Lagrangian analysis and quantum mechanics: a mathematical structure related to asymptotic expansions and the Maslov index
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English French |
| Veröffentlicht: |
Cambridge, Mass.
MIT Pr.
1981
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Aus dem Franz. übers. |
| Beschreibung: | XVI, 271 S. |
| ISBN: | 0262120879 |
Internformat
MARC
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| 100 | 1 | |a Leray, Jean |e Verfasser |4 aut | |
| 240 | 1 | 0 | |a Analyse lagrangienne et mecanique quantique |
| 245 | 1 | 0 | |a Lagrangian analysis and quantum mechanics |b a mathematical structure related to asymptotic expansions and the Maslov index |
| 264 | 1 | |a Cambridge, Mass. |b MIT Pr. |c 1981 | |
| 300 | |a XVI, 271 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Aus dem Franz. übers. | ||
| 650 | 7 | |a Equacoes Diferenciais |2 larpcal | |
| 650 | 4 | |a Lagrange, Fonctions de | |
| 650 | 4 | |a Maslov, Indice de | |
| 650 | 7 | |a Mecanica Quantica (Teoria Quantica) |2 larpcal | |
| 650 | 4 | |a Théorie quantique | |
| 650 | 4 | |a Équations aux dérivées partielles - Théorie asymptotique | |
| 650 | 4 | |a Quantentheorie | |
| 650 | 4 | |a Differential equations, Partial |x Asymptotic theory | |
| 650 | 4 | |a Lagrangian functions | |
| 650 | 4 | |a Maslov index | |
| 650 | 4 | |a Quantum theory | |
| 650 | 0 | 7 | |a Partielle Differentialgleichung |0 (DE-588)4044779-0 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Lagrange-Funktion |0 (DE-588)4166459-0 |2 gnd |9 rswk-swf |
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| 650 | 0 | 7 | |a Maslov-Index |0 (DE-588)4169023-0 |2 gnd |9 rswk-swf |
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Datensatz im Suchindex
| _version_ | 1804116727549657088 |
|---|---|
| adam_text | Contents
Preface xi
Index of Symbols xiii
Index of Concepts xvii
I. The Fourier Transform and Symplectic Group
Introduction 1
§1. Differential Operators, The Metaplectic and Symplectic Groups 1
0. Introduction 1
1. The Metaplectic Group Mp(Z) 1
2. The Subgroup Sp2(/) of Mp(/) 9
3. Differential Operators with Polynomial Coefficients 20
§2. Maslov Indices; Indices of Inertia; Lagrangian Manifolds and
Their Orientations 25
0. Introduction 25
1. Choice of Hermitian Structures on Z(/) 26
2. The Lagrangian Grassmannian A(/) of Z(l) 27
3. The Covering Groups of Sp(/) and the Covering Spaces of A(/) 31
4. Indices of Inertia 37
5. The Maslov Index m on Ax(/) 42
6. The Jump of the Maslov Index m(Ax, A x) at a Point (A, A )
Where dim/n A = 1 47
7. The Maslov Index on Spw(Z); the Mixed Inertia 51
8. Maslov Indices on Aq(l) and Sp?(?) 53
9. Lagrangian Manifolds 55
10. q-Orientation (q = 1, 2, 3, ..., oo) 56
§3. Symplectic Spaces 58
0. Introduction 58
1. Symplectic Space Z 58
2. The Frames of Z 60
3. The q- Frames of Z 61
4. q-Symplectic Geometries 65
Conclusion 65
viii Contents
II. Lagrangian Functions; Lagrangian Differential Operators
Introduction 67
§1. Formal Analysis 68
0. Summary 68
1. The Algebra ^(X) of Asymptotic Equivalence Classes 68
2. Formal Numbers; Formal Functions 73
3. Integration of Elements of #0(Jf) 80
4. Transformation of Formal Functions by Elements of Sp2(/) 86
5. Norm and Scalar product of Formal Functions with Compact
Support 91
6. Formal Differential Operators 97
7. Formal Distributions 102
§2. Lagrangian Analysis 104
0. Summary 104
1. Lagrangian Operators 105
2. Lagrangian Functions on V 109
3. Lagrangian Functions on V 115
4. The Group Sp2(Z) 123
5. Lagrangian Distributions 123
§3. Homogeneous Lagrangian Systems in One Unknown 124
0. Summary 124
1. Lagrangian Manifolds on Which Lagrangian Solutions of all = 0
Are Defined 124
2. Review of E. Cartan s Theory of Pfaffian Forms 125
3. Lagrangian Manifolds in the Symplectic Space Z and in Its
Hypersurfaces 129
4. Calculation of a U 135
5. Resolution of the Lagrangian Equation all = 0 139
