Mathematical concepts of quantum mechanics:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin ; Heidelberg
Springer
2011
|
| Edition: | 2. ed. |
| Series: | Universitext
|
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Item Description: | Literaturangaben |
| Physical Description: | XIII, 382 S. graph. Darst. 24 cm |
| ISBN: | 9783642218651 3642218652 |
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Record in the Search Index
| _version_ | 1805145350126698496 |
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| adam_text |
IMAGE 1
CONTENTS
1 PHYSICAL BACKGROUND 1
1.1 THE DOUBLE-SLIT EXPERIMENT 2
1.2 WAVE FUNCTIONS 4
1.3 STATE SPACE 5
1.4 THE SCHROEDINGER EQUATION 5
2 DYNAMICS 9
2.1 CONSERVATION OF PROBABILITY 9
2.2 SELF-ADJOINTNESS 10
2.3 EXISTENCE OF DYNAMICS 14
2.4 THE FREE PROPAGATOR 16
3 OBSERVABLES 19
3.1 MEAN VALUES AND THE MOMENTUM OPERATOR 19
3.2 OBSERVABLES 20
3.3 THE HEISENBERG REPRESENTATION 21
3.4 SPIN 22
3.5 CONSERVATION LAWS 23
3.5.1 PROBABILITY CURRENT 24
4 QUANTIZATION 27
4.1 QUANTIZATION 27
4.2 QUANTIZATION AND CORRESPONDENCE PRINCIPLE 29
4.3 COMPLEX QUANTUM SYSTEMS 32
4.4 SUPPLEMENT: HAMILTONIAN FORMULATION OF CLASSICAL MECHANICS 36
5 UNCERTAINTY PRINCIPLE AND STABILITY OF ATOMS AND MOLECULES 41
5.1 THE HEISENBERG UNCERTAINTY PRINCIPLE 41
5.2 A REFINED UNCERTAINTY PRINCIPLE 42
5.3 APPLICATION: STABILITY OF ATOMS AND MOLECULES 43
BIBLIOGRAFISCHE INFORMATIONEN HTTP://D-NB.INFO/1011958880
DIGITALISIERT DURCH
IMAGE 2
X CONTENTS
6 SPECTRUM AND DYNAMICS 47
6.1 THE SPECTRUM OF AN OPERATOR 47
6.2 BOUND AND DECAYING STATES 50
6.3 SPECTRA OF SCHROEDINGER OPERATORS 53
6.4 SUPPLEMENT: PARTICLE IN A PERIODIC POTENTIAL 57
7 SPECIAL CASES 61
7.1 THE INFINITE WELL 61
7.2 THE TORUS 62
7.3 THE SQUARE WELL 62
7.4 A PARTICLE ON A SPHERE 63
7.5 THE HYDROGEN ATOM 64
7.6 THE HARMONIC OSCILLATOR 66
7.7 A PARTICLE IN A CONSTANT MAGNETIC FIELD 69
7.8 LINEARIZED GINZBURG-LANDAU EQUATIONS OF SUPERCONDUCTIVITY 70
8 BOUND STATES AND VARIATIONAL PRINCIPLE 75
8.1 VARIATIONAL CHARACTERIZATION OF EIGENVALUES 75
8.2 EXPONENTIAL DECAY OF BOUND STATES 81
8.3 NUMBER OF BOUND STATES 81
9 SCATTERING STATES 89
9.1 SHORT-RANGE INTERACTIONS: SS 1 90
9.2 LONG-RANGE INTERACTIONS: /I 1 92
9.3 WAVE OPERATORS 93
9.4 APPENDIX: THE POTENTIAL STEP AND SQUARE WELL 96
10 EXISTENCE OF ATOMS AND MOLECULES 99
10.1 ESSENTIAL SPECTRA OF ATOMS AND MOLECULES 99
10.2 BOUND STATES OF ATOMS AND BO MOLECULES 101
10.3 OPEN PROBLEMS 104
11 PERTURBATION THEORY: FESHBACH-SCHUR METHOD 107 11.1 THE
FESHBACH-SCHUR METHOD 108
