Phase integral method: allowing nearlying transition points
This book is the result of two decades spent by the authors developing and refining the phase-integral method to a high level of precision. The efficiency of the phase-integral method has been shown both analytically and numerically. With the inclusion of supplementary quantities closely related to...
Gespeichert in:
| Format: | Buch |
|---|---|
| Sprache: | English |
| Veröffentlicht: |
New York [u.a.]
Springer
1996
|
| Schriftenreihe: | Springer tracts in natural philosophy
40 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Zusammenfassung: | This book is the result of two decades spent by the authors developing and refining the phase-integral method to a high level of precision. The efficiency of the phase-integral method has been shown both analytically and numerically. With the inclusion of supplementary quantities closely related to new Stokes constants and obtained with the aid of comparison equation techniques, important classes of problems in which transition points may approach each other become accessible to accurate analytical treatment. The treatment of material is mathematically rigorous but it has important physical applications that are found in the adjoined papers. This book will be useful to researchers in theoretical physics, where the problems can be reduced to one-dimensional second-order differential equations of the Schrodinger type for which phase-integral solutions are required. |
| Beschreibung: | X, 250 S. |
| ISBN: | 0387945202 |
Internformat
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| 245 | 1 | 0 | |a Phase integral method |b allowing nearlying transition points |c Nanny Fröman ; Per Olof Fröman |
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| 490 | 1 | |a Springer tracts in natural philosophy |v 40 | |
| 520 | 3 | |a This book is the result of two decades spent by the authors developing and refining the phase-integral method to a high level of precision. The efficiency of the phase-integral method has been shown both analytically and numerically. With the inclusion of supplementary quantities closely related to new Stokes constants and obtained with the aid of comparison equation techniques, important classes of problems in which transition points may approach each other become accessible to accurate analytical treatment. The treatment of material is mathematically rigorous but it has important physical applications that are found in the adjoined papers. This book will be useful to researchers in theoretical physics, where the problems can be reduced to one-dimensional second-order differential equations of the Schrodinger type for which phase-integral solutions are required. | |
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Datensatz im Suchindex
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| adam_text | NANNY FROMAN PER OLOF FROMAN PHASE-INTEGRAL METHOD ALLOWING NEARLYING
TRANSITION POINTS WITH ADJOINED PAPERS BY: A. DZIECIOL, N. FROMAN, P.O.
FROMAN, A. HOKBACK S. LINNAEUS, B. LUNDBORG, AND E. WALLES WITH 17
ILLUSTRATIONS SPRINGER CONTENTS PHASE-INTEGRAL APPROXIMATION OF
