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Physical Problems Solved by the Phase-Integral Method PDF

230 Pages·2005·0.888 MB·English
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This page intentionally left blank 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

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