Advanced modern physics: theoretical foundations
Gespeichert in:
| 1. Verfasser: | |
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| Format: | Buch |
| Sprache: | English |
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New Jersey [u.a.]
World Scientific
2010
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| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XVIII, 479 S. graph. Darst. |
| ISBN: | 9814291528 981429151X 9789814291521 9789814291514 |
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MARC
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| 245 | 1 | 0 | |a Advanced modern physics |b theoretical foundations |c John Dirk Walecka |
| 264 | 1 | |a New Jersey [u.a.] |b World Scientific |c 2010 | |
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Datensatz im Suchindex
| _version_ | 1804141094632423424 |
|---|---|
| adam_text | Titel: Advanced modern physics
Autor: Walecka, John Dirk
Jahr: 2010
Contents
Preface vii
1. Introduction 1
2. Quantum Mechanics (Revisited) 9
2.1 Linear Vector Spaces...................... 9
2.1.1 Three-Dimensional Vectors.............. 9
2.1.2 n-Dimensions...................... 10
2.2 Hubert Space.......................... 11
2.2.1 Example........................ 11
2.2.2 Definition........................ 12
2.2.3 Relation to Linear Vector Space........... 13
2.2.4 Abstract State Vector................. 13
2.3 Linear Hermitian Operators.................. 14
2.3.1 Eigenfunctions..................... 15
2.3.2 Eigenstates of Position ................ 16
2.4 Abstract Hubert Space .................... 18
2.4.1 Inner Product..................... 18
2.4.2 Completeness...................... 19
2.4.3 Linear Hermitian Operators ............. 19
2.4.3.1 Eigenstates.................. 20
2.4.3.2 Adjoint Operators.............. 20
2.4.4 Schrödinger Equation................. 20
2.4.4.1 Stationary States............... 23
2.5 Measurements......................... 23
2.5.1 Coordinate Space................... 23
Advanced Modern Physics
2.5.2 Abstract Form..................... 25
2.5.3 Reduction of the Wave Packet............ 26
2.5.4 Stern-Gerlach Experiment .............. 27
2.6 Quantum Mechanics Postulates................ 29
2.7 Many-Particle Hubert Space................. 30
2.7.1 Simple Harmonie Oscillator.............. 30
2.7.2 Bosons......................... 31
2.7.3 Fermions........................ 33
Angular Momentum 35
3.1 Translations .......................... 35
3.2 Rotations............................ 37
3.3 Angular Momentum Operator ................ 39
3.4 Eigenvalue Spectrum...................... 40
3.5 Coupling of Angular Momenta................ 46
3.6 Recoupling........................... 53
3.7 Irreducible Tensor Operators................. 55
3.8 The Wigner-Eckart Theorem................. 57
3.9 Finite Rotations........................ 59
3.9.1 Properties ....................... 61
3.9.2 Tensor Operators ................... 63
3.9.3 Wigner-Eckart Theorem (Completed)........ 64
3.10 Tensor Products........................ 65
3.11 Vector Model.......................... 66
Scattering Theory 69
4.1 Interaction Picture....................... 69
4.2 Adiabatic Approach...................... 70
4.3 {/-Operator........................... 72
4.4 ^/-Operator for Finite Times................. 76
4.5 The 5-Matrix.......................... 77
4.6 Time-Independent Analysis.................. 79
4.7 Scattering State........................ 83
4.8 Transition Rate......................... 86
4.9 Unitarity............................ 89
4.10 Example: Potential Scattering................ 90
4.10.1 Green s Function (Propagator)............ 90
4.10.2 Scattering Wave Function............... 92
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
Advanced Modern Physics
6.3.3.1 DiracTriplet................. 139
6.3.3.2 Scalar Octet ................. 141
6.3.3.3 Interacting Fields (Dirac-Scalar)...... 142
6.4 Phase Invariance........................ 142
6.4.1 Global Phase Invariance................ 142
6.4.2 Local Phase Invariance................ 143
6.5 Yang-Mills Theories...................... 145
6.6 Chiral Symmetry........................ 151
6.6.1 cr-Model......................... 156
6.6.2 Spontaneous Symmetry Breaking .......... 158
6.7 Lorentz Transformations ................... 162
Feynman Rules 163
7.1 Wick s Theorem........................ 164
7.2 Example (Dirac-Scalar).................... 171
7.2.1 Scattering Amplitudes................. 172
7.2.2 Self-Energies...................... 175
7.2.3 Vacuum Amplitude.................. 176
7.3 Feynman Diagrams ...................... 177
7.4 Feynman Rules......................... 180
7.5 Cancellation of Disconnected Diagrams........... 182
7.6 Mass Renormalization..................... 185
Quantum Electrodynamics (QED) 187
8.1 Classical Theory........................ 187
