Wave Equations in Higher Dimensions:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Dordrecht [u.a.]
Springer
2011
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 269-287 |
| Physical Description: | XXV, 295 S. |
| ISBN: | 9789400719163 |
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MARC
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Record in the Search Index
| _version_ | 1804148493139312640 |
|---|---|
| adam_text | Titel: Wave equations in higher dimensions
Autor: Dong, Shi-Hai
Jahr: 2011
Contents
Part I Introduction
1 Introduction................................ 3
1 Basic Review............................. 3
2 Motivations and Aims........................ 8
Part II Theory
2 Special Orthogonal Group SQ(N) ................... 13
1 Introduction............................. 13
2 Abstract Groups........................... 15
3 Orthogonal Group SO(AT)...................... 16
4 Tensor Representations of the Orthogonal Group SO( N)...... 16
4.1 Tensors of the Orthogonal Group SO(N).......... 16
4.2 Irreducible Basis Tensors of the SO(21 + 1) ........ 19
4.3 Irreducible Basis Tensors of the SO(21)........... 21
4.4 Dimensions of Irreducible Tensor Representations..... 24
4.5 Adjoint Representation of the SO(N)............ 25
4.6 Tensor Representations of the Groups O(N)........ 26
5 T Matrix Groups........................... 27
5.1 Fundamental Property of r Matrix Groups......... 27
5.2 Case N = 21......................... 28
5.3 Case N = 2l+1....................... 30
6 Spinor Representations of the SO(N)................ 30
6.1 Covering Groups of the SO(N)............... 30
6.2 Fundamental Spinors of the SO(N)............. 33
6.3 Direct Products of Spinor Representations......... 34
6.4 Spinor Representations of Higher Ranks.......... 35
6.5 Dimensions of the Spinor Representations......... 37
7 Concluding Remarks......................... 38
3 Rotational Symmetry and Schrodinger Equation in D-Dimensional
Space ................................... 39
1 Introduction............................. 39
2 Rotation Operator.......................... 39
3 Orbital Angular Momentum Operators............... 41
4 The Linear Momentum Operators.................. 45
5 Radial Momentum Operator..................... 46
6 Spherical Harmonic Polynomials.................. 48
7 Schrodinger Equation for a Two-Body System........... 49
8 Concluding Remarks......................... 50
4 Dirac Equation in Higher Dimensions ................. 51
1 Introduction............................. 51
2 Dirac Equation in (D + 1) Dimensions............... 51
3 The Radial Equations ........................ 52
3.1 The SO(2N+ l)Case.................... 53
3.2 The SO(2N) Case...................... 55
4 Application to Hydrogen Atom................... 57
5 Concluding Remarks......................... 59
5 Klein-Gordon Equation in Higher Dimensions............. 61
1 Introduction............................. 61
2 The Radial Equations ........................ 61
3 Application to Hydrogen-like Atom................. 62
4 Concluding Remarks......................... 64
Part III Applications in Non-relativistic Quantum Mechanics
6 Harmonic Oscillator........................... 67
1 Introduction............................. 67
2 Exact Solutions of Harmonic Oscillator............... 68
3 Recurrence Relations for the Radial Function............ 69
4 Realization of Dynamic Group SU(1, 1) .............. 72
5 Generalization to Pseudoharmonic Oscillator............ 73
5.1 Introduction......................... 73
5.2 Exact Solutions ....................... 74
5.3 Ladder Operators ...................... 75
5.4 Recurrence Relation..................... 78
6 Position and Momentum Information Entropy........... 78
7 Conclusions............................. 79
7 Coulomb Potential............................ 81
1 Introduction............................. 81
2 Exact Solutions ........................... 82
3 Shift Operators............................ 84
4 Mapping Between the Coulomb and Harmonic Oscillator Radial
Functions............................... 87
5 Realization of Dynamic Group SU(1, 1) .............. 88
6 Generalization to Kratzer Potential................. 90
7 Concluding Remarks......................... 96
8 Wavefunction Ansatz Method...................... 97
1 Introduction............................. 97
2 Sextic Potential ........................... 97
3 Singular One Fraction Power Potential............... 100
4 Mixture Potential .......................... 101
5 Non-polynomial Potential...................... 103
6 Screened Coulomb Potential..................... 104
7 Morse Potential ........................... 106
8 Conclusions............................. 108
9 The Levinson Theorem for Schrodinger Equation........... 109
1 Introduction............................. 109
2 Scattering States and Phase Shifts.................. 109
3 Bound States............................. 111
4 The Sturm-Liouville Theorem.................... 111
5 The Levinson Theorem ....................... 112
6 Discussions on General Case .................... 114
7 Comparison Theorem........................ 117
8 Conclusions............................. 117
10 Generalized Hypervirial Theorem ................... 119
1 Introduction............................. 119
2 Generalized Blanchard s and Kramers Recurrence Relations in
Arbitrary Dimensions D....................... 120
3 Applications to Certain Central Potentials.............. 123
3.1 Coulomb-like Potential Case ................ 123
3.2 Harmonic Oscillator..................... 125
3.3 Kratzer Oscillator...................... 126
4 Concluding Remarks......................... 128
11 Exact and Proper Quantization Rules and Langer Modification ... 129
1 Introduction............................. 129
2 WKB Approximation........................ 130
3 Exact Quantization Rule....................... 133
4 Application to Trigonometric Rosen-Morse Potential........ 134
5 Proper Quantization Rule...................... 136
6 Illustrations of Proper Quantization Rule.............. 137
6.1 Energy Spectra for Modified Rosen-Morse Potential .... 137
