Semiclassical Dynamics and Relaxation

347 Pages·2008·4.305 MB·English

by  D.S.F. Crothers (auth.)

SpringerSerieson
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47
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47 SemiclassicalDynamicsandRelaxation
By:D.S.F.Crothers
Vols.10-35oftheformerSpringerSeriesonAtomsandPlasmasarelistedattheendofthebook
D.S.F. Crothers
Semiclassical
Dynamics and
Relaxation
With56Figures
D.S.F.Crothers
DepartmentofAppliedMathematics
andTheoreticalPhysics
Queen’sUniversityofBelfast,UK
UniversityRoad
BelfastBT71NN
E-mail:[email protected]
ISBN:978-0-387-74312-7 e-ISBN:978-0-387-74313-4
LibraryofCongressControlNumber:2007940870
(cid:2)c 2008SpringerScience+BusinessMedia,LLC
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Idedicatethisbooktothememoryofmymentorandsupervisor,thelateProfes-
sorSirDavidBatesFRS.Igratefullyacknowledgemyteachers:atRaineyEndowed
School: the late Mr Thomas Fazackerley (applied mathematics), the late Dr Arthur
Gwilliam (pure mathematics), and the late Mr James McAteer (physics), at Balliol
College Oxford: the late Professor Jacobus Stephanus de Wet (applied mathemat-
ics)andthelateDrKennethGravett(puremathematics),andatQueen’sUniversity
BelfastmyothersupervisorProfessorRonMcCarroll.
I warmly acknowledge fruitful collaboration with Professors Anders Ba´ra´ny,
Alex Devdariani, Bill Coffey, Yura Kalmykov, Kanika Roy, and Vladimir Gaiduk.
Ialsothankmy32PhDstudentsfortheirinspiringhardworkandcollaborationand
mywifeEithneforherlovingcare.Ofmy32PhDstudentsIparticularlythankmy
colleaguesDrJimMcCannandDrFrancescaO’Rourke,eachofwhomIhavecol-
laborated with over the years. I also thank my former postdocs: Dr Narayan Deb,
DrGeoffreyBrown,DrP.J.Cregg,DrLawrenceGeoghegan,DrElaineKennedy,Dr
ArleneLoughan,DrPierre-MichelDejardin,DrElenaBichoutskaia,andDrSergei
Titov.IthankMiss(soontobeDr)CarlaMcGrathforherwonderfullyprecisetyp-
ingofthisbookinSpringer-Latex.Lastbutnotleast,IprofoundlythankCarlaand
Elizabeth(DrO’Sullivan)fortheirindustriousapplicationofSpringercorrectionsto
thefinalversion.
Preface
Theeclecticchoiceoftopicsinthebookreflectstheauthor’sresearchinterestsover
fortyfouryears,beforewhichhewasWarMemorialOpenScholarinMathematics
at Balliol College Oxford (1960–1963). Accordingly Chapter 1 covers some good
oldfashioned applied mathematics of relevance to Chapters 2–4 concerning atomic
andmolecularphysicsinthegaseousphase(singlecollisionsatlowpressures)and
to Chapter 5 concerning condensed-matter physics in the liquid and solid phases
(dielectricsandferromagnetics).Thefivechaptersarebasedonasetoffivespecial
lectures given to postgraduate PhD students in the Centre for Atomic, Molecular
andOpticalPhysics,intheSchoolofMathematicsandPhysics,Queen’sUniversity
Belfast, in May and June 2003. The author was appointed to a Personal Chair in
Theoretical Physics at Queen’s University Belfast (1985), and elected as Member
of the Royal Irish Academy (1991), Fellowship of the American Physical Society
(1994),HonoraryProfessorofPhysicsatStPetersburgStateUniversity(2003)and
HonoraryFellowofTrinityCollegeDublin(2006).
A good introduction to Chapters 3 and 4 is given by Chapter 52 (Continuum
Distorted Waves and Wannier Methods by D.S.F. Crothers et al) of the Springer
HandbookofAtomic,MolecularandOpticalPhysics(edG.W.F.Drake),2006.
Belfast, DerrickCrothers
NorthernIreland April2007
Contents
1 MathematicsfortheSemiclassicist ............................... 1
1.1 Single-ValuedAnalyticFunctions............................. 1
1.2 MethodofSteepestDescentandAsymptoticMethods............ 2
1.2.1 Stationary-PhaseVersion.............................. 3
1.3 GeneralizedVariationandPerturbationTheories ................ 4
1.4 HypergeometricSeries ...................................... 6
1.5 ContourIntegralTransforms ................................. 11
1.6 Combinatorics ............................................. 14
1.6.1 ProofviaSisterCeline’sTechnique ..................... 15
1.7 GeneralizedHypergeometricFunctions ........................ 16
1.8 FourierandLaplaceTransforms .............................. 19
1.8.1 CriticalFourierTransformRelation..................... 19
1.8.2 CriticalLaplaceTransformRelation .................... 20
2 SemiclassicalPhaseIntegrals .................................... 21
2.1 Approximation............................................. 21
2.1.1 JWKBApproximation................................ 21
2.1.2 Gans–JeffreysAsymptoticConnectionFormula........... 24
2.2 PhaseIntegrals............................................. 25
2.2.1 StokesPhenomenon:OneTransitionPoint ............... 25
2.2.2 ApplicationofJWKBtoCoupledWaveEquations ........ 29
2.3 TwoandFourTransitionPoints:CrossingandNoncrossing ....... 44
2.3.1 Introduction......................................... 44
2.3.2 ExactResummingofAsymptoticRelationsforParabolic
CylinderFunctionsofLargeOrderandArgument......... 45
2.3.3 TheCrossingParabolicModel ......................... 58
2.3.4 Connection to Ba´ra´ny-Crothers Phase-Integral
Nikitin-ModelAnalysis............................... 61
2.3.5 ConnectionstoNakamuraandZhuPhase-IntegralAnalysis. 62