8.2 Hamiltonian.......................... 188
8.3 Quantization.......................... 189
8.4 Photon Propagator....................... 191
8.5 Second-Order Processes.................... 196
8.5.1 Scattering Amplitudes................. 196
8.5.2 Self-Energies...................... 198
8.6 QED With Two Leptons................... 200
8.7 Cross Sections......................... 200
8.7.1 e-+n--+e-+p-.................. 200
8.7.1.1 Scattering Amplitude............ 201
8.7.1.2 Cross Section................. 201
8.7.1.3 Traces..................... 205
8.7.1.4 Cross Section (Continued).......... 206
Contents xiii
8.7.1.5 Limiting Cases................ 208
8.7.1.6 Miller Scattering .............. 209
8.7.2 e+ + er - p+ + p~.................. 210
8.7.2.1 Scattering Amplitude............ 210
8.7.2.2 Cross Section................. 211
8.7.2.3 Limiting Cases................ 214
8.7.2.4 Colliding Beams............... 214
8.8 QED in External Field .................... 214
8.8.1 Nuclear Coulomb Field................ 216
8.8.2 Bremsstrahlung .................... 217
8.8.3 Pair Production.................... 218
8.9 Scattering Operator Sfxt in Order e3............. 219
8.10 Feynman Rules for QED ................... 221
8.10.1 General Scattering Operator............. 221
8.10.2 Feynman Diagrams.................. 222
8.10.3 Feynman Rules..................... 223
8.10.3.1 Coordinate Space .............. 223
8.10.3.2 Momentum Space.............. 223
9. Higher-Order Processes 225
9.1 Example-Scattering in External Field............ 225
9.1.1 Feynman Diagrams.................. 226
9.1.2 First-Order..............?........ 226
9.1.3 Vertex Insertion.................... 226
9.1.4 Vacuum Polarization ................. 226
9.1.5 Self-Energy Insertions................. 227
9.2 Ward s Identity......................... 228
9.3 Electron Self-Energy...................... 228
9.3.1 General Form ..................... 229
9.3.2 Evaluation....................... 230
9.3.3 Mass Renormalization................. 232
9.4 Vertex.............................. 234
9.4.1 General Form ..................... 234
9.4.2 Ward s Identity .................... 235
9.4.3 Evaluation....................... 236
9.4.4 The Constant L.................... 238
9.4.5 The Infrared Problem................. 239
9.4.6 Schwinger Moment................... 240
9.5 External Lines and Wavefunction Renormalization..... 243
xiv Advanced Modern Physics
9.5.1 Cancellation of Divergences.............. 247
9.6 Vacuum Polarization...................... 247
9.6.1 Evaluation....................... 248
9.6.2 General Form ..................... 249
9.6.3 Limiting Cases..................... 250
9.6.4 Insertion........................ 250
9.6.5 Charge Renormalization................ 252
9.6.6 Charge Strength.................... 254
9.7 Renormalization Theory.................... 255
9.7.1 Proper Self-Energies.................. 255
9.7.2 Proper Vertex..................... 256
9.7.3 Ward s Identity .................... 257
9.7.4 Ward s Vertex Construct............... 259
9.7.5 Finite Parts ...................... 260
9.7.6 Proof of Renormalization............... 262
9.7.7 The Renormalization Group............. 264
10. Path Integrals 265
10.1 Non-Relativistic Quantum Mechanics with One Degree of
Freedom............................. 265
10.1.1 General Relations................... 266
10.1.2 Infinitesimals...................... 268
10.1.3 Transition Amplitude and Path Integral....... 270
10.1.4 Classical Limit..................... 272
10.1.5 Superposition...................... 272
10.1.6 Matrix Elements.................... 273
10.1.7 Crucial Theorem of Abers and Lee.......... 275
10.1.8 Functional Derivative................. 276
10.1.9 Generating Functional................. 279
10.2 Many Degrees of Freedom................... 279
10.2.1 Gaussian Integrals................... 280
10.3 Field Theory.......................... 282
10.3.1 Fields as Coordinates................. 282
10.3.2 Measure......................... 283
10.3.3 Generating Functional................. 283
10.3.4 Convergence...................... 283
10.3.4.1 Euclidicity Postulate............. 283
10.3.4.2 Adiabatic Damping............. 284
10.4 Scalar Field........................... 285
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 Böse 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 Superßuid4He..................... 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 Kq............ 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 Modern Physics
11.2.3.6 Quasiparticle Spectrum........... 319
11.2.3.7 Calculation of the Energy Gap A...... 320
11.2.3.8 Quasiparticle Interactions.......... 322
12. Problems 323
Appendix A Multipole Analysis of the Radiation Field 369
A.l Vector Spherical Harmonics.................. 370
A.2 Plane-Wave Expansion .................... 373
A.3 Transition Rate......................... 375
