . . . . . . . . . . . . . 243 11.14Hole Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 11.14.1Compton Scattering . . . . . . . . . . . . . . . . . . . . . 248 11.14.2 Charge Conjugation . . . . . . . . . . . . . . . . . . . . . 252 Charge Conjugation . . . . . . . . . . . . . . . . . . . . . 252 12 The Many-Particle Dirac Field 255 12.1 Second Quantization of Fermions . . . . . . . . . . . . . . . . . . 255 12.2 Quantizing the Dirac Field . . . . . . . . . . . . . . . . . . . . . 255 12.3 The Connection between Spin and Statics . . . . . . . . . . . . . 259 13 Massive Gauge Field Theory 260 13.1 The Gauge Symmetry . . . . . . . . . . . . . . . . . . . . . . . . 260 13.2 The Coupling to the Gauge Field . . . . . . . . . . . . . . . . . . 262 13.3 The Free Gauge Fields . . . . . . . . . . . . . . . . . . . . . . . . 263 13.4 Breaking the Symmetry . . . . . . . . . . . . . . . . . . . . . . . 2
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