11.2 EXAMPLE: THE ZEEMAN EFFECT 112
11.3 EXAMPLE: TIME-DEPENDENT PERTURBATIONS 114
11.4 BORN-OPPENHEIMER APPROXIMATION 119
11.5 APPENDIX: PROJECTING-OUT PROCEDURE 122
11.6 APPENDIX: PROOF OF THEOREM 11.1 123
12 GENERAL THEORY OF MANY-PARTICLE SYSTEMS 127
12.1 MANY-PARTICLE SCHROEDINGER OPERATORS 127
12.2 SEPARATION OF THE CENTRE-OF-MASS MOTION 129
12.3 BREAK-UPS 131
IMAGE 3
CONTENTS XI
12.4 THE HVZ THEOREM 133
12.5 INTRA- VS. INTER-CLUSTER MOTION 135
12.6 EXPONENTIAL DECAY OF BOUND STATES 136
12.7 REMARKS ON DISCRETE SPECTRUM 137
12.8 SCATTERING STATES 138
13 SELF-CONSISTENT APPROXIMATIONS 141
13.1 HARTREE, HARTREE - FOCK AND GROSS-PITAEVSKI EQUATIONS 141 13.2
APPENDIX: BEC AT T=0 146
14 THE FEYNMAN PATH INTEGRAL 149
14.1 THE FEYNMAN PATH INTEGRAL 149
14.2 GENERALIZATIONS OF THE PATH INTEGRAL 152
14.3 MATHEMATICAL SUPPLEMENT: THE TROTTER PRODUCT FORMULA 153
15 QUASI-CLASSICAL ANALYSIS 155
15.1 QUASI-CLASSICAL ASYMPTOTICS OF THE PROPAGATOR 156
15.2 QUASI-CLASSICAL ASYMPTOTICS OF GREEN'S FUNCTION 160 15.2.1 APPENDIX
163
15.3 BOHR-SOMMERFELD SEMI-CLASSICAL QUANTIZATION 163 15.4
QUASI-CLASSICAL ASYMPTOTICS FOR THE GROUND STATE ENERGY. 165 15.5
MATHEMATICAL SUPPLEMENT: THE ACTION OF THE CRITICAL PATH 167
15.6 APPENDIX: CONNECTION TO GEODESIES . 170
16 RESONANCES 173
16.1 COMPLEX DEFORMATION AND RESONANCES 173
16.2 TUNNELING AND RESONANCES 177
16.3 THE FREE RESONANCE ENERGY 178
16.4 INSTANTONS 180
16.5 POSITIVE TEMPERATURES 182
16.6 PRE-EXPONENTIAL FACTOR FOR THE BOUNCE 184
16.7 CONTRIBUTION OF THE ZERO-MODE 185
16.8 BOHR-SOMMERFELD QUANTIZATION FOR RESONANCES 186
17 QUANTUM STATISTICS 191
17.1 INFORMATION REDUCTION 191
17.2 STATIONARY STATES 195
17.3 QUANTUM STATISTICS: GENERAL FRAMEWORK 197
17.4 HUBERT SPACE APPROACH 199
17.5 QUASI-CLASSICAL LIMIT 200
17.6 REDUCED DYNAMICS 203
17.7 IRREVERSIBILITY 207
IMAGE 4
XII CONTENTS
18 THE SECOND QUANTIZATION 209
18.1 FOCK SPACE AND CREATION AND ANNIHILATION OPERATORS 209 18.2
MANY-BODY HAMILTONIAN 212
18.3 EVOLUTION OF QUANTUM FIELDS 214
18.4 RELATION TO QUANTUM HARMONIC OSCILLATOR 215
18.5 SCALAR FERMIONS 216
18.6 MEAN FIELD REGIME 217
18.7 APPENDIX: THE IDEAL BOSE GAS 220
18.7.1 BOSE-EINSTEIN CONDENSATION 223
19 QUANTUM ELECTRO-MAGNETIC FIELD - PHOTONS 227
19.1 KLEIN-GORDON CLASSICAL FIELD THEORY 227
19.1.1 PRINCIPLE OF MINIMUM ACTION 227
19.1.2 HAMILTONIANS 229
19.1.3 HAMILTONIAN SYSTEM 230
19.1.4 COMPLEXIFICATION OF THE KLEIN-GORDON EQUATION . . . 231 19.2
QUANTIZATION OF THE KLEIN-GORDON EQUATION 232
19.3 THE GAUSSIAN SPACES 234
