Rotations, quaternions, and double groups:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Mineola, NY
Dover
2005
|
| Edition: | corr. republ. |
| Subjects: | |
| Online Access: | Publisher description Inhaltsverzeichnis |
| Item Description: | Originally published: Oxford : Clarendon Press ; New York, NY : Oxford University Press, 1986. |
| Physical Description: | XIV, 317 S. graph. Darst. |
| ISBN: | 0486445186 9780486445182 |
Staff View
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Record in the Search Index
| _version_ | 1804135447356506112 |
|---|---|
| adam_text | CONTENTS 0 NOTATION-CONVENTIONS-HOW TO USE THIS BOOK 1 1 INTRODUCTION The Rodrigues programme Rotations by 2-л· Spinor representations 9 18 22 24 2 ALL YOU NEED TO KNOW ABOUT SYMMETRIES, MATRICES, AND GROUPS 1 Symmetry operators in configuration space Description of the point-symmetry operations Specification of the symmetry operations Composition of symmetry operations 2 Eigenvectors of configuration space operators 3 Symmetry operators in function space 4 Matrices and operators 5 Groups 6 All about matrix properties Orthogonal matrices Unitary matrices Hermitian and skew-Hermitian matrices Supermatrices and the direct product Commutation of matrices Matrix functions 7 Quantal symmetry. Observables and infinitesimal operators Symmetries and observables Infinitesimal operators and observables 29 29 30 32 33 35 36 38 42 50 52 55 56 56 58 58 60 62 63 3 A PRIMER ON ROTATIONS AND ROTATION MATRICES 1 Euler angles Rotation matrices in terms of the Euler angles 2 Angle and axis of an orthogonal matrix 3 The matrix of a rotation Я(фп) 4 Euler angles in terms of the angle and axis of rotation 5 A rotation in terms of rotations about orthogonal axes 6 Comments on the parametrization of rotations 65 66 69 70 73 75 76 79 xi
CONTENTS CONTENTS 4 ROTATIONS AND ANGULAR MOMENTUM 1 2 3 4 5 6 7 Infinitesimal rotations The infinitesimal generator: angularmomentum Rotation matrices Commutation Shift operators The eigenfunctions of Iz The irreducible bases for SO(3) Spherical and solid harmonics 8 The Condon and Shortley convention 9 Applications. Matrices for j = 1 andj = į (Pauli matrices) The Pauli matrices, j = į 5 TENSOR BASES: INTRODUCTION TO SPINORS 1 Vectors and spherical vectors 2 Tensor bases and tensor products Symmetrization of tensors 3 Half-integral bases: spinors 6 THE BILINEAR TRANSFORMATION: INTRODUCTION TO SU(2), SU (2), AND ROTATIONS. MORE ABOUT SPINORS 1 The bilinear transformation The inverse 2 Special unitary matrices. The SU(2) group 3 Rotations and SU(2): a first contact 4 Binary rotations as the group generators 5 Do we have a representation of SO(3)? 6 SU(2) plus the inversion: SU (2) Inversion and parity 7 Spinors and their invariants 7 ROTATIONS AND SU(2). THE STEREOGRAPHIC PROJECTION 1 The stereographic projection 2 Geometry and coordinates of the projection 3 The homomorphism between SU(2) and SO(3) The spinor components 8 PROJECTIVE REPRESENTATIONS 80 80 82 84 85 86 87 90 93 94 96 98 99 100 103 104 106 109 110 111 112 113 116 117 118 120 121 124 124 126 128 132 135 1 The group D2 and its SU(2) matrices. Definition of projec tive representations 135 2 Bases of the projective representations 138 Bases and energy levels 140 3 The factor system 142 xii 4 The representations Characters 5 Direct products of representations 6 The covering group Remarks 144 144 145 146 149 9 THE