ARBITRARY ORDER GENERATED FROM AN UNSPECIFIED BASE FUNCTION 1 1.1
INTRODUCTION 1 1.2 THE SO-CALLED WKB APPROXIMATION, ITS DEFICIENCIES IN
HIGHER ORDER, AND EARLY ATTEMPTS TO REMEDY THESE DEFICIENCIES 4 1.2.1
DERIVATION OF THE WKB APPROXIMATION 4 1.2.2 DEFICIENCIES OF THE WKB
APPROXIMATION IN HIGHER ORDER 8 1.2.3 PHASE-INTEGRAL APPROXIMATION OF
ARBITRARY ORDER, FREED FROM THE FIRST DEFICIENCY 11 1.3 PHASE-INTEGRAL
APPROXIMATION OF ARBITRARY ORDER, GENERATED FROM AN UNSPECIFIED BASE
FUNCTION 15 1.3.1 DIRECT PROCEDURE 15 1.3.2 TRANSFORMATION PROCEDURE 22
1.4 ADVANTAGE OF PHASE-INTEGRAL APPROXIMATION VERSUS WKB APPROXIMATION
IN HIGHER ORDER 23 1.5 RELATIONS BETWEEN SOLUTIONS OF THE SCHRODINGER
EQUATION AND THE G-EQUATION 26 1.5.1 SOLUTIONS OF THE SCHRODINGER
EQUATION AND SOLUTIONS OF THE G-EQUATION EXPRESSED IN TERMS OF EACH
OTHER 27 1.5.2 ERMAKOV-LEWIS INVARIANT 29 1.6 PHASE-INTEGRAL METHOD 30
APPENDIX: PHASE-AMPLITUDE RELATION 31 REFERENCES 34 TECHNIQUE OF THE
COMPARISON EQUATION ADAPTED TO THE PHASE-INTEGRAL METHOD 37 2.1
BACKGROUND 38 2.2 COMPARISON EQUATION TECHNIQUE 42 2.2.1 DIFFERENTIAL
EQUATION FOR 0 46 2.2.2 DETERMINATION OF THE COEFFICIENTS A N FI AND B Q
46 2.2.3 DIFFERENTIAL EQUATION FOR 2 N WHEN N 0 51 2.2.4 REGULARITY
PROPERTIES OF I 2 N AND 2 N WHEN N 0 55 2.2.5 DETERMINATION OF THE
COEFFICIENTS A N ,2N WHEN N 0 58 2.2.6 EXPRESSIONS FOR 2 AND VIII
CONTENTS 2.2.7 BEHAVIOR OF 2 N{Z) M THE NEIGHBORHOOD OF A FIRST- OR
SECOND-ORDER POLE OF Q 2 (Z) WHEN N 0 61 2.3 DERIVATION OF THE
ARBITRARY-ORDER PHASE-INTEGRAL APPROXIMATION FROM THE COMPARISON
EQUATION SOLUTION 66 2.4 SUMMARY OF THE PROCEDURE AND THE RESULTS 68
REFERENCES 70 ADJOINED PAPERS 73 3 PROBLEM INVOLVING ONE TRANSITION ZERO
BY NANNY FROMAN AND PER OLOF FROMAN 75 3.1 INTRODUCTION 75 3.2
COMPARISON EQUATION SOLUTION 76 3.3 PHASE-INTEGRAL APPROXIMATION
OBTAINED FROM THE COMPARISON EQUATION SOLUTION 80 REFERENCES 84 4
RELATIONS BETWEEN DIFFERENT NONOSCILLATING SOLUTIONS OF THE Q-EQUATION
CLOSE TO A TRANSITION ZERO BY ALEKSANDER DZIECIOL, PER OLOF FROMAN, AND
NANNY FROMAN 85 4.1 INTRODUCTION 85 4.2 COMPARISON EQUATION SOLUTIONS 87
4.3 COMPARISON EQUATION EXPRESSIONS FOR NONOSCILLATING SOLUTIONS OF THE
G-EQUATION 90 4.3.1 THE CASE WHEN RE C, INCREASES AS Z MOVES AWAY FROM T
IN THE NEIGHBORHOOD OF THE ANTI-STOKES LINE A 91 4.3.2 THE CASE WHEN RE
DECREASES AS Z MOVES AWAY FROM T IN THE NEIGHBORHOOD OF THE
ANTI-STOKES LINE A 96 4.3.3 SUMMARY OF THE RESULTS FOR THE TWO CASES IN
SECTIONS 4.3.1 AND 4.3.2 97 4.3.4 APPLICATION ILLUSTRATING THE
CONSISTENCY OF THE FORMULAS OBTAINED 99 4.4 SIMPLE FIRST-ORDER FORMULAS
100 4.5 RELATIONS BETWEEN THE A-COEFFICIENTS ASSOCIATED WITH DIFFERENT
Q-FUNCTIONS, IN TERMS OF WHICH A GIVEN SOLUTION TP(Z) IS EXPRESSED 102
4.6 CONDITION FOR DETERMINATION OF REGGE POLE POSITIONS 104 REFERENCES
108 5 CLUSTER OF TWO SIMPLE TRANSITIONS ZEROS BY NANNY FROMAN, PER OLOF
FROMAN, AND BENGT LUNDBORG 109 5.1 INTRODUCTION 109 5.2 WAVE EQUATION
AND PHASE-INTEGRAL APPROXIMATION 111 CONTENTS IX 5.3 COMPARISON EQUATION
119 5.4 COMPARISON EQUATION SOLUTION 120 5.4.1 DETERMINATION OF /?O(Z)
AND KQ 121 5.4.2 DETERMINATION OF IP 2 P AND K 2 P FOR (3 0 123 5.5