19.4 WICK QUANTIZATION 236
19.5 FOCK SPACE 238
19.6 QUANTIZATION OF MAXWELL'S EQUATIONS 241
20 STANDARD MODEL OF NON-RELATIVISTIC MATTER AND RADIATION 247
20.1 CLASSICAL PARTICLE SYSTEM INTERACTING WITH AN ELECTRO-MAGNETIC
FIELD 247
20.2 QUANTUM HAMILTONIAN OF NON-RELATIVISTIC QED 250
20.2.1 TRANSLATION INVARIANCE 253
20.2.2 FIBER DECOMPOSITION WITH RESPECT TO TOTAL MOMENTUM 253
20.3 RESCALING AND DECOUPLING SCALAR AND VECTOR POTENTIALS 255 20.3.1
SELF-ADJOINTNESS OF H(E) 256
20.4 MASS RENORMALIZATION 258
20.5 APPENDIX: RELATIVE BOUND ON I(E) AND PULL-THROUGH FORMULAE 260
21 THEORY OF RADIATION 263
21.1 SPECTRUM OF UNCOUPLED SYSTEM 264
21.2 COMPLEX DEFORMATIONS AND RESONANCES 265
21.3 RESULTS 268
21.4 IDEA OF THE PROOF OF THEOREM 21.1 269
21.5 GENERALIZED PAULI-FIERZ TRANSFORMATION 270
21.6 ELIMINATION OF PARTICLE AND HIGH PHOTON ENERGY DEGREES OF FREEDOM
273
21.7 THE HAMILTONIAN H 0 (E, Z) 276
21.8 ESTIMATES ON THE OPERATOR H 0 (E, Z) 277
IMAGE 5
CONTENTS XIII
21.9 GROUND STATE OF H(E) 278
21.10 APPENDIX: ESTIMATES ON I S AND HP (E) 280
21.11 APPENDIX: KEY BOUND ." 282
22 RENORMALIZATION GROUP 285
22.1 MAIN RESULT 286
22.2 A BANACH SPACE OF OPERATORS 287
22.3 THE DECIMATION MAP 289
22.4 THE RENORMALIZATION MAP 291
22.5 DYNAMICS OF RG AND SPECTRA OF HAMILTONIANS 293
22.6 SUPPLEMENT: GROUP PROPERTY OF H P 298
23 MATHEMATICAL SUPPLEMENT: SPECTRAL ANALYSIS 299
23.1 SPACES 299
23.2 OPERATORS ON HILBERT SPACES 303
23.3 INTEGRAL OPERATORS 306
23.4 INVERSES AND THEIR ESTIMATES 307
23.5 SELF-ADJOINTNESS 308
23.6 EXPONENTIAL OF AN OPERATOR 311
23.7 PROJECTIONS 315
23.8 THE SPECTRUM OF AN OPERATOR 316
23.9 FUNCTIONS OF OPERATORS AND THE SPECTRAL MAPPING THEOREM 318
23.10 WEYL SEQUENCES AND WEYL SPECTRUM 321
23.11 THE TRACE, AND TRACE CLASS OPERATORS 327
23.12 OPERATOR DETERMINANTS 332
23.13 TENSOR PRODUCTS 334
23.14 THE FOURIER TRANSFORM 335
24 MATHEMATICAL SUPPLEMENT: THE CALCULUS OF VARIATIONS . . . 339 24.1
FUNCTIONALS 339
24.2 THE FIRST VARIATION AND CRITICAL POINTS 341
24.3 THE SECOND VARIATION 345
24.4 CONJUGATE POINTS AND JACOBI FIELDS 346
24.5 CONSTRAINED VARIATIONAL PROBLEMS 350
24.6 LEGENDRE TRANSFORM AND POISSON BRACKET 351
24.7 COMPLEX HAMILTONIAN SYSTEMS 354
24.8 CONSERVATION LAWS 356
25 COMMENTS ON LITERATURE, AND FURTHER READING 359
REFERENCES 365
INDEX 377 |
| any_adam_object | 1 |
| author | Gustafson, Stephen J. 1972- Sigal, Israel Michael 1945- |
| author_GND | (DE-588)124948987 (DE-588)110612043 |
| author_facet | Gustafson, Stephen J. 1972- Sigal, Israel Michael 1945- |