GEOMETRY OF ROTATIONS 151 1 The unit sphere and therotation poles Conjugate poles Improper rotations 2 The Euler construction 3 Spherical trigonometry revisited 4 The Euler construction in formulae. The Euler-Rodrigues parameters Remarks 5 The conical transformation 10 THE TOPOLOGY OF ROTATIONS 1 2 3 4 5 6 The parametric ball Paths Programme: continuity Homotopy The projective factors Operations, turns, and connectivity 11 THE SPINOR REPRESENTATIONS 1 2 3 4 5 6 Determination of the projective factors The intertwining theorem The character theorem The irreducible representations The projective factors from the Euler-Rodrigues parameters Inverses and conjugates in the Euler-Rodrigues parametrization Conjugation and the choice of the positive hemisphere 7 The character theorem proved in the Euler-Rodrigues parametrization 8 The SU(2) representation of SO(3) 9 Cļ and the irreducible representations of 0(3). The SU (2) representation of 0(3) The representations of C, The irreducible representations of 0(3) The SU (2) representation of 0(3) The factor system for 0(3) 10 Improper point groups xiii 152 154 154 155 157 159 160 162 164 165 167 170 170 174 174 177 177 179 182 182 183 187 189 190 191 194 194 196 197 197 199
1 CONTENTS 12 THE ALGEBRA OF ROTATIONS: QUATERNIONS 1 An entertainment on binary rotations 2 The definition of quaternions 3 Inversion of quaternions. Characterization of their scalar and vector parts 4 Conjugate and normalized quaternions. Inverse quaternions 5 The quaternion units 6 SO(3), SU(2), and quaternions 7 Exponential form of quaternions 8 The conical transformation 9 The rectangular transformation 10 Quaternion algebra and the Clifford algebra In praise of mirrors 11 Applications: angle and axis of rotation and SU(2) matrices in terms of Euler angles 13 DOUBLE GROUPS 1 Introduction and example 2 The double group in the quaternion parametrization 3 Notation and operational rules Intertwining 4 Class structure: Opechowski Theorem 14 THE IRREDUCIBLE REPRESENTATIONS OF SO(3) 201 201 202 205 206 209 211 213 214 217 219 221 223 225 226 230 231 233 235 237 1 More about spinor bases 2 The irreducible representation 3 The bases of the representations 237 241 246 15 EXAMPLES AND APPLICATIONS 248 1 The choice of the positive hemisphere 2 Parametrization of the group elements for D6 D3, C3u. Multiplication tables and factor systems 3 The standard representation 4 The irreducible projective and vector representations The representations of D3 The representations of C3„ 5 The double group Ď3 6 Some applications 248 251 253 255 259 260 261 262 16 SOLUTIONS TO PROBLEMS 266 References 298 Index 305 xiv
|
| adam_txt |
CONTENTS 0 NOTATION-CONVENTIONS-HOW TO USE THIS BOOK 1 1 INTRODUCTION The Rodrigues programme Rotations by 2-л· Spinor representations 9 18 22 24 2 ALL YOU NEED TO KNOW ABOUT SYMMETRIES, MATRICES, AND GROUPS 1 Symmetry operators in configuration space Description of the point-symmetry operations Specification of the symmetry operations Composition of symmetry operations 2 Eigenvectors of configuration space operators 3 Symmetry operators in function space 4 Matrices and operators 5 Groups 6 All about matrix properties Orthogonal matrices Unitary matrices Hermitian and skew-Hermitian matrices Supermatrices and the direct product Commutation of matrices Matrix functions 7 Quantal symmetry. Observables and infinitesimal operators Symmetries and observables Infinitesimal operators and observables 29 29 30 32 33 35 36 38 42 50 52 55 56 56 58 58 60 62 63 3 A PRIMER ON ROTATIONS AND ROTATION MATRICES 1 Euler angles Rotation matrices in terms of the Euler angles 2 Angle and axis of an orthogonal matrix 3 The matrix of a rotation Я(фп) 4 Euler angles in terms of the angle and axis of rotation 5 A rotation in terms of rotations about orthogonal axes 6 Comments on the parametrization of rotations 65 66 69 70 73 75 76 79 xi
CONTENTS CONTENTS 4 ROTATIONS AND ANGULAR MOMENTUM 1 2 3 4 5 6 7 Infinitesimal rotations The infinitesimal generator: angularmomentum Rotation matrices Commutation Shift operators The eigenfunctions of Iz The irreducible bases for SO(3) Spherical and solid harmonics 8 The Condon and Shortley convention 9 Applications. Matrices for j = 1 andj = į (Pauli matrices) The Pauli matrices, j = į 5 TENSOR BASES: INTRODUCTION TO SPINORS 1 Vectors and spherical vectors 2 Tensor bases and tensor products Symmetrization of tensors 3 Half-integral bases: spinors 6 THE BILINEAR TRANSFORMATION: INTRODUCTION TO SU(2), SU'(2), AND ROTATIONS. MORE ABOUT SPINORS 1 The bilinear transformation The inverse 2 Special unitary matrices. The SU(2) group 3 Rotations and SU(2): a first contact 4 Binary rotations as the group generators 5 Do we have a representation of SO(3)? 