PHASE-INTEGRAL SOLUTION OBTAINED FROM THE COMPARISON EQUATION SOLUTION
125 5.6 STOKES CONSTANTS 136 5.7 APPLICATION TO COMPLEX POTENTIAL
BARRIER 138 5.8 APPLICATION TO REGGE POLE THEORY 139 APPENDIX:
PHASE-INTEGRAL SOLUTION OBTAINED FROM THE COMPARISON EQUATION SOLUTION
BY STRAIGHTFORWARD CALCULATION 140 REFERENCES 145 PHASE-INTEGRAL
FORMULAS FOR THE REGULAR WAVE FUNCTION WHEN THERE ARE TURNING POINTS
CLOSE TO A POLE OF THE POTENTIAL BY NANNY FROMAN, PER OLOF FROMAN, AND
STAFFAN LINNAEUS 149 6.1 INTRODUCTION 149 6.2 DEFINITIONS AND
PREPARATORY CALCULATIONS 151 6.2.1 DETERMINATION OF AND A T O 154 6.2.2
DETERMINATION OF IP 2 0 AND AI^P FOR FT 0 156 6.3 COMPARISON EQUATION
CORRESPONDING TO SCATTERING STATES 162 6.3.1 COMPARISON EQUATION
SOLUTION 162 6.3.2 PHASE-INTEGRAL APPROXIMATION OBTAINED FROM THE
COMPARISON EQUATION SOLUTION 164 6.3.3 BEHAVIOR OF THE WAVE FUNCTION
CLOSE TO THE ORIGIN 171 6.3.4 SUMMARY OF FORMULAS IN SECTION 6.3 171 6.4
COMPARISON EQUATION CORRESPONDING TO BOUND STATES 172 6.4.1 QUANTIZATION
CONDITION 172 6.4.2 NORMALIZED WAVE FUNCTION 175 APPENDIX: CALCULATION
OF Q(Z) AND 5 2 + 1 ) 177 REFERENCES 181 NORMALIZED WAVE FUNCTION OF
THE RADIAL SCHRODINGER EQUATION CLOSE TO THE ORIGIN BY NANNY FROMAN, PER
OLOF FROMAN, ERIK WALLES, AND STAFFAN LINNAEUS 183 7.1 INTRODUCTION 183
7.2 0 0 187 7.3 CO = O, NO ^ 0 195 7.4 SUMMARY OF THE RESULTS
OBTAINED IN THE PRESENT CHAPTER AND DISCUSSION OF RESULTS OBTAINED BY
PREVIOUS AUTHORS 198 X CONTENTS REFERENCES 199 8 PHASE-AMPLITUDE METHOD
COMBINED WITH COMPARISON EQUATION TECHNIQUE APPLIED TO AN IMPORTANT
SPECIAL PROBLEM BY PER OLOF FROMAN, ANDERS HOKBACK, AND NANNY FROMAN 201
8.1 INTRODUCTION 201 8.2 QUANTIZATION CONDITION 202 8.3 SOLUTION OF THE
DIFFICULTY AT THE ORIGIN BY MEANS OF COMPARISON EQUATION SOLUTIONS
EXPRESSED IN TERMS OF COULOMB WAVE FUNCTIONS 203 8.4 APPLICATION TO A
TWO-DIMENSIONAL ANHARMONIC OSCILLATOR 206 REFERENCES 209 9 IMPROVED
PHASE-INTEGRAL TREATMENT OF THE COMBINED LINEAR AND COULOMB POTENTIAL BY
STAFFAN LINNAEUS 211 9.1 INTRODUCTION 211 9.2 ENERGY LEVELS 212 9.3
EXPECTATION VALUES 215 APPENDIX: EXPRESSIONS FOR PHASE-INTEGRAL
QUANTITIES IN TERMS OF COMPLETE ELLIPTIC INTEGRALS 220 REFERENCES 222 10
HIGH-ENERGY SCATTERING FROM A YUKAWA POTENTIAL BY STAFFAN LINNAEUS 223
10.1 INTRODUCTION 223 10.2 PHASE SHIFTS 224 10.3 PROBABILITY DENSITY AT
THE ORIGIN 225 APPENDIX: NUMERICAL SOLUTION OF THE SCHRODINGER EQUATION
230 REFERENCES 231 11 PROBABILITIES FOR TRANSITIONS BETWEEN BOUND STATES
IN A YUKAWA POTENTIAL, CALCULATED WITH COMPARISON EQUATION TECHNIQUE BY
STAFFAN LINNAEUS 233 11.1 INTRODUCTION 233 11.2 PHASE-INTEGRAL FORMULAS
234 11.3 COMPARISON EQUATION FORMULAS 236 REFERENCES 240 AUTHOR INDEX
243 SUBJECT INDEX 245
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| id | DE-604.BV010707289 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T17:57:33Z |
| institution | BVB |
| isbn | 0387945202 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-007148082 |
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| physical | X, 250 S. |
| publishDate | 1996 |
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| series | Springer tracts in natural philosophy |
