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PHYSICAL PROBLEMS SOLVED BY
THE PHASE-INTEGRAL METHOD
This book provides a thorough introduction to one of the most efficient approx-
imation methods for the analysis and solution of problems in theoretical physics
andappliedmathematics.Itiswrittenwithpracticalneedsinmindandcontainsa
discussionof50problemswithsolutions,ofvaryingdegreesofdifficulty.Theprob-
lemsaretakenfromquantummechanics,butthemethodhasimportantapplications
inanyfieldofscienceinvolvingsecond-orderordinarydifferentialequations.The
poweroftheasymptoticsolutionofsecond-orderdifferentialequationsisdemon-
strated,andineachcasetheauthorsclearlyindicatewhichconceptsandresultsof
thegeneraltheoryareneededtosolveaparticularproblem.Thisbookwillbeideal
asamanualforusersofthephase-integralmethod,aswellasavaluablereference
textforexperiencedresearchworkers.Itcanalsobeusedasatextbookforgraduate
students.
Nanny Fro¨man obtained her PhD in theoretical physics from Uppsala University.
She was head of the Institute of Theoretical Physics in Uppsala for ten years and
becameProfessorofTheoreticalPhysicsin1981.In1990shewaselectedamember
oftheAustrianAcademyofSciences,andin1995shewasthethefirstwomantobe
electedpresidentoftheRoyalSocietyofSciencesofUppsala,theoldestacademy
inSweden.NannyFro¨manhaspublishednumerouspapersandreviewarticles,and
coauthoredtwopreviousbookswithPerOlofFro¨man.
PerOlofFro¨manobtainedhisPhDintheoreticalphysicsfromUppsalaUniversity.
HespenttwoyearsasavisitingresearcherattheNielsBohrInstituteinCopenhagen.
In 1960 he became Professor of Classical Mechanics at the Royal Institute of
TechnologyinStockholmandin1964becameProfessorofTheoreticalPhysicsat
UppsalaUniversity.HewaselectedamemberoftheAustrianAcademyofSciences
in1972.PerOlofFro¨manhasauthoredandcoauthorednumerousscientificpapers,
andpublishedtwopreviousbookswithNannyFro¨man.
The authors began a joint investigation of the so-called WKB method in 1960,
which resulted in their classic book JWKB Approximation, Contributions to the
Theory from 1965. They went on to develop an efficient phase-integral method,
based on the use of a phase-integral approximation of arbitrary order generated
fromanunspecifiedbasefunction.
PHYSICAL PROBLEMS SOLVED BY
THE PHASE-INTEGRAL METHOD
NANNY FRO¨MAN AND PER OLOF FRO¨MAN
UniversityofUppsala,Sweden
The Pitt Building, Trumpington Street, Cambridge, United Kingdom
The Edinburgh Building, Cambridge CB2 2RU, UK
40 West 20th Street, New York, NY 10011-4211, USA
477 Williamstown Road, Port Melbourne, VIC 3207, Australia
Ruiz de Alarcón 13, 28014 Madrid, Spain
Dock House, The Waterfront, Cape Town 8001, South Africa
http://www.cambridge.org
©Nanny Froman and Per Olof Froman 2004
First published in printed format 2002
ISBN 0-511-02995-0 eBook (Adobe Reader)
ISBN 0-521-81209-7 hardback
Contents
Preface page xi
1 Historical survey 1
1.1 Development from 1817 to 1926 1
1.1.1 Carlini’s pioneering work 1
1.1.2 The work by Liouville and Green 3
1.1.3 Jacobi’scontributiontowardsmakingCarlini’s
work known 4
1.1.4 Scheibner’salternativetoCarlini’streatment
of planetary motion 4
1.1.5 Publications 1895–1912 5
1.1.6 First traces of a connection formula 5
1.1.7 Publications 1915–1921 6
1.1.8 Both connection formulas are derived in explicit form 7
1.1.9 The method is rediscovered in quantum mechanics 7
1.2 Development after 1926 8
2 Description of the phase-integral method 12
2.1 Form of the wave function and the q-equation 12
2.2 Phase-integral approximation generated from an unspecified
base function 13
2.3 F-matrix method 21
2.3.1 Exact solution expressed in terms of the F-matrix 22
2.3.2 General relations satisfied by the F-matrix 25
2.3.3 F-matrix corresponding to the encircling of a simple
zero of Q2(z) 26
2.3.4 Basic estimates 26
2.3.5 Stokes and anti-Stokes lines 28
2.3.6 Symbolsfacilitatingthetracingofawavefunction
in the complex z-plane 29
v
vi Contents
2.3.7 Removalofaboundaryconditionfromtherealz-axis
to an anti-Stokes line 30
2.3.8 Dependenceofthe F-matrixonthelowerlimitof
integration in the phase integral 32
2.3.9 F-matrixexpressedintermsoftwolinearlyindependent
solutions of the differential equation 33
2.4 F-matrixconnectingpointsonoppositesidesofawell-isolated
turningpoint,andexpressionsforthewavefunction
in these regions 35
2.4.1 Symmetryrelationsandestimatesofthe F-matrix
elements 36
2.4.2 Parameterization of the matrix F(x ,x ) 38
1 2
2.4.2.1 Changes ofα,β andγ when x moves in the
1
classically forbidden region 40
2.4.2.2 Changesofα,β andγ when x movesinthe
2
classically allowed region 41
2.4.2.3 Limiting values ofα,β andγ 42
2.4.3 Wave function on opposite sides of a well-isolated
turning point 43
2.4.4 Power and limitation of the parameterization method 45
2.5 Phase-integral connection formulas for a real, smooth,