6 SU(2) plus the inversion: SU'(2) Inversion and parity 7 Spinors and their invariants 7 ROTATIONS AND SU(2). THE STEREOGRAPHIC PROJECTION 1 The stereographic projection 2 Geometry and coordinates of the projection 3 The homomorphism between SU(2) and SO(3) The spinor components 8 PROJECTIVE REPRESENTATIONS 80 80 82 84 85 86 87 90 93 94 96 98 99 100 103 104 106 109 110 111 112 113 116 117 118 120 121 124 124 126 128 132 135 1 The group D2 and its SU(2) matrices. Definition of projec tive representations 135 2 Bases of the projective representations 138 Bases and energy levels 140 3 The factor system 142 xii 4 The representations Characters 5 Direct products of representations 6 The covering group Remarks 144 144 145 146 149 9 THE
GEOMETRY OF ROTATIONS 151 1 The unit sphere and therotation poles Conjugate poles Improper rotations 2 The Euler construction 3 Spherical trigonometry revisited 4 The Euler construction in formulae. The Euler-Rodrigues parameters Remarks 5 The conical transformation 10 THE TOPOLOGY OF ROTATIONS 1 2 3 4 5 6 The parametric ball Paths Programme: continuity Homotopy The projective factors Operations, turns, and connectivity 11 THE SPINOR REPRESENTATIONS 1 2 3 4 5 6 Determination of the projective factors The intertwining theorem The character theorem The irreducible representations The projective factors from the Euler-Rodrigues parameters Inverses and conjugates in the Euler-Rodrigues parametrization Conjugation and the choice of the positive hemisphere 7 The character theorem proved in the Euler-Rodrigues parametrization 8 The SU(2) representation of SO(3) 9 Cļ and the irreducible representations of 0(3). The SU'(2) representation of 0(3) The representations of C, The irreducible representations of 0(3) The SU'(2) representation of 0(3) The factor system for 0(3) 10 Improper point groups xiii 152 154 154 155 157 159 160 162 164 165 167 170 170 174 174 177 177 179 182 182 183 187 189 190 191 194 194 196 197 197 199
1 CONTENTS 12 THE ALGEBRA OF ROTATIONS: QUATERNIONS 1 An entertainment on binary rotations 2 The definition of quaternions 3 Inversion of quaternions. Characterization of their scalar and vector parts 4 Conjugate and normalized quaternions. Inverse quaternions 5 The quaternion units 6 SO(3), SU(2), and quaternions 7 Exponential form of quaternions 8 The conical transformation 9 The rectangular transformation 10 Quaternion algebra and the Clifford algebra In praise of mirrors 11 Applications: angle and axis of rotation and SU(2) matrices in terms of Euler angles 13 DOUBLE GROUPS 1 Introduction and example 2 The double group in the quaternion parametrization 3 Notation and operational rules Intertwining 4 Class structure: Opechowski Theorem 14 THE IRREDUCIBLE REPRESENTATIONS OF SO(3) 201 201 202 205 206 209 211 213 214 217 219 221 223 225 226 230 231 233 235 237 1 More about spinor bases 2 The irreducible representation 3 The bases of the representations 237 241 246 15 EXAMPLES AND APPLICATIONS 248 1 The choice of the positive hemisphere 2 Parametrization of the group elements for D6 D3, C3u. Multiplication tables and factor systems 3 The standard representation 4 The irreducible projective and vector representations The representations of D3 The representations of C3„ 5 The double group Ď3 6 Some applications 248 251 253 255 259 260 261 262 16 SOLUTIONS TO PROBLEMS 266 References 298 Index 305 xiv |
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| discipline | Chemie / Pharmazie Physik Mathematik |
| discipline_str_mv | Chemie / Pharmazie Physik Mathematik |
| edition | corr. republ. |
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| id | DE-604.BV021644037 |
| illustrated | Illustrated |