6. Solutions of the Lagrangian Equation all = 0 mod(l/v2) with
Positive Lagrangian Amplitude: Maslov s Quantization 143
7. Solution of Some Lagrangian Systems in One Unknown 145
Contents ix
8. Lagrangian Distributions That Are Solutions of a Homogeneous
Lagrangian System 151
Conclusion 151
§4. Homogeneous Lagrangian Systems in Several Unknowns 152
1. Calculation of 2£=1 C/m 152
2. Resolution of the Lagrangian System all = 0 in Which the Zeros of
det a% Are Simple Zeros 156
3. A Special Lagrangian System all = 0 in Which the Zeros of det a%
Are Multiple Zeros 159
III. Schrodinger and Klein-Gordon Equations for One-Electron
Atoms in a Magnetic Field
Introduction 163
§1. A Hamiltonian H to Which Theorem 7.1 (Chapter II, §3)
Applies Easily; the Energy Levels of One-Electron Atoms with the
Zeeman Effect 166
1. Four Functions Whose Pairs Are All in Involution on E3 © E3
Except for One 166
2. Choice of a Hamiltonian H 170
3. The Quantized Tori T(l, m, ri) Characterizing Solutions, Defined
mod(l/v) on Compact Manifolds, of the Lagrangian System
aU = (aL* - L20)U = (aM - M0)U = 0 mod(l/v2) 174
4. Examples: The Schrodinger and Klein-Gordon Operators 179
§2. The Lagrangian Equestion aU = 0 mod(l/v2) (a Associated to H,
U Having Lagrangian Amplitude 0 Defined on a Compact V) 184
0. Introduction 184
1. Solutions of the Equation aU = 0 mod(l/v2) with Lagrangian
Amplitude ^0 Defined on the Tori V[L0, Mo] 185
2. Compact Lagrangian Manifolds V, Other Than the Tori V[L0, Mo],
on Which Solutions of the Equation all = 0 mod(l/v2) with
Lagrangian Amplitude ^0 Exist 190
3. Example: The Schrodinger-Klein-Gordon Operator 204
Conclusion 207
x Contents
§3. The Lagrangian System
aU = (aM — const.) U = {aLi — const.) U = 0
When a Is the Schrddinger-Klein-Gordon Operator 207
0. Introduction 207
1. Commutivity of the Operators a, ajj, and aM Associated to the
Hamiltonians H (§1, Section 2), L2, and M (§1, Section 1) 207
2. Case of an Operator a Commuting with aL2 and aM 210
3. A Special Case 221
4. The Schrodinger-Klein-Gordon Case 226
Conclusion 230
§4. The Schrodinger-Kleiiv-Gordon Equation 230
0. Introduction 230
1. Study of Problem (0.1) without Assumption (0.4) 231
2. The Schrodinger-Klein-Gordon Case 234
Conclusion 237
IV. Dirac Equation with the Zeeman Effect
Introduction 238
§1. A Lagrangian Problem in Two Unknowns 238
1. Choice of Operators Commuting mod(l/v3) 238
2. Resolution of a Lagrangian Problem in Two Unknowns 240
§2. The Dirac Equation 248
0. Summary 248
1. Reduction of the Dirac Equation in Lagrangian Analysis 248
2. The Reduced Dirac Equation for a One-Electron Atom in a
Constant Magnetic Field 254
3. The Energy Levels 258
4. Crude Interpretation of the Spin in Lagrangian Analysis 262
5. The Probability of the Presence of the Electron 264
Conclusion 266
Bibliography 269
|
| any_adam_object | 1 |
| author | Leray, Jean |
| author_facet | Leray, Jean |
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| author_sort | Leray, Jean |
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| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 540 |
| ctrlnum | (OCoLC)7924566 (DE-599)BVBBV002269854 |
| dewey-full | 515.3/53 530.1/2 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis 530 - Physics |
| dewey-raw | 515.3/53 530.1/2 |
| dewey-search | 515.3/53 530.1/2 |
| dewey-sort | 3515.3 253 |
| dewey-tens | 510 - Mathematics 530 - Physics |
| discipline | Physik Mathematik |
| format | Book |
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| genre_facet | Asymptotenentwicklung |
| id | DE-604.BV002269854 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T15:43:06Z |
| institution | BVB |
| isbn | 0262120879 |
| language | English French |
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| physical | XVI, 271 S. |
| publishDate | 1981 |
| publishDateSearch | 1981 |
| publishDateSort | 1981 |
| publisher | MIT Pr. |
| record_format | marc |