A.4 Arbitrary Photon Direction.................. 379
Appendix B Functions of a Complex Variable 383
B.l Convergence.......................... 383
B.2 Analytic Functions....................... 383
B.3 Integration........................... 384
B.4 Cauchy s Theorem....................... 385
B.5 Cauchy s Integral........................ 385
B.6 Taylor s Theorem ....................... 387
B.7 Laurent Series......................... 387
B.8 Theory of Residues....................... 387
B.9 Zeros of an Analytic Function................. 389
B.10 Analytic Continuation..................... 389
B.10.1 Standard Method................... 390
B.10.2 Uniqueness....................... 391
Appendix C Electromagnetic Field 393
C.l Lagrangian Field Theory................... 394
C.2 Stress Tensor.......................... 395
C.3 Free Fields........................... 395
C.4 Quantization.......................... 397
C.5 Commutation Relations.................... 399
C.6 Interaction With External Current.............. 401
C.6.1 Hamiltonian...................... 402
C.6.2 Quantization...................... 404
Appendix D Irreducible Representations of SU(n) 407
D.l Young Tableaux and Young Operators............ 409
Contents xvii
D.2 Adjoint Representation.................... 410
D.3 Dimension of the Representation............... 412
D.4 Outer Product......................... 412
D.5 SU(n-l) Content of SU(n)................... 413
D.6 Some Examples......................... 414
D.6.1 Angular Momentum-SU(2).............. 414
D.6.2 Sakata Model-SU(3).................. 415
D.6.3 Giant Resonances-SU(4)............... 417
Appendix E Lorentz Transformations in Quantum Field Theory 419
E.l Scalar Field........................... 420
E.l.l States.......................... 420
E.l.2 Lorentz Transformation................ 422
E.1.3 Generators....................... 423
E.l.4 Commutation Rules.................. 426
E.2 Dirac Field........................... 427
E.2.1 Lorentz Transformation................ 427
Appendix F Green s Functions and Other Singular Functions 429
F.l Commutator at Unequal Times................ 429
F.2 Green s Functions....................... 434
F.2.1 Boundary Conditions................. 437
F.3 Time-Ordered Products.................... 439
F.3.1 Scalar Field....................... 439
F.3.2 Electromagnetic Field................. 440
F.3.3 Dirac Field....................... 440
F.3.4 Vector Field...................... 442
Appendix G Dimensional Regularization 443
G.l Dirichlet Integral........................ 444
G.2 Basic Relation......................... 445
G.3 Complex n-plane........................ 446
G.4 Algebra............................. 447
G.5 Lorentz Metrie......................... 448
G.6 7-Matrix Algebra........................ 449
G.7 Examples............................ 449
G.7.1 Convergent Momentum Integrals........... 450
G.7.2 Vacuum Tadpoles................... 450
xviii Advanced Modern Physics
G.7.3 Vacuum Polarization ................. 450
Appendix H Path Integrals and the Electromagnetic Field 453
H.l Faddeev-Popov Identity.................... 453
H.2 Application........................... 455
H.3 Generating Functional..................... 457
H.4 Ghosts ............................. 458
H.5 Photon Propagator....................... 458
Appendix I Metrie Conversion 461
Bibliography 463
Index 467
|
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| spelling | Walecka, John Dirk 1932- Verfasser (DE-588)132844788 aut Advanced modern physics theoretical foundations John Dirk Walecka New Jersey [u.a.] World Scientific 2010 XVIII, 479 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Physics Theoretische Physik (DE-588)4117202-4 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Theoretische Physik (DE-588)4117202-4 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018946360&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Walecka, John Dirk 1932- Advanced modern physics theoretical foundations Physics Theoretische Physik (DE-588)4117202-4 gnd |
| subject_GND | (DE-588)4117202-4 (DE-588)4123623-3 |
| title | Advanced modern physics theoretical foundations |
| title_auth | Advanced modern physics theoretical foundations |
| title_exact_search | Advanced modern physics theoretical foundations |
| title_full | Advanced modern physics theoretical foundations John Dirk Walecka |
| title_fullStr | Advanced modern physics theoretical foundations John Dirk Walecka |
| title_full_unstemmed | Advanced modern physics theoretical foundations John Dirk Walecka |
| title_short | Advanced modern physics |
| title_sort | advanced modern physics theoretical foundations |
| title_sub | theoretical foundations |
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| topic_facet | Physics Theoretische Physik Lehrbuch |
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