Symmetry and quantum mechanics:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Boca Raton ; London ; New York
CRC Press
[2017]
|
| Schriftenreihe: | Monographs and research notes in mathematics
A Chapman & Hall book |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis Klappentext |
| Beschreibung: | Includes bibliographical references and index |
| Beschreibung: | xxv, 256 Seiten Diagramme |
| ISBN: | 9781498701167 |
Internformat
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| 650 | 4 | |a Symmetry (Physics) | |
| 650 | 4 | |a Quantum theory | |
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Datensatz im Suchindex
| _version_ | 1804177630339006464 |
|---|---|
| adam_text | Contents
A
Author Biography xiii
Preface xv
Plan of the Book xix
List of Figures xxiii
I Spin 1
1 Physical Space 3
1.1 Modeling space............................................. 3
1.2 Real linear operators and matrix groups.................... 7
1.3 SO(3) is the group of rotations........................... 11
2 Spinor Space 17
2.1 Angular momentum in classical mechanics .................. 17
2.2 Modeling spin ............................................ 22
2.3 Complex linear operators and matrix groups ............... 27
2.4 The geometry of 5(7(2) 31
2.4.1 The tangent space to the circle (7(1) = 51......... 31
2.4.2 The tangent space to the sphere 5(7(2) — 53........ 33
2.4.3 The exponential of a matrix........................ 34
2.4.4 5(7(2) is the universal cover of 50(3).............. 40
2.5 Back to spinor space...................................... 43
3 Observables and Uncertainty 47
3.1 Spin observables.......................................... 47
3.2 The Lie algebra au(2) 50
3.3 Commutation relations and uncertainty..................... 55
3.4 Some related Lie algebras................................. 60
3.4.1 Warmup: The Lie algebra u(l)...................... 60
3.4.2 The Lie algebra 012(C)............................. 61
3.4.3 The Lie algebra u(2)............................... 65
ix
x Contents
3.4.4 The Lie algebra gl2(C).............................. 66
4 Dynamics 69
4.1 Time-independent external fields........................... 70
4.2 Time-dependent external fields............................. 74
4.3 The energy-time uncertainty principle...................... 75
4.3.1 Conserved quantities................................ 77
5 Higher Spin 79
5.1 Group representations...................................... 80
5.2 Representations of SU(2)................................... 83
5.3 Lie algebra representations ............................... 87
5.4 Representations of su(2)c = sl2(C)......................... 92
5.5 Spin-5 particles........................................... 98
5.6 Representations of 50(3).................................. 106
5.6.1 The 5o(3)-action................................... 109
5.6.2 Comments about analysis............................ 114
6 Multiple Particles 121
6.1 Tensor products of representations........................ 121
6.2 The Clebsch-Gordan problem................................ 128
6.3 Identical particles—spin only............................. 130
II Position Momentum 133
7 A One-Dimensional World 135
7.1 Position.................................................. 136
7.2 Momentum.................................................. 140
7.3 The Heisenberg Lie algebra and Lie group .............. 143
7.3.1 The meaning of the Heisenberg group action........ 145
7.4 Time-evolution............................................ 147
7.4.1 The free particle................................. 149
7.4.2 The infinite square well.......................... 150
7.4.3 The simple harmonic oscillator.................... 152
8 A Three-Dimensional World 161
8.1 Position.................................................. 161
8.2 Linear momentum........................................... 164
8.2.1 The Heisenberg group H$ and its algebra f)3........ 166
8.3 Angular momentum..................................... 168
8.4 The Lie group G = H3 x 50(3) and its Lie algebra $ .... 171
Contents x*
8.5 Time-evolution................................................ 173
8.5.1 The free particle...................................... 174
8.5.2 The three-dimensional harmonic oscillator............. 174
8.5.3 Central potentials .................................... 176
8.5.4 The infinite spherical well............................ 178
8.6 Two-particle systems ......................................... 180
8.6.1 The Coulomb potential ................................. 182
8.7 Particles with spin........................................... 185
8.7.1 The hydrogen atom ..................................... 189
8.8 Identical particles .......................................... 191
9 Toward a Relativistic Theory 197
9.1 Galilean relativity .......................................... 197
9.2 Special relativity............................................ 206
9.3 SZ/2(C) is the universal cover of SO^ (1,3) 216
9.4 The Dirac equation ........................................... 220
A Appendices 231
A.l Linear algebra ............................................... 231
A. 1.1 Vector spaces and linear transformations............... 231
A. 1.2 Inner product spaces and adjoints...................... 236
A.2 Multivariable calculus........................................ 239
A.3 Analysis .................................................... 242
A.3.1 Hilbert spaces and adjoints............................. 242
A.3.2 Some big theorems....................................... 243
A.4 Solutions to selected exercises .............................. 245
Bibliography 251
Index 253
Mathematics
MONOGRAPHS AND RESEARCH NOTES IN MATHEMATICS
Structured as a dialogue between a mathematician and a physicist,
Symmetry and Quantum Mechanics unites the mathematical topics
of this field into a compelling and physically-motivated narrative that
focuses on the central role of symmetry.
