Photons and atoms: Introduction to quantum electrodynamics/
Claude Cohen-Tannoudji
- UK: Wiley-VCH, 1989.
- 468 p.
A. The Fundamental Equations in Real Space 7 1. The Maxwell-Lorentz Equations 7 2. Some Important Constants of the Motion g 3. Potentials—Gauge Invariance g B. Electrodynamics in Reciprocal Space U 1. The Fourier Spatial Transformation—Notation 11 2. The Field Equations in Reciprocal Space 12 3. Longitudinal and Transverse Vector Fields I3 4. Longitudinal Electric and Magnetic Fields I5 5. Contribution of the Longitudinal Electric Field to the Total Energy,to the Total Momentum,and to the Total Angular Momentum—a. The Total Energy, b. The Total Momentum, c. The Total Angular Momentum 17 6. Equations of Motion for the Transverse Fields 21 C. Normal Variables 23 1. Introduction 23 2. Definition of the Normal Variables 23 3. Evolution of the Normal Variables 24 4. The Expressions for the Physical Observables of the Transverse Field as a Function of the Normal Variables—a. The Energy i/trans Transverse Field, b. The Momentum and the Angular Momentum J, trans of the Transverse Field, c. Transverse Electric and Magnetic Fields in RealSpace, d. The Transverse Vector Potential\ t) ... 26 VVI Contents 5. Similarities and Differences between the Normal Variables and the Wave Function of a Spin-1 Particle in Reciprocal Space 30 6. Periodic Boundary Conditions.Simplified Notation 31 D. Conclusion: Discussion of Various Possible Quantization Schemes 33 1. Elementary Approach 2. Lagrangian and Hamiltonian Approach 34 Complement A,—The"Transverse" Delta Function 1. Definition in Reciprocal Space—a. Cartesian Coordinates. Transverse and Longitudinal Components, b. Projection on the Subspace of Transverse Fields • • • 2. The Expression for the Transverse Delta Function in Real Space— a. Regularization of 5,^ (p). b. Calculation of g(p). c. Evaluation of the Derivatives of(p). d. Discussion of the Expression for (p) 38 3. Application to the Evaluation of the Magnetic Field Created by a Magnetization Distribution. Contact Interaction Complement B,—Angular Momentum of the Electromagnetic Field. Multipole Waves Introduction 1. Contribution of the Longitudinal Electric Field to the Total Angular Momentum o' 2 Angular Momentum of the Transverse Field—a. J, ran Reciprocal Space, b. Jtr&RS in Terms of Normal Variables, ^ f c. An analogy with the Mean Value of the total angular Momentum ofa Spin-l Particle 3. Set of Vector Functions of k "Adapted" to the Angular Momentum. General Idea. b. Method for Constructing Vector Eigenfunctions for J-and J,. c. LongitudinalEigenfunctions. d. Transverse Eigenfunctions.. 51 4. Application: Multiple Waves in Real Space-a. Evaluation of Some Fourier Transforms, b. Electric Multipole Waves, c. Magnetic Multipole Waves complement Exercises 1. //and P as Constants of the Motion 61 2. Transformation from the Coulomb Gauge to the Lorentz Gauge 63 3 Cancellation of the Longitudinal Electric Field by the Instantaneous Transverse FieldContents * VII 4. Normal Variables and Retarded Potentials 66 5. Field Created by a Charged Particle at Its Own Position. Radiation Reaction 68 6. Field Produced by an Oscillating Electric Dipole 71 7. Cross-section for Scattering of Radiation by a Classical Elastically Bound Electron 74
LAGRANGIAN AND HAMILTON APPROACH TO ELECTRODYNAMICS.THE STANDARD LAGRANGIAN ANDTHECOULOMB GAUGE Introduction 79 A. Review of the Lagrangian and Hamiltonian Formalism 81 1. Systems Having a Finite Number of Degrees of Freedom— a. Dynamical Variables, the Lagrangian,and the Action, b. Lagrange's Equations, c. Equivalent Lagrangians. d. Conjugate Momenta and the Hamiltonian. e. Change of Dynamical Variables, f. Use of Complex Generalized Coordinates, g. Coordinates, Momenta, and Hamilton nian in Quantum Mechanics. 81 2. A System with a Continuous Ensemble of Degrees of Freedom— a. Dynamical Variables, b. the Lagrangian. c. Lagrange equations d. Conjugate Momenta and the Hamiltonian. e. Quantization, f Lagrangian Formalism with Complex Fields, g. Hamiltonian Formalism and Quantization with Complex Fields 90 B. The Standard Lagrangian of Classical Electrodynamics ICQ 1. The Expression for the Standard Lagrangian—a. The Standard Lagrangian in Real Space, b. The Standard Lagrangian in Reciprocal Space 100 2. The Derivation of the Classical Electrodynamic Equations from the Standard Lagangian—a. Lagrange's Equation for Particles, b. The Ldgrange Equation Relative to the Scalar Potential, c. The Lagrange Equation Relative to the Vector Potential IO3 3. General Properties of the Standard Lagrangian—a. Global Symmetries. b. Gauge