single-hump potential barrier 46
2.5.1 Exact expressions for the wave function on both sides
of the barrier 48
2.5.2 Phase-integral connection formulas for a real barrier 50
2.5.2.1 Wave function given as an outgoing wave
to the left of the barrier 53
2.5.2.2 Wave function given as a standing wave
to the left of the barrier 54
3 Problems with solutions 59
3.1 BasefunctionfortheradialSchro¨dingerequationwhen
thephysicalpotentialhasatthemostaCoulomb
singularity at the origin 59
3.2 Basefunctionandwavefunctionclosetotheoriginwhen
thephysicalpotentialisrepulsiveandstronglysingular
at the origin 61
3.3 Reflectionless potential 62
3.4 Stokes and anti-Stokes lines 63
3.5 Propertiesofthephase-integralapproximationalong
an anti-Stokes line 66
Contents vii
3.6 Propertiesofthephase-integralapproximationalongapathon
whichtheabsolutevalueofexp[iw(z)]ismonotonic
in the strict sense, in particular along a Stokes line 66
3.7 DeterminationoftheStokesconstantsassociatedwiththethree
anti-Stokeslinesthatemergefromawellisolated,simple
transition zero 69
3.8 Connectionformulafortracingaphase-integralwavefunction
fromaStokeslineemergingfromasimpletransitionzerot to
the anti-Stokes line emerging from t in the opposite direction 72
3.9 Connection formula for tracing a phase-integral wave function
fromananti-Stokeslineemergingfromasimpletransition
zerot totheStokeslineemergingfromt intheopposite
direction 73
3.10 Connectionformulafortracingaphase-integralwavefunction
from a classically forbidden to a classically allowed region 74
3.11 One-directional nature of the connection formula for tracing
aphase-integralwavefunctionfromaclassicallyforbidden
to a classically allowed region 77
3.12 Connection formulas for tracing a phase-integral wave function
from a classically allowed to a classically forbidden region 79
3.13 One-directional nature of the connection formulas for tracing
aphase-integralwavefunctionfromaclassicallyallowed
to a classically forbidden region 81
3.14 Valueattheturningpointofthewavefunctionassociated
withtheconnectionformulafortracingaphase-integralwave
functionfromtheclassicallyforbiddentotheclassically
allowed region 83
3.15 Value at the turning point of the wave function associated with a
connection formula for tracing the phase-integral wave function
from the classically allowed to the classically forbidden region 87
3.16 Illustration of the accuracy of the approximate formulas for
the value of the wave function at a turning point 88
3.17 Expressionsforthea-coefficientsassociatedwith
the Airy functions 91
3.18 Expressionsfortheparametersα,β andγ when
Q2(z) = R(z) = −z 96
3.19 SolutionsoftheAirydifferentialequationthatatafixedpoint
ononesideoftheturningpointarerepresentedbyasingle,
purephase-integralfunction,andtheirrepresentationon
the other side of the turning point 98
viii Contents
3.20 Connectionformulasandtheirone-directionalnature
demonstrated for the Airy differential equation 102
3.21 Dependenceofthephaseofthewavefunctioninaclassically
allowedregiononthevalueofthelogarithmicderivative
ofthewavefunctionatafixedpoint x inanadjacent
1
classically forbidden region 105
3.22 Phase of the wave function in the classically allowed regions
adjacenttoareal,symmetricpotentialbarrier,when
thelogarithmicderivativeofthewavefunctionisgiven
at the centre of the barrier 107
3.23 Eigenvalueproblemforaquantalparticleinabroad,symmetric
potentialwellbetweentwosymmetricpotentialbarriersof
equalshape,withboundaryconditionsimposedinthe
middle of each barrier 115
3.24 Dependenceofthephaseofthewavefunctioninaclassically
allowedregiononthepositionofthepoint x inanadjacent
1
classically forbidden region where the boundary condition
ψ(x ) = 0 is imposed 117
1
3.25 Phase-shift formula 121
3.26 Distance between near-lying energy levels in different types
ofphysicalsystems,expressedeitherintermsofthe
frequencyofclassicaloscillationsinapotentialwell
or in terms of the derivative of the energy with respect to
a quantum number 123
3.27 Arbitrary-orderquantizationconditionforaparticlein
a single-well potential, derived on the assumption
thattheclassicallyallowedregionisbroadenough
to allow the use of a connection formula 125
3.28 Arbitrary-orderquantizationconditionforaparticlein
asingle-wellpotential,derivedwithouttheassumption
that the classically allowed region is broad 127
3.29 Displacementoftheenergylevelsduetocompression
of an atom (simple treatment) 130
3.30 Displacement of the energy levels due to compression
of an atom (alternative treatment) 133
3.31 Quantizationconditionforaparticleinasmoothpotentialwell,
limited on one side by an impenetrable wall and on the other
sidebyasmooth,infinitelythickpotentialbarrier,andin
particularforaparticleinauniformgravitationalfield
limited from below by an impenetrable plane surface 137
