Phase-Integral Method: Allowing Nearlying Transition Points

257 Pages·1996·5.88 MB·English

by  Nanny Fröman

Springer Tracts in Natural
Philosophy
Volume 40
Edited by C. Truesdell
Springer
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Nanny Froman Per Olaf 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
Nanny Froman Per Olof Froman
Department of Theoretical Physics Department of Theoretical Physics
University of Uppsala University of Uppsala
Box 803 Box 803
S-7S1 08 Uppsala S-7Sl 08 Uppsala
Sweden Sweden
Mathematical Subject Classification (1991): 810xx
Library of Congress Cataloging in Publication Data
Froman, Nanny.
Phase-integral method: allowing nearlying transition
points 1 Nanny Froman, Per Olof Froman.
p. cm. - (Springer tracts in natural philosophy; v. 40)
Includes bibliographical references and indexes.
ISBN-13 :978-1-4612-7511-4 e-ISBN -13 :978-1-4612-2342-9
DOl: 10.1007/978-1-4612-2342-9
1. WKB approximation. 2. Wave equation. I. Froman, Per OIof.
II. Title. III. Series.
QC20.7.W53F76 1995
530.1'24-dc20 95-12919
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Preface
The efficiency of the phase-integral method developed by the present au
thors has been shown both analytically and numerically in many publica
tions. With the inclusion of supplementary quantities, closely related to
new Stokes constants and obtained with the aid of comparison equation
technique, important classes of problems in which transition points may
approach each other become accessible to accurate analytical treatment.
The exposition in this monograph is of a mathematical nature but has
important physical applications, some examples of which are found in the
adjoined papers. Thus, we would like to emphasize that, although we aim
at mathematical rigor, our treatment is made primarily with physical needs
in mind.
To introduce the reader into the background of this book, we start by de
scribing the phase-integral approximation of arbitrary order generated from
an unspecified base function. This is done in Chapter 1, which is reprinted,
after minor changes, from a review article. Chapter 2 is the result of re
search work that was pursued during more than two decades, interrupted
at times. It started in the sixties, when we were still using a phase-integral
approximation, which in our present terminology corresponds to a special
choice of the base function. At the time our primary aim was to derive
expressions for the supplementary quantities needed in order to obtain an
accurate connection formula for a real potential barrier, when the energy
lies in the neighborhood of the top of the barrier. In 1972 we published
analytic expressions, without derivation, for such quantities up to the fifth
order of the phase-integral approximation. These results were then used in
a number of applications. After our derivation in 1974 of the more general
and flexible phase-integral approximation generated from an unspecified
base function, described in Chapter 1, we had to generalize the compari
son equation technique. We then also decided to elaborate that technique
into a general scheme for obtaining the Stokes constants, pertaining to an
arbitrary order of the phase-integral approximation used, and valid also
when transition points, not specified a priori, approach each other. We
wished to formulate the comparison equation technique, developed chiefly
by Cherry, Erdelyi, and others, in such a way that, when the resulting solu
tion sufficiently far from transition points is expanded in terms of a "small"
bookkeeping parameter, the result is the phase-integral approximation of
arbitrary order generated from an unspecified base function, with expres
sions for phase and amplitude that remain valid also when transition points
approach each other. In addition, our goal was to perform the rather lengthy
vi Preface
calculations once and for all, in order to obtain formulas that can readily
be particularized to specific situations encountered in various applications.
Thus, the comparison equation technique described in Chapter 2 is de
veloped in general terms with the aim of facilitating its application to
particular problems as much as possible. The exposition displays both the
power and the limitations of comparison equation technique. Although the
general ideas that we exploit can be found in papers by earlier authors, our
aim required a nontrivial adaptation of comparison equation technique to
phase-integral technique with a number of new steps needed in the proce
dure. The resulting phase-integral formulas, involving supplementary quan
tities, are new. The accomplishment of our aims turned out to be a long
toil; we had to rewrite the manuscript numerous times over the years, in
order to make the exposition clear step by step.
To illustrate the application of the results in Chapters 1 and 2, we have
added a number of adjoined papers (Chapters 3 through 11), which were
originally intended to be published in scientific-journals. The papers in
Chapters 3 through 8 are essentially of a mathematical nature, but with a