6.2 Energy Spectra for the Coulombic Ring-Shaped Hartmann
Potential........................... 138
6.3 Energy Spectra for the Manning-Rosen Effective
Potential........................... 141
7 The Langer Modification and Maslov Index in D Dimensions ... 142
8 Calculations of Logarithmic Derivatives of Wavefunction.....144
9 Conclusions.............................146
12 Schrodinger Equation with Position-Dependent Mass.........149
1 Introduction............................. 149
2 Formalism.............................. 149
3 Applications to Harmonic Oscillator and Coulomb Potential .... 151
4 Conclusions............................. 152
Part IV Applications in Relativistic Quantum Mechanics
13 Dirac Equation with the Coulomb Potential..............157
1 Introduction............................. 157
2 Exact Solutions of Hydrogen-like Atoms.............. 157
3 Analysis of the Eigenvalues..................... 162
4 Generalization to the Dirac Equation with a Coulomb Potential
Plus a Scalar Potential........................ 167
4.1 Introduction......................... 167
4.2 Exact Solutions ....................... 167
4.3 Analysis of the Energy Level................ 172
5 Concluding Remarks......................... 177
14 Klein-Gordon Equation with the Coulomb Potential.........181
1 Introduction............................. 181
2 Eigenfunctions and Eigenvalues................... 181
3 Analysis of the Eigenvalues..................... 183
4 Generalization to the Klein-Gordon Equation with a Coulomb
Potential Plus a Scalar Potential................... 188
4.1 Introduction......................... 188
4.2 Eigenfunctions and Eigenvalues............... 188
4.3 Analysis of the Energy Levels................ 191
5 Comparison Theorem........................ 200
6 Concluding Remarks......................... 201
15 The Levinson Theorem for Dirac Equation...............203
1 Introduction............................. 203
2 Generalized Sturm-Liouville Theorem............... 204
3 The Number of Bound States.................... 206
4 The Relativistic Levinson Theorem................. 208
5 Discussions on General Case .................... 213
6 Friedel Theorem........................... 216
7 Comparison Theorem........................ 217
8 Concluding Remarks......................... 218
16 Generalized Hypervirial Theorem for Dirac Equation........219
1 Introduction............................. 219
2 Relativistic Recurrence Relation................... 219
3 Diagonal Case............................ 223
4 Conclusions............................. 224
17 Kaluza-Klein Theory...........................225
1 Introduction.............................225
2 (4 + D)-Dimensional Kaluza-Klein Theories............229
3 Particle Spectrum of Kaluza-Klein Theories for Fermions.....231
4 Warped Extra Dimensions......................233
5 Conclusions.............................233
Part V Conclusions and Outlooks
18 Conclusions and Outlooks........................237
1 Conclusions.............................237
2 Outlooks...............................237
Appendix A Introduction to Group Theory................239
1 Subgroups..............................240
2 Cosets................................242
3 Conjugate Classes..........................243
4 Invariant Subgroups.........................244
Appendix B Group Representations.................... 247
1 Characters.............................. 248
2 Construction of Representations................... 248
3 Reducible and Irreducible Representations............. 250
4 Schur s Lemmas........................... 251
5 Criteria for Irreducibility....................... 251
6 Expansion of Functions in Basis Functions of Irreducible
Representations ........................... 252
Appendix C Fundamental Properties of Lie Groups and Lie
Algebras..................................255
1 Continuous Groups .........................255
2 Infinitesimal Transformations and Lie Algebra...........257
3 Structure of Compact Semisimple Lie Groups and Their
Algebras...............................259
4 Irreducible Representations of Lie Groups and Lie Algebras .... 259
Appendix D Angular Momentum Operators in Spherical
Coordinates................................261
Appendix E Confluent Hypergeometric Functions............265
References...................................269
Index......................................289
|
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| isbn | 9789400719163 |
| language | English |
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| physical | XXV, 295 S. |
| publishDate | 2011 |
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| publisher | Springer |
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| spelling | Dong, Shi-Hai 1969- Verfasser (DE-588)1018145125 aut Wave Equations in Higher Dimensions Shi-Hai Dong Dordrecht [u.a.] Springer 2011 XXV, 295 S. txt rdacontent n rdamedia nc rdacarrier Literaturverz. S. 269-287 Quantentheorie Functional equations Quantum theory Physics Erscheint auch als Online-Ausgabe Wave Equations in Higher Dimensions HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024489145&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Dong, Shi-Hai 1969- Wave Equations in Higher Dimensions Quantentheorie Functional equations Quantum theory Physics |
| title | Wave Equations in Higher Dimensions |
| title_auth | Wave Equations in Higher Dimensions |
| title_exact_search | Wave Equations in Higher Dimensions |
| title_full | Wave Equations in Higher Dimensions Shi-Hai Dong |
| title_fullStr | Wave Equations in Higher Dimensions Shi-Hai Dong |
| title_full_unstemmed | Wave Equations in Higher Dimensions Shi-Hai Dong |
| title_short | Wave Equations in Higher Dimensions |
| title_sort | wave equations in higher dimensions |
| topic | Quantentheorie Functional equations Quantum theory Physics |
| topic_facet | Quantentheorie Functional equations Quantum theory Physics |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024489145&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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