The Schrödinger and Riccati equations:
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
1999
|
| Series: | Lecture notes in chemistry
70 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIV, 218 S. graph. Darst. |
| ISBN: | 3540651055 |
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Record in the Search Index
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| adam_text | S. FRAGA J.M. GARCIA DE LA VEGA E.S. FRAGA THE SCHROEDINGER AND RICCATI
EQUATIONS SPRINGER TABLE OF CONTENTS 1 INTRODUCTION 1 THE LINEAR
SCHROEDINGER EQUATION 7 2 DERIVATION OF THE SCHROEDINGER EQUATION 8 2.1
THE DIRAC EQUATION 8 2.2 THE BREIT CORRECTION 10 2.3 THE GENERALIZED
DIRAC-BREIT EQUATION 10 2.3.1 SEPARATION OF THE COORDINATES OF THE
CENTRE OF MASS 11 2.3.2 REDUCTION TO NON-RELATIVISTIC FORM 14 2.4 THE
SCHROEDINGER HAMILTONIAN OPERATOR 15 2.4.1 ATOMS 16 2.4.2 MOLECULES 18
2.5 THE INVARIANT FORM OF THE HAMILTONIAN OPERATOR 18 3 THE SCHROEDINGER
EQUATION IN POSITION SPACE 24 3.1 THE HAMILTONIAN OPERATOR AND ITS
EIGENFUNCTIONS 24 3.1.1 THE HAMILTONIAN OPERATOR 26 3.1.2. THE
EIGENFUNCTIONS 26 3.2 LOCAL PROPERTIES 31 3.2.1 NODES AND NODAL PLANES
32 3.2.2 LOCAL ENERGIES 33 3.2.3 COALESCENCE AND CUSP CONDITIONS 36
3.2.4 NULL KINETIC- AND POTENTIAL-ENERGY REGIONS 37 3.3 GLOBAL
PROPERTIES 38 3.3.1 THE POTENTIAL ENERGY HYPERSURFACE 39 3.3.2 THE
HELLMANN-FEYNMAN THEOREM 40 3.3.3 THE HYPERVIRIAL THEOREM 41 3.3.4 THE
VIRIAL THEOREM 43 4 THE SCHROEDINGER EQUATION IN MOMENTUM SPACE 46 4.1
INTRODUCTION 46 4.1.1 THE R-REPRESENTATION VERSUS THE P-REPRESENTATION
46 4.1.2 THE FOURIER TRANSFORM METHOD 47 4.2 THE TRANSFORMED EQUATION 49
VIII 4.2.1 GENERAL FORMULATION 49 4.2.2 THE HYDROGEN ATOM 52 4.3 THE
TRANSFORMED FUNCTIONS 58 4.3.1 HYDROGENIC ORBITALS 58 4.4 PROPERTIES AND
EXPECTATION VALUES 60 4.4.1 SYMMETRY 60 4.4.2 MOMENTUM DENSITY
DISTRIBUTIONS 63 4.4.3 EXPECTATION VALUES 65 5 THE LOCAL SCHROEDINGER
EQUATION 80 5.1 ALTERNATIVES FOR THE COMPUTATIONAL SOLUTION OF THE
SCHROEDINGER EQUATION 80 5.1.1 COMPUTATIONAL SOLUTIONS OF THE GLOBAL
PROBLEM 80 5.1.2 LOCAL VS. GLOBAL COMPUTATIONAL SOLUTIONS 83 5.2 THE
LOCAL ENERGY METHODS 83 5.2.1 THE FROST EQUATION 83 5.2.2 REDUCED LOCAL
ENERGIES 88 5.3 LOCAL ORBITAL ENERGIES 90 6 THE TIME-DEPENDENT
SCHROEDINGER EQUATION 92 6.1 THE EQUATION AND ITS SOLUTIONS 92 6.2
TIME-EVOLUTION OPERATORS 94 6.3 TIME-EVOLUTION REPRESENTATIONS 95 6.3 *
THE SCHROEDINGER REPRESENTATION 95 6.3.2 THE HEISENBERG REPRESENTATION 96
6.3.3 THE INTERACTION REPRESENTATION 97 6.4 THE SOLUTION OF THE
INTERACTION EVOLUTION EQUATION 98 6.4.1 THE PERTURBATION SERIES
EXPANSION 98 6.4.2 THE MAGNUS FORMULA 100 6.5 EVOLUTION OF THE SYSTEM:
TRANSITION PROBABILITIES 103 THE NON-LINEAR SCHROEDINGER EQUATION 105 7
