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[资源] Advanced Modern Physics... Theoretical Foundations

Contents
Preface
1. Introduction
vii
1
2. Quantum Mechanics (Revisited)
2.1 Linear Vector Spaces .
2.1.1 Three- Dimensional Vectors
2.1.2 n- Dimensions .
2.2 Hilbert Space .
2.2.1 Example .
2.2.2 Definition .
2.2.3 Relation to Linear Vector Space
2.2.4 Abstract State Vector
2.3 Linear Hermitian Operators ..
2.3.1 Eigenfunctions .....
2.3.2 Eigenstates of Position
2.4 Abstract Hilbert Space
2.4.1 Inner Product . . . . .
2.4.2 Completeness......
2.4.3 Linear Hermitian Operators
2.4.3.1 Eigenstates ....
2.4.3.2 Adjoint Operators
2.4.4 Schrodinger Equation . . . .
2.4.4.1 Stationary States ..
2.5 Measurements .
2.5.1 Coordinate Space
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x Advanced Modern Physics
2.5.2 Abstract Form . . . . . . . . . .
2.5.3 Reduction of the Wave Packet
2.5.4 Stern-Gerlach Experiment
2.6 Quantum Mechanics Postulates. . .
2.7 Many-Particle Hilbert Space ....
2.7.1 Simple Harmonic Oscillator .
2.7.2 Bosons .
2.7.3 Fermions
3. Angular Momentum
3.1 'franslations .
3.2 Rotations....... . . . . .
3.3 Angular Momentum Operator
3.4 Eigenvalue Spectrum. . . . . .
3.5 Coupling of Angular Momenta
3.6 Recoupling . . . . . . . . . .
3.7 Irreducible Tensor Operators
3.8 The Wigner-Eckart Theorem
3.9 Finite Rotations .
3.9.1 Properties .
3.9.2 Tensor Operators .
3.9.3 Wigner-Eckart Theorem (Completed)
3.10 Tensor Products
3.11 Vector Model
4. Scattering Theory
4.1 Interaction Picture.
4.2 Adiabatic Approach
4.3 U -Operator . . . . . .
4.4 (; -Operator for Finite Times
4.5 The S-Matrix. . . . . . . .
4.6 Time-Independent Analysis
4.7 Scattering State
4.8 'fransition Rate. . . . . . .
4.9 Unitarity .
4.10 Example: Potential Scattering
4.10.1 Green's Function (Propagator)
4.10.2 Scattering Wave Function ...
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Contents xi
4.10.3 T-matrix .... 93
4.10.4 Cross Section . 93
4.10.5 Unitarity .... 94
5. Lagrangian Field Theory 95
5.1 Particle Mechanics. · .... 96
5.1.1 Hamilton's Principle . . .... 96
5.1.2 Lagrange's Equations 97
5.1.3 Hamiltonian · .... 98
5.2 Continuum Mechanics (String-a Review) 101
5.2.1 Lagrangian Density 101
5.2.2 Hamilton's Principle . 102
5.2.3 Lagrange's Equation. . 103
5.2.4 Two-Vectors · ..... 103
5.2.5 Momentum Density 104
5.2.6 Hamiltonian Density. . 104
5.3 Quantization ......... 104
5.3.1 Particle Mechanics ... 104
5.3.2 Continuum Mechanics (String) 106
5.4 Relativistic Field Theory 108
5.4.1 Scalar Field. . . 109
5.4.2 Stress Tensor . . . 111
5.4.3 Dirac Field ..... 117
5.4.4 Noether's Theorem 120
5.4.4.1 Normal-Ordered Current 122
5.4.5 Electromagnetic Field . . . . . . 123
5.4.6 Interacting Fields (Dirac-Scalar) 123
6. Symmetries 125
6.1 Lorentz Invariance 125
6.2 Rotational Invariance 126
6.3 Internal Symmetries 127
6.3.1 Isospin-SU(2) 127
6.3.1.1 Isovector 127
6.3.1.2 Isospinor 131
6.3.1.3 Transformation Law 133
6.3.2 Lie Groups ...... 134
6.3.3 Sakata Model-SU (3) . . . . . 138
xii Advanced Modern Physics
6.3.3.1 Dirac 'friplet . . . . . . . . . . .
6.3.3.2 Scalar Octet .
6.3.3.3 Interacting Fields (Dirac-Scalar)
6.4 Phase Invariance . . . . . . . . . .
6.4.1 Global Phase Invariance .
6.4.2 Local Phase Invariance
6.5 Yang-Mills Theories
6.6 Chiral Symmetry. . . . . . . . .
6.6.1 a-Model .