| author_role | aut aut |
| author_sort | Gustafson, Stephen J. 1972- |
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| dewey-ones | 530 - Physics |
| dewey-raw | 530.12 |
| dewey-search | 530.12 |
| dewey-sort | 3530.12 |
| dewey-tens | 530 - Physics |
| discipline | Physik Mathematik |
| edition | 2. ed. |
| format | Book |
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| spelling | Gustafson, Stephen J. 1972- Verfasser (DE-588)124948987 aut Mathematical concepts of quantum mechanics Stephen J. Gustafson ; Israel Michael Sigal 2. ed. Berlin ; Heidelberg Springer 2011 XIII, 382 S. graph. Darst. 24 cm txt rdacontent n rdamedia nc rdacarrier Universitext Literaturangaben Mathematische Physik (DE-588)4037952-8 gnd rswk-swf Quantenmechanik (DE-588)4047989-4 gnd rswk-swf 1\p (DE-588)4123623-3 Lehrbuch gnd-content Quantenmechanik (DE-588)4047989-4 s Mathematische Physik (DE-588)4037952-8 s DE-604 Sigal, Israel Michael 1945- Verfasser (DE-588)110612043 aut X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=3826539&prov=M&dok_var=1&dok_ext=htm Inhaltstext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024484040&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Gustafson, Stephen J. 1972- Sigal, Israel Michael 1945- Mathematical concepts of quantum mechanics Mathematische Physik (DE-588)4037952-8 gnd Quantenmechanik (DE-588)4047989-4 gnd |
| subject_GND | (DE-588)4037952-8 (DE-588)4047989-4 (DE-588)4123623-3 |
| title | Mathematical concepts of quantum mechanics |
| title_auth | Mathematical concepts of quantum mechanics |
| title_exact_search | Mathematical concepts of quantum mechanics |
| title_full | Mathematical concepts of quantum mechanics Stephen J. Gustafson ; Israel Michael Sigal |
| title_fullStr | Mathematical concepts of quantum mechanics Stephen J. Gustafson ; Israel Michael Sigal |
| title_full_unstemmed | Mathematical concepts of quantum mechanics Stephen J. Gustafson ; Israel Michael Sigal |
| title_short | Mathematical concepts of quantum mechanics |
| title_sort | mathematical concepts of quantum mechanics |
| topic | Mathematische Physik (DE-588)4037952-8 gnd Quantenmechanik (DE-588)4047989-4 gnd |
| topic_facet | Mathematische Physik Quantenmechanik Lehrbuch |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=3826539&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024484040&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT gustafsonstephenj mathematicalconceptsofquantummechanics AT sigalisraelmichael mathematicalconceptsofquantummechanics |