THE NON-LINEAR SCHROEDINGER EQUATION 106 7.1 SOLITARY WAVES AND SOLITONS
106 7.2 THE 1-DIMENSIONAL NLS EQUATION 107 7.2.1 CONSERVATION LAWS AND
CONSERVED QUANTITIES 108 7.2.2 ANALYTIC SOLUTION 110 7.3 NUMERICAL
SOLUTION OF THE 1-DIMENSIONAL NLS EQUATION 111 7.3.1 THE FINITE ELEMENT
METHOD 111 7.3.2 MESH REFINEMENT TECHNIQUES 113 7.4 THE GENERALIZED
1-DIMENSIONAL NLS EQUATIONS 114 7.4.1 EQUATION I T + * ** + Q C IOI 2 0
+ Q Q IOI 4 0 + I Q M *L^ + I * * 1*1 2 * * = 0 1 15 7.4.2 EQUATION [* *
= - * ** + *(*)* + V(X,T)0 + WO 118 7.5 THE PROBLEM OF HIGHER DIMENSIONS
121 IX THE RICCATI EQUATION 123 8 THE RICCATI EQUATION AND ITS SOLUTION
12 4 8.1 DERIVATION OF THE RICCATI EQUATION 124 8.2 SOLUTION OF THE
RICCATI EQUATION 127 8.2.1 SOLUTION OF THE GENERAL RICCATI EQUATION 127
8.2.2 SOLUTION OF THE TRANSFORMED RICCATI EQUATION 130 8.2.3 SOLUTION OF
THE SPECIAL RICCATI EQUATION 131 9 QUANTUM-MECHANICAL APPLICATIONS OF
THE RICCATI EQUATION 132 9.1 INTRODUCTION 132 9.1.1 THE STURM-LIOUVILLE
EQUATION 132 9.1.2 THE 1-DIMENSIONAL SCHROEDINGER EQUATION 133 9.1.3 THE
RICCATI EQUATION FOR THE LOGARITHMIC DERIVATIVE.. 134 9.2 SOLUTION OF
STURM-LIOUVILLE EQUATIONS 136 9.2.1 EQUATIONS * (*,*) + *(*)* (*,*) +
*(*,*)* (*,*) = * 136 9.2.2 EQUATIONS 4 (E,X) + B(E,X)4 (E,X) = 0 139
THE SCHROEDINGER-RICCATI EQUATION 145 10 THE SCHROEDINGER-RICCATI EQUATION
146 10.1 FORMULATION FOR SPIN-FREE FUNCTIONS 146 10.1.1 BASIC
DEFINITIONS AND RELATIONSHIPS 146 10.1.2 THE RICCATI LOCAL ENERGIES 148
10.1.3 THE SCHROEDINGER-RICCATI EQUATION 150 10.1.4 EVALUATION OF THE
CORRECTION FUNCTION
10.1.5 EVALUATION OF THE ENERGY 155 10.2 FORMULATION FOR ANTISYMMETRIC
FUNCTIONS 157 10.2.1 TWO-ELECTRON SYSTEMS 157 10.2.2 GENERAL CASE 158
10.3 APPLICABILITY OF THE SCHROEDINGER-RICCATI EQUATION 159 11 NUMERICAL
EXPERIENCE WITH THE SCHROEDINGER-RICCATI EQUATION ... 160 11.1
1-DIMENSIONAL SCHROEDINGER EQUATION 161 11.2 THE GROUNDSTATE OF THE
HYDROGEN ATOM 164 11.2.1 PRELIMINARY TEST 165 11.2.2 EVALUATION OF * AND
E
11.3 GROUNDSTATESOF IONS OF THE HE-ISOELECTRONIC SERIES 171 11.3.1
EVALUATION OF THE BASIC QUANTITIES 173 11.3.2 EVALUATION OF THE
QUANTITIES TW 177 11.3.3 EVALUATION OF THE ENERGY CONTRIBUTIONS 184 12
REFERENCES AND BIBLIOGRAPHY 187 12.1 REFERENCES 187 12.2 BIBLIOGRAPHY
213 APPENDIX. MATRIX NOTATION 219
|
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| author | Fraga, Serafin 1931- García de la Vega, José Manuel 1952- Fraga, Eric S. 1959- |
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| indexdate | 2024-07-09T18:24:35Z |
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| isbn | 3540651055 |
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| physical | XIV, 218 S. graph. Darst. |