| index_date | 2024-07-02T15:00:57Z |
| indexdate | 2024-07-09T20:40:39Z |
| institution | BVB |
| isbn | 0486445186 9780486445182 |
| language | English |
| lccn | 2005049630 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-014858778 |
| oclc_num | 60605288 |
| open_access_boolean | |
| owner | DE-29T DE-860 DE-355 DE-BY-UBR DE-739 |
| owner_facet | DE-29T DE-860 DE-355 DE-BY-UBR DE-739 |
| physical | XIV, 317 S. graph. Darst. |
| publishDate | 2005 |
| publishDateSearch | 2005 |
| publishDateSort | 2005 |
| publisher | Dover |
| record_format | marc |
| spelling | Altmann, Simon L. 1924- Verfasser (DE-588)174096615 aut Rotations, quaternions, and double groups Simon L. Altmann corr. republ. Mineola, NY Dover 2005 XIV, 317 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Originally published: Oxford : Clarendon Press ; New York, NY : Oxford University Press, 1986. Rotation groups Quaternions Finite groups Representations of groups Gruppentheorie (DE-588)4072157-7 gnd rswk-swf Physik (DE-588)4045956-1 gnd rswk-swf Drehgruppe (DE-588)4150571-2 gnd rswk-swf Darstellungstheorie (DE-588)4148816-7 gnd rswk-swf Quantentheorie (DE-588)4047992-4 gnd rswk-swf Endliche Gruppe (DE-588)4014651-0 gnd rswk-swf Chemie (DE-588)4009816-3 gnd rswk-swf Gruppe Mathematik (DE-588)4022379-6 gnd rswk-swf Quaternion (DE-588)4176653-2 gnd rswk-swf Gruppe Mathematik (DE-588)4022379-6 s Darstellungstheorie (DE-588)4148816-7 s DE-604 Quaternion (DE-588)4176653-2 s Quantentheorie (DE-588)4047992-4 s Gruppentheorie (DE-588)4072157-7 s Physik (DE-588)4045956-1 s Chemie (DE-588)4009816-3 s Drehgruppe (DE-588)4150571-2 s Endliche Gruppe (DE-588)4014651-0 s 1\p DE-604 http://www.loc.gov/catdir/enhancements/fy0625/2005049630-d.html Publisher description Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014858778&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 | Altmann, Simon L. 1924- Rotations, quaternions, and double groups Rotation groups Quaternions Finite groups Representations of groups Gruppentheorie (DE-588)4072157-7 gnd Physik (DE-588)4045956-1 gnd Drehgruppe (DE-588)4150571-2 gnd Darstellungstheorie (DE-588)4148816-7 gnd Quantentheorie (DE-588)4047992-4 gnd Endliche Gruppe (DE-588)4014651-0 gnd Chemie (DE-588)4009816-3 gnd Gruppe Mathematik (DE-588)4022379-6 gnd Quaternion (DE-588)4176653-2 gnd |
| subject_GND | (DE-588)4072157-7 (DE-588)4045956-1 (DE-588)4150571-2 (DE-588)4148816-7 (DE-588)4047992-4 (DE-588)4014651-0 (DE-588)4009816-3 (DE-588)4022379-6 (DE-588)4176653-2 |
| title | Rotations, quaternions, and double groups |
| title_auth | Rotations, quaternions, and double groups |
| title_exact_search | Rotations, quaternions, and double groups |
| title_exact_search_txtP | Rotations, quaternions, and double groups |
| title_full | Rotations, quaternions, and double groups Simon L. Altmann |
| title_fullStr | Rotations, quaternions, and double groups Simon L. Altmann |
| title_full_unstemmed | Rotations, quaternions, and double groups Simon L. Altmann |
| title_short | Rotations, quaternions, and double groups |
| title_sort | rotations quaternions and double groups |
| topic | Rotation groups Quaternions Finite groups Representations of groups Gruppentheorie (DE-588)4072157-7 gnd Physik (DE-588)4045956-1 gnd Drehgruppe (DE-588)4150571-2 gnd Darstellungstheorie (DE-588)4148816-7 gnd Quantentheorie (DE-588)4047992-4 gnd Endliche Gruppe (DE-588)4014651-0 gnd Chemie (DE-588)4009816-3 gnd Gruppe Mathematik (DE-588)4022379-6 gnd Quaternion (DE-588)4176653-2 gnd |
| topic_facet | Rotation groups Quaternions Finite groups Representations of groups Gruppentheorie Physik Drehgruppe Darstellungstheorie Quantentheorie Endliche Gruppe Chemie Gruppe Mathematik Quaternion |
| url | http://www.loc.gov/catdir/enhancements/fy0625/2005049630-d.html http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014858778&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT altmannsimonl rotationsquaternionsanddoublegroups |