Aimed at advanced undergraduate and beginning graduate students
in mathematics with only a minimal background in physics, this title
is also useful to physicists seeking a mathematical introduction to the
subject. Part I focuses on spin, and covers such topics as Lie groups
and algebras, while part II offers an account of position and momentum
in the context of the representation theory of the Heisenberg group,
along the way providing an informal discussion of fundamental con-
cepts from analysis such as self-adjoint operators oh Hilbert space and
the Stone-von Neumann Theorem. Mathematical theory is applied to
physical examples such as spin-precession in a magnetic field, the har-
monic oscillator, the infinite spherical well, and the hydrogen atom.
Key Features
• Provides an appreciation of quantum mechanics and its many
connections to the field of pure mathematics
• Emphasizes the underlying mathematical structure of the physical
theory
• Offers a motivating introduction to Lie groups and their represen-
tations with a focus on the quantum mechanically relevant Heisen-
berg group H3 and special unitary group SU(2)
• Contains exercises interspersed in each chapter to provide addi-
tional understanding and practice for the reader
Scott Corry is an Associate Professor of Mathematics at Lawrence
University in Appleton, Wisconsin. His research interests include Galois
theory, algebraic geometry, combinatorics, and mathematical physics.
CRC Press
Taylor 8* Francis Croup
an informa business
www.crcpress.com
6000 Broken Sound Parkway, NW
Suite 300, Boca Raton, FL 33487
711 Third Avenue
New York, NV 10017
2 Park Square, Milton Park
Abmgdon, Oxon 0X14 4RN, UK
|
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| isbn | 9781498701167 |
| language | English |
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| series2 | Monographs and research notes in mathematics A Chapman & Hall book |
| spelling | Corry, Scott 1978- Verfasser (DE-588)1163410675 aut Symmetry and quantum mechanics Scott Corry Boca Raton ; London ; New York CRC Press [2017] © 2017 xxv, 256 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Monographs and research notes in mathematics A Chapman & Hall book Includes bibliographical references and index Quantentheorie Symmetry (Physics) Quantum theory Gruppentheorie (DE-588)4072157-7 gnd rswk-swf Quantenmechanik (DE-588)4047989-4 gnd rswk-swf Symmetrie (DE-588)4058724-1 gnd rswk-swf Quantenmechanik (DE-588)4047989-4 s DE-604 Gruppentheorie (DE-588)4072157-7 s Symmetrie (DE-588)4058724-1 s Erscheint auch als 978-1-4987-0117-4 Online-Ausgabe, PDF Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029775714&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029775714&sequence=000002&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Corry, Scott 1978- Symmetry and quantum mechanics Quantentheorie Symmetry (Physics) Quantum theory Gruppentheorie (DE-588)4072157-7 gnd Quantenmechanik (DE-588)4047989-4 gnd Symmetrie (DE-588)4058724-1 gnd |
| subject_GND | (DE-588)4072157-7 (DE-588)4047989-4 (DE-588)4058724-1 |
| title | Symmetry and quantum mechanics |
| title_auth | Symmetry and quantum mechanics |
| title_exact_search | Symmetry and quantum mechanics |
| title_full | Symmetry and quantum mechanics Scott Corry |
| title_fullStr | Symmetry and quantum mechanics Scott Corry |
| title_full_unstemmed | Symmetry and quantum mechanics Scott Corry |
| title_short | Symmetry and quantum mechanics |
| title_sort | symmetry and quantum mechanics |
| topic | Quantentheorie Symmetry (Physics) Quantum theory Gruppentheorie (DE-588)4072157-7 gnd Quantenmechanik (DE-588)4047989-4 gnd Symmetrie (DE-588)4058724-1 gnd |
| topic_facet | Quantentheorie Symmetry (Physics) Quantum theory Gruppentheorie Quantenmechanik Symmetrie |
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