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| spelling | Fraga, Serafin 1931- Verfasser (DE-588)120518120 aut The Schrödinger and Riccati equations S. Fraga ; J. M. García de la Vega ; E. S. Fraga Berlin [u.a.] Springer 1999 XIV, 218 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Lecture notes in chemistry 70 Chimie quantique - Mathématiques ram Riccati, Équation de ram Schrödinger, Équation de ram Mathematik Quantum chemistry Mathematics Riccati equation Schrödinger equation Schrödinger-Gleichung (DE-588)4053332-3 gnd rswk-swf Riccati-Differentialgleichung (DE-588)4230752-1 gnd rswk-swf (DE-588)4143413-4 Aufsatzsammlung gnd-content Riccati-Differentialgleichung (DE-588)4230752-1 s DE-604 Schrödinger-Gleichung (DE-588)4053332-3 s García de la Vega, José Manuel 1952- Verfasser (DE-588)120518163 aut Fraga, Eric S. 1959- Verfasser (DE-588)120518171 aut Lecture notes in chemistry 70 (DE-604)BV000000216 70 GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008314321&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Fraga, Serafin 1931- García de la Vega, José Manuel 1952- Fraga, Eric S. 1959- The Schrödinger and Riccati equations Lecture notes in chemistry Chimie quantique - Mathématiques ram Riccati, Équation de ram Schrödinger, Équation de ram Mathematik Quantum chemistry Mathematics Riccati equation Schrödinger equation Schrödinger-Gleichung (DE-588)4053332-3 gnd Riccati-Differentialgleichung (DE-588)4230752-1 gnd |
| subject_GND | (DE-588)4053332-3 (DE-588)4230752-1 (DE-588)4143413-4 |
| title | The Schrödinger and Riccati equations |
| title_auth | The Schrödinger and Riccati equations |
| title_exact_search | The Schrödinger and Riccati equations |
| title_full | The Schrödinger and Riccati equations S. Fraga ; J. M. García de la Vega ; E. S. Fraga |
| title_fullStr | The Schrödinger and Riccati equations S. Fraga ; J. M. García de la Vega ; E. S. Fraga |
| title_full_unstemmed | The Schrödinger and Riccati equations S. Fraga ; J. M. García de la Vega ; E. S. Fraga |
| title_short | The Schrödinger and Riccati equations |
| title_sort | the schrodinger and riccati equations |
| topic | Chimie quantique - Mathématiques ram Riccati, Équation de ram Schrödinger, Équation de ram Mathematik Quantum chemistry Mathematics Riccati equation Schrödinger equation Schrödinger-Gleichung (DE-588)4053332-3 gnd Riccati-Differentialgleichung (DE-588)4230752-1 gnd |
| topic_facet | Chimie quantique - Mathématiques Riccati, Équation de Schrödinger, Équation de Mathematik Quantum chemistry Mathematics Riccati equation Schrödinger equation Schrödinger-Gleichung Riccati-Differentialgleichung Aufsatzsammlung |
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