| series2 | Springer tracts in natural philosophy |
| spelling | Phase integral method allowing nearlying transition points Nanny Fröman ; Per Olof Fröman Phase-integral method New York [u.a.] Springer 1996 X, 250 S. txt rdacontent n rdamedia nc rdacarrier Springer tracts in natural philosophy 40 This book is the result of two decades spent by the authors developing and refining the phase-integral method to a high level of precision. The efficiency of the phase-integral method has been shown both analytically and numerically. With the inclusion of supplementary quantities closely related to new Stokes constants and obtained with the aid of comparison equation techniques, important classes of problems in which transition points may approach each other become accessible to accurate analytical treatment. The treatment of material is mathematically rigorous but it has important physical applications that are found in the adjoined papers. This book will be useful to researchers in theoretical physics, where the problems can be reduced to one-dimensional second-order differential equations of the Schrodinger type for which phase-integral solutions are required. WKB approximation Wave equation Wellengleichung (DE-588)4065315-8 gnd rswk-swf WKB-Methode (DE-588)4190133-2 gnd rswk-swf Pfadintegral (DE-588)4173973-5 gnd rswk-swf Wellengleichung (DE-588)4065315-8 s WKB-Methode (DE-588)4190133-2 s DE-604 Pfadintegral (DE-588)4173973-5 s Fröman, Nanny Sonstige oth Fröman, Per Olof Sonstige (DE-588)128102977 oth Springer tracts in natural philosophy 40 (DE-604)BV000692404 40 GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007148082&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Phase integral method allowing nearlying transition points Springer tracts in natural philosophy WKB approximation Wave equation Wellengleichung (DE-588)4065315-8 gnd WKB-Methode (DE-588)4190133-2 gnd Pfadintegral (DE-588)4173973-5 gnd |
| subject_GND | (DE-588)4065315-8 (DE-588)4190133-2 (DE-588)4173973-5 |
| title | Phase integral method allowing nearlying transition points |
| title_alt | Phase-integral method |
| title_auth | Phase integral method allowing nearlying transition points |
| title_exact_search | Phase integral method allowing nearlying transition points |
| title_full | Phase integral method allowing nearlying transition points Nanny Fröman ; Per Olof Fröman |
| title_fullStr | Phase integral method allowing nearlying transition points Nanny Fröman ; Per Olof Fröman |
| title_full_unstemmed | Phase integral method allowing nearlying transition points Nanny Fröman ; Per Olof Fröman |
| title_short | Phase integral method |
| title_sort | phase integral method allowing nearlying transition points |
| title_sub | allowing nearlying transition points |
| topic | WKB approximation Wave equation Wellengleichung (DE-588)4065315-8 gnd WKB-Methode (DE-588)4190133-2 gnd Pfadintegral (DE-588)4173973-5 gnd |
| topic_facet | WKB approximation Wave equation Wellengleichung WKB-Methode Pfadintegral |
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| volume_link | (DE-604)BV000692404 |
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