| spelling | Leray, Jean Verfasser aut Analyse lagrangienne et mecanique quantique Lagrangian analysis and quantum mechanics a mathematical structure related to asymptotic expansions and the Maslov index Cambridge, Mass. MIT Pr. 1981 XVI, 271 S. txt rdacontent n rdamedia nc rdacarrier Aus dem Franz. übers. Equacoes Diferenciais larpcal Lagrange, Fonctions de Maslov, Indice de Mecanica Quantica (Teoria Quantica) larpcal Théorie quantique Équations aux dérivées partielles - Théorie asymptotique Quantentheorie Differential equations, Partial Asymptotic theory Lagrangian functions Maslov index Quantum theory Partielle Differentialgleichung (DE-588)4044779-0 gnd rswk-swf Lagrange-Funktion (DE-588)4166459-0 gnd rswk-swf Quantenmechanik (DE-588)4047989-4 gnd rswk-swf Maslov-Index (DE-588)4169023-0 gnd rswk-swf Asymptotenentwicklung gnd rswk-swf Quantenmechanik (DE-588)4047989-4 s DE-604 Maslov-Index (DE-588)4169023-0 s Lagrange-Funktion (DE-588)4166459-0 s Partielle Differentialgleichung (DE-588)4044779-0 s Asymptotenentwicklung f HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001491409&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Leray, Jean Lagrangian analysis and quantum mechanics a mathematical structure related to asymptotic expansions and the Maslov index Equacoes Diferenciais larpcal Lagrange, Fonctions de Maslov, Indice de Mecanica Quantica (Teoria Quantica) larpcal Théorie quantique Équations aux dérivées partielles - Théorie asymptotique Quantentheorie Differential equations, Partial Asymptotic theory Lagrangian functions Maslov index Quantum theory Partielle Differentialgleichung (DE-588)4044779-0 gnd Lagrange-Funktion (DE-588)4166459-0 gnd Quantenmechanik (DE-588)4047989-4 gnd Maslov-Index (DE-588)4169023-0 gnd |
| subject_GND | (DE-588)4044779-0 (DE-588)4166459-0 (DE-588)4047989-4 (DE-588)4169023-0 |
| title | Lagrangian analysis and quantum mechanics a mathematical structure related to asymptotic expansions and the Maslov index |
| title_alt | Analyse lagrangienne et mecanique quantique |
| title_auth | Lagrangian analysis and quantum mechanics a mathematical structure related to asymptotic expansions and the Maslov index |
| title_exact_search | Lagrangian analysis and quantum mechanics a mathematical structure related to asymptotic expansions and the Maslov index |
| title_full | Lagrangian analysis and quantum mechanics a mathematical structure related to asymptotic expansions and the Maslov index |
| title_fullStr | Lagrangian analysis and quantum mechanics a mathematical structure related to asymptotic expansions and the Maslov index |
| title_full_unstemmed | Lagrangian analysis and quantum mechanics a mathematical structure related to asymptotic expansions and the Maslov index |
| title_short | Lagrangian analysis and quantum mechanics |
| title_sort | lagrangian analysis and quantum mechanics a mathematical structure related to asymptotic expansions and the maslov index |
| title_sub | a mathematical structure related to asymptotic expansions and the Maslov index |
| topic | Equacoes Diferenciais larpcal Lagrange, Fonctions de Maslov, Indice de Mecanica Quantica (Teoria Quantica) larpcal Théorie quantique Équations aux dérivées partielles - Théorie asymptotique Quantentheorie Differential equations, Partial Asymptotic theory Lagrangian functions Maslov index Quantum theory Partielle Differentialgleichung (DE-588)4044779-0 gnd Lagrange-Funktion (DE-588)4166459-0 gnd Quantenmechanik (DE-588)4047989-4 gnd Maslov-Index (DE-588)4169023-0 gnd |
| topic_facet | Equacoes Diferenciais Lagrange, Fonctions de Maslov, Indice de Mecanica Quantica (Teoria Quantica) Théorie quantique Équations aux dérivées partielles - Théorie asymptotique Quantentheorie Differential equations, Partial Asymptotic theory Lagrangian functions Maslov index Quantum theory Partielle Differentialgleichung Lagrange-Funktion Quantenmechanik Maslov-Index Asymptotenentwicklung |
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