Invariance. c. Redundancy of the Dynamical Variables 1Q5 C. Electrodynamics in the Coulomb Gauge HI 1. Elimination of the Redundant Dynamical Variables from the Standard Lagrangian—fl. Elimination of the Scalar Potential, b. The Choice of the Longitudinal Component of the Vector Potential HI 2. The Lagrangian in the Coulomb Gauge 113Vm Contents 3. Hamiltonian Formalism—a Conjugate Particle Momenta, b. Conjugate Momenta for the Field Variables, c. The Hamiltonian in the Coulomb Gauge, d. The Physical Variables 115 4. Canonical Quantization in the Coulomb Gauge—a Fundamental Commutation Relations, b. The Importance of Transversality in the Case of the Electromagnetic Field, c. Creation and Annihilation Operators 118 5. Conclusion: Some Important Characteristics of Electrodynamics in the Coulomb Gauge—a. The Dynamical Variables Are Independent, b. The Electric Field Is Split into a Coulomb Field and a Transverse Field, c. The Formalism Is Not Manifestly Covariant. d. The Interaction of the particles with Relativistic ModesIs Not Correctly Described.. 121 ComplementA Functional Derivative.Introduction AND A Few Applications 1. From a Discrete to a Continuous System. The Limit of Partial Derivatives 126 2. Functional Derivative 128 3. Functional Derivative of the Action and the Lagrange Equations 128 4. Functional Derivative of the Lagrangian for a Continuous System 130 5. Functional Derivative of the Hamiltonian for a Continuous System 132 Complement Bh—Symmetriesof the Lagrangian in the Coulomb Gauge and the Constants of the Motion 1. The Variation of the Action between Two Infinitesimally Close Real Motions 134 2. Constants of the Motion in a Simple Case 136 3. Conservation of Energy for the System Charges + Field 137 4. Conservation of the Total Momentum 138 5. Conservation of the Total Angular Momentum 139 Complement C„—Electrodynamicsin the Presence of an External Field 1. Separation of the External Field 141 2. The Lagrangian in the Presence of an External Field—a. Introduction of a Lagrangian. b. The Lagrangian in the Coulomb Gauge 142 3. The Hamiltonian in the Presence of an External Field—a. Conjugate Momenta, b. The Hamiltonian. c. Quantization 143Contents , IX Complement Du—Exercises 1. An Example of a Hamiltonian Different from the Energy 146 2. From a Discrete to a Continuous System: Introduction of the Lagrangian and Hamiltonian Densities 147 3. Lagrange's Equations for the Components of the Electromagnetic Field in Real Space 150 4. Lagrange's Equations for the Standard Lagrangian in the Coulomb Gauge 151 5. Momentum and Angular Momentum of an Arbitrary Field 152 6. A Lagrangian Using Complex Variables and Linear in Velocity 154 7. Lagrangian and Hamiltonian Descriptions of the Schrodinger Matter Field 157 8. Quantization of the Schrodinger Field 161 9. Schrodinger Equation of a Particle in an Electromagnetic Field: Arbitrariness of Phase and Gauge Invariance 167 111 QUANTUM ELECTRODYNAMICS THE COULOMB GAUGE Introduction 169 A. The General Framework 171 1. Fundamental Dynamical Variables. Commutation Relations 171 2. The Operators Associated with the Various Physical Variables of the System 171 3. State Space 175 B. Time Evolution 176 1. The Schrodinger Picture 176 2. The Heisenberg Picture. The Quantized Maxwell-Lorentz Equations—a. The Heisenberg Equations for Particles, b. The Heisenberg Equations for Fields, c. The Advantages of the Heisenberg Point of View 176 C. Observables and States of the Quantized Free Field 183 1. Review of Various Observables of the Free Field—a. Total Energy and Total Momentum of the Field b. The Fields at a Given Point r of Space, c. Observables Corresponding to Photoelectric Measurements.. 183 2. Elementary Excitations of the Quantized Free Field. Photons— a. Eigenstates of the Total Energy and the Total Momentum. b. The Interpretation in Terms of Photons, c. Single-Photon States. Propagation 3. Some Properties of the Vacuum—a. Qualitative Discussion, b. Mean Values and Variances of the Vacuum Field, c. Vacuum Fluctuations .. 189 4. Quasi-classical States— a. Introducing the Quasi-classical States, b. Characterization of the Quasi-classical States, c. Some Properties of the Quasi-classical States, d. The Translation Operator 192 D. The Hamiltonian for the Interaction between Particles and Fields. 197 1. Particle Hamiltonian, Radiation Field Hamiltonian, Interaction 197 Hamilton 2. Orders of Magnitude of the Various Interactions Terms for Systems of Bound Particles 3. Selection Rules 4. Introduction of a Cutoff Complement Am—The Analysis of Interference Phenomena in the Quantum Theory of Radiation