physical aim, whereas those in Chapters 9 through 11 are of a more physical
nature.
In Chapters 3 and 4, the formulas obtained in Chapter 2 are applied to
the case in which there is one transition zero; in Chapter 5 they are applied
to the case in which there are two transition zeros, and in Chapters 6 and
7 they are applied to the case in which there are two or three transition
points (one first- or second-order pole and one or two simple zeros of the
square of the base function). In Chapter 8 the phase-amplitude method
for numerical solution of Schrodinger-like differential equations is combined
with comparison equation technique in order to master an important special
problem.
Chapters 9 through 11 concern physical applications in which transition
points may lie close together. Numerical results illustrate the accuracy that
can be achieved by means of the phase-integral method when supplemen
tary quantities are included.
We are much indebted to the Editor, Professor C. Truesdell, for his gen
erous help with the publication of this book. We are also very happy to
have enjoyed the friendship of Charlotte and Clifford Truesdell during many
years.
We are grateful to Miss Ebba Johansson, who has typed several early
versions of some of the chapters in this book, and to Mrs. Maud Hogberg,
who has typed several later versions as well as other chapters. For their
unfailing and generous help we would like to give them our heartfelt thanks.
Uppsala Nanny Froman
July 1995 Per Olof Froman
Contents
1 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 q-Equation 26
1.5.1 Solutions of the Schrodinger Equation and
Solutions of the q-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
2 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 'Po 46
2.2.2 Determination of the Coefficients An,o and Bq 46
2.2.3 Differential Equation for 'P2N When N > 0 51
2.2.4 Regularity Properties of 12N and 'P2N When N > 0 55
2.2.5 Determination of the Coefficients An,2N When N > 0 58
2.2.6 Expressions for 'P2 and 'P4 60
viii Contents
2.2.7 Behavior of 'P2N(Z) in the Neighborhood of a
First- or Second-Order Pole of Q2(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 q-Equation 90
4.3.1 The Case When Re (Increases as z Moves
Away from t in the Neighborhood ofthe
Anti-Stokes Line A 91
4.3.2 The Case When Re (Decreases as z Moves
Away from t in the Neighborhood ofthe
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 'l/J(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 be
5.3 Comparison Equation 119
5.4 Comparison Equation Solution 120
5.4.1 Determination of <Po{z) and Ko 121
5.4.2 Determination of <P2{3 and K 2{3 for f3 > 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
6 Phase-Integral Formulas for the Regular Wave
Function When Tohere 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 <Po and A1,o 154
6.2.2 Determination of <P2{3 and A1,2{3 for f3 > 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 8(2n+1) 177
References 181
7 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 eo> 0 187
7.3 eo = 0, JLo =F 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
1
Phase-Integral Approximation of
Arbitrary Order Generated from an
Unspecified Base Function
Abstract. We begin with a brief review of the so-called WKB ap
proximation, its deficiencies in higher order, and attempts by several
authors to remedy them. It is then shown that these deficiencies do
not appear in the phase-integral approximation generated from an
a priori unspecified base function, which was originally devised by
the present authors in 1974 and is presented here in a way that clar
ifies the role of the "small" parameter in the differential equation.
The advantage of this approximation versus the WKB approxima
tion in higher order is also discussed. In a discussion of relations
between solutions of the SchrOdinger equation and the q-equation,
the Ermakov-Lewis invariant is considered. In the concluding section
we mention other items that constitute the phase-integral method
beside the phase-integral approximation in question.
1.1 Introduction
The mathematical approximation method that since the breakthrough of
quantum mechanics has usually been called the WKB method, has in real
ity been known for a very long time.1 The method describes various kinds of
wave motion in an inhomogeneous medium, in which the properties change
only slightly over one wavelength; in addition, the method provides the
connection between classical mechanics and quantum mechanics. To a sur
prisingly large extent it can already be found in an investigation by Carlini
(1817) on the motion of a planet in an unperturbed elliptic orbit. After
that the method was independently developed and used by many people.
However, the important connection formulas were missing until Rayleigh
(1912'), very implicitly, and Gans (1915), somewhat more explicitly, derived
lChapter 1 is reprinted after minor changes from Forty More Years of Rami
fications (Discourses in Mathematics and Its Applications, No.1, 1991), by per
mission of the Department of Mathematics of Texas A & M University.