6.6.2 Spontaneous Symmetry Breaking
6.7 Lorentz 'fransformations .
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7. Feynman Rules
7.1 Wick's Theorem .
7.2 Example (Dirac-Scalar) .
7.2.1 Scattering Amplitudes.
7.2.2 Self-Energies ....
7.2.3 Vacuum Amplitude
7.3 Feynman Diagrams .
7.4 Feynman Rules .
7.5 Cancellation of Disconnected Diagrams
7.6 Mass Renormalization . . . . . . . . . .
8. Quantum Electrodynamics (QED)
8.1 Classical Theory
8.2 Hamiltonian ..
8.3 Quantization ..
8.4 Photon Propagator.
8.5 Second-Order Processes
8.5.1 Scattering Amplitudes .
8.5.2 Self-Energies ....
8.6 QED With Two Leptons
8.7 Cross Sections . . . . . . .
8.7.1 e- + p,-
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8.7.1.1 Scattering Amplitude
8.7.1.2 Cross Section . . . . .
8.7.1.3 'fraces .
8.7.1.4 Cross Section (Continued) .
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Contents
8.7.1.5 Limiting Cases ..
8.7.1.6 M￠ller Scattering
872 ++ - ++- .. e e
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8.7.2.1 Scattering Amplitude
8.7.2.2 Cross Section ..
8.7.2.3 Limiting Cases .
8.7.2.4 Colliding Beams
8.8 QED in External Field ....
8.8.1 Nuclear Coulomb Field
8.8.2 Bremsstrahlung ....
8.8.3 Pair Production . . . . .
8.9 Scattering Operator Si xt in Order e 3 ..
8.10 Feynman Rules for QED .
8.10.1 General Scattering Operator
8.10.2 Feynman Diagrams .
8.10.3 Feynman Rules .
8.10.3.1 Coordinate Space .
8.10.3.2 Momentum Space
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9. Higher-Order Processes
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Example-Scattering in External Field ....
9.1.1 Feynman Diagrams
9.1.2 First-Order.......
9.1.3 Vertex Insertion ...
9.1.4 Vacuum Polarization
9.1.5 Self-Energy Insertions
Ward's Identity. . . . .
Electron Self-Energy . .
9.3.1 General Form
9.3.2 Evaluation . .
9.3.3 Mass Renormalization .
Vertex .
9.4.1 General Form .
9.4.2 Ward's Identity
9.4.3 Evaluation ... .
9.4.4 The Constant L
9.4.5 The Infrared Problem .
9.4.6 Schwinger Moment. . .
External Lines and Wavefunction Renormalization
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xiv
9.6
9.7
Advanced Modern Physics
9.5.1 Cancellation of Divergences.
Vacuum Polarization. .
9.6.1 Evaluation ..
9.6.2 General Form ....
9.6.3 Limiting Cases .
9.6.4 Insertion ....
9.6.5 Charge Renormalization. .
9.6.6 Charge Strength . . . .
Renormalization Theory . . .
9.7.1 Proper Self-Energies .
9.7.2 Proper Vertex .
9.7.3 Ward's Identity .
9.7.4 Ward's Vertex Construct
9.7.5 Finite Parts .
9.7.6 Proof of Renormalization
9.7.7 The Renormalization Group
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10. Path Integrals
10.1 Non-Relativistic Quantum Mechanics with One Degree of
Freedom .
10.1.1 General Relations .
10.1.2 Infinitesimals . . . . . . . . . . . . . . . .
10.1.3 Thansition Amplitude and Path Integral .
10.1.4 Classical Limit .
10.1.5 Superposition .
10.1.6 Matrix Elements . . . . . . . . . . . .
10.1.7 Crucial Theorem of Abers and Lee .
10.1.8 Functional Derivative .
10.1.9 Generating Functional.
10.2 Many Degrees of Freedom ..
10.2.1 Gaussian Integrals ...
10.3 Field Theory . . . . . . . . . .
10.3.1 Fields as Coordinates
10.3.2 Measure. . . . . . . .
10.3.3 Generating Functional.
10.3.4 Convergence . . . . . .
10.3.4.1 Euclidicity Postulate.
10.3.4.2 Adiabatic Damping
10.4 Scalar Field . . . . . . . . . . . . . . . .
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Contents xv
10.4.1 Generating Functional for Free Scalar Field. 285
10.4.1.1 Applications 287
10.4.2 Interactions. . . . . 289
10.5 Fermions . . . . . . . . . . . 290
10.5.1 Grassmann Algebra . . 290
10.5.2 Functional Derivative 291
10.5.3 Functional Integration. 291
10.5.4 Integrals . . 292
10.5.5 Basic Results. . . . . . . . . . 292
10.5.6 Generating Functional for Free Dirac Field 294
10.5.6.1 Applications ..... 296
10.5.7 Interactions (Dirac-Scalar) 297
10.6 Electromagnetic Field . . . . . . . 298
11. Canonical Transformations for Quantum Systems 299
11.1 Interacting Bose System . . . . . . . . . . 300
11.1.1 Pseudopotential . . . . . . . . . . 300
11.1.2 Special Role of the Ground State. . 301
11.1.3 Effective Hamiltonian . . . . 302
11.1.4 Bogoliubov Transformation . . 303
11.1.5 Discussion of Results 305
11.1.5.1 Excitation Spectrum . 305
11.1.5.2 Depletion . . . . . . . 306
11.1.5.3 Ground-State Energy 308
11.1.6 Superfiuid 4He .... 309
11.2 Superconductors . . . . . . . . . . . . 309
11.2.1 Cooper Pairs. . . . . . . . . . . 309
11.2.2 Bogoliubov- Valatin Transformation 310
11.2.2.1 Pairing . . . . . . . . . . . 310
11.2.2.2 Thermodynamic Potential. 311
11.2.2.3 Wick's Theorem . . . 312
11.2.2.4 Diagonalization of K o 314
11.2.2.5 Gap Equation .. 316
11.2.3 Discussion of Results 316
11.2.3.1 Particle Number . . . 316
11.2.3.2 Ground-State Thermodynamic Potential.. 317
11.2.3.3 Ground-State Energy 318
11.2.3.4 Excitation Spectrum . 318
11.2.3.5 Momentum Operator 319
xvi Advanced Modem Physics
11.2.3.6 Quasiparticle Spectrum . . . . . . .
11.2.3.7 Calculation of the Energy Gap
~
..
11.2.3.8 Quasiparticle Interactions . . . . .
12 . Problems
Appendix A Multipole Analysis of the Radiation Field
A.1 Vector Spherical Harmonics . .
A.2 Plane-Wave Expansion ..
A.3 'fransition Rate. . . . . . .
A.4 Arbitrary Photon Direction .
Appendix B Functions of a Complex Variable
B.1 Convergence .....
B.2 Analytic Functions . .
B.3 Integration ....
B.4 Cauchy's Theorem
B.5 Cauchy's Integral.
B.6 Taylor's Theorem
B.7 Laurent Series ..
B.8 Theory of Residues.
B.9 Zeros of an Analytic Function .
B.10 Analytic Continuation . .
B.10.1 Standard Method
B.10.2 Uniqueness .....
Appendix C Electromagnetic Field
C.1 Lagrangian Field Theory
C.2 Stress Tensor .
C.3 Free Fields .
C.4 Quantization .
C.5 Commutation Relations
C.6 Interaction With External Current
C.6.1 Hamiltonian
C.6.2 Quantization........
Appendix D Irreducible Representations of SU(n)
D.1 Young Tableaux and Young Operators ..
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Contents
D.2 Adjoint Representation .....
D.3 Dimension of the Representation
D.4 Outer Product . . . . . . .
D.5 SU(n-1) Content of SU(n) .
D.6 Some Examples .
D.6.1 Angular Momentum-SU(2)
D.6.2 Sakata Model-SU(3) ....
D.6.3 Giant Resonances-SU(4) ..
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Appendix E Lorentz 'fransformations in Quantum Field Theory 419
E.1 Scalar Field .
E.1.1 States .
E.1.2 Lorentz 'fransformation
E.1.3 Generators .....
E.1.4 Commutation Rules ..
E.2 Dirac Field . . . . . . . . . . .
E.2.1 Lorentz 'fransformation
Appendix F Green's Functions and Other Singular Functions
F.! Commutator at Unequal Times ..
F.2 Green's Functions . . . . . . .
F.2.1 Boundary Conditions
F.3 Time-Ordered Products ....
F.3.1 Scalar Field. . . . . .
F. 3.2 Electromagnetic Field
F.3.3 Dirac Field .
F.3.4 Vector Field . . . . .
Appendix G Dimensional Regularization
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G.1 Dirichlet Integral . 444
G.2 Basic Relation . . 445
G.3 Complex n-plane . 446
G.4 Algebra . . . . . . 447
G.5 Lorentz Metric . . 448
G.6 l'-Matrix Algebra. 449
G.7 Examples . . . . . . . . . . . . . . . 449
G.7.1 Convergent Momentum Integrals.. 450
G.7.2 Vacuum Tadpoles . . . . . . . . . . . . .. 450
xviii Advanced Modern Physics
Go703 Vacuum Polarization 0 0 0 0 0 0 0 0 0 0 0 0
Appendix H Path Integrals and the Electromagnetic Field
HoI Faddeev-Popov Identity
Ho2 Application 0 0 0 0 0 0 0
Ho3 Generating Functional 0
Ho4 Ghosts 0 0 0 0 0 0 0
Ho5 Photon Propagator 0 0 0
Appendix I Metric Conversion
Bibliography
Index
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