2.3.6 Connections to the Fro¨mans-Lundborg Phase-Integral
Analysis............................................ 64
X Contents
2.3.7 Conclusions......................................... 65
2.3.8 CurveCrossingReflectionProbabilitiesinOneDimension . 66
2.4 AdditionofaSimplePole ................................... 71
2.4.1 Introduction......................................... 71
2.4.2 TheSemiclassicalScatteringMatrix .................... 74
2.4.3 Phase-IntegralTreatment.............................. 75
2.4.4 ComparisonEquation................................. 80
2.4.5 GeneralPhase-IntegralAbstraction ..................... 83
2.4.6 Discussion.......................................... 83
2.5 OtherGeneralizations....................................... 85
2.5.1 FourCloseCurve-CrossingTransitionPoints............. 85
2.5.2 Circuit-DependentAdiabaticPhaseFactorsfromPhase
IntegralTheory...................................... 88
3 SemiclassicalMethodforHypersphericalCoordinateSystems ...... 93
3.1 Wannier’sClassicalTreatmentofElectronCorrelation ........... 93
3.2 DifferentialandIntegratedWannierCrossSections .............. 98
3.2.1 Conclusions......................................... 115
3.3 DoublyExcitedStatesandTheirLifetimes ..................... 116
3.3.1 Results............................................. 123
3.3.2 DoublyExcitedStatesofHe........................... 125
3.4 DivergentExponents........................................ 128
3.4.1 Wannier’sTheory.................................... 129
3.4.2 TheSemiclassicalJWKBApproximation ................ 130
3.4.3 SemiclassicalTheorywhentheExponentDiverges ........ 131
3.4.4 Results,Discussion,andConclusions ................... 137
4 Ion–AtomCollisions............................................ 139
4.1 TheSemiclassicalImpactParameterTreatment ................. 139
4.2 TravelingAtomicandMolecularOrbitals ...................... 144
4.2.1 TravelingMolecularH+Orbitals ....................... 145
2
4.2.2 TravelingMolecularHeH2+Orbitals .................... 155
4.2.3 TravelingAtomicOrbitals............................. 171
4.3 ContinuumDistortedWavesandTheirGeneralizations ........... 172
4.3.1 Introduction......................................... 172
4.3.2 ChargeTransfer ..................................... 173
4.3.3 Ionization .......................................... 182
4.3.4 Fullydifferentialcrosssectionsforionization............. 197
4.3.5 GeneralizedContinuumDistortedWaves ................ 210
4.3.6 DoubleIonization.................................... 215
4.4 RelativisticCDW .......................................... 219
4.4.1 AntihydrogenProduction ............................. 231
4.5 SemiclassicalAcausality .................................... 234
4.5.1 Introduction......................................... 234
4.5.2 GeneralizedImpact-ParameterTreatment ................ 236
Contents XI
4.5.3 PerturbationTheory .................................. 238
4.5.4 DiscussionandConclusions ........................... 240
5 DiffusioninLiquidsandSolids .................................. 243
5.1 Single-DomainFerromagneticParticles........................ 243
5.2 TheFokker–PlanckandLangevinEquations.................... 267
5.2.1 DriftandDiffusionCoefficients ........................ 273
5.3 DieletricRelaxation,AnomalousDiffusion,Fractals,andAfter
Effects.................................................... 284
5.3.1 NumericalCalculationandPhysicalUnderstanding........ 289
5.4 NonlinearResponseofPermanentDipolesandAfterEffects ...... 292
5.4.1 ComplexSusceptibilityfortheDebyeandDebye-Fro¨hlich
ModelsofRelaxation................................. 294
5.4.2 LinearDielectricResponse ............................ 297
5.4.3 DynamicKerrEffect ................................. 299
5.4.4 NonlinearDielectricRelaxation ........................ 300
5.4.5 ApproximateAnalyticalFormulafortheDynamicKerr
Effectfora
PureCosinusoid ..................................... 301
A ContinuedFractionSolutionsofEq.(5.301) ....................... 305
B Mittag–LefflerFunctions........................................ 309
B.0.1 PropertiesofMittag–LefflerFunctions .................. 309
B.0.2 AsymptoticsofMittag–Lefflerfunctions................. 309
B.1 CheckonNormofx2(τ)..................................... 311
C NonlinearResponsetoAlternatingFields ......................... 313
References......................................................... 321
Index ............................................................. 337
1
Mathematics for the Semiclassicist
1.1 Single-ValuedAnalytic Functions
1 1 1
x= z+ z∗ , y= (z−z∗) (1.1)
2 2 2i
Considerafunction f ofz= x+iyandz∗ = x−iy.Clearlyifxandyareindependent,
thensoingeneralarezandz∗.Thenwehave,with∗ascomplexconjugate,
∂f ∂x∂f ∂y∂f 1∂f 1 ∂f
= + = + (1.2)
∂z ∂z ∂x ∂z ∂y 2∂x 2i ∂y
and
∂f ∂x ∂f ∂y ∂f 1∂f 1 ∂f
= + = − (1.3)
∂z∗ ∂z∗ ∂x ∂z∗ ∂y 2∂x 2i ∂y
However,ifandonlyif∂f/∂z∗ =0,then
∂f ∂f
i = ⇒ (1.4)
∂x ∂y
∂f df ∂f ∂f
≡ = =−i (1.5)
∂z dz ∂x ∂y
andsetting
f(z)=u(z)+iv(z) (1.6)
(whereuandvarerealfunctionsofacomplexvariablez),wehave
∂u ∂v ∂u ∂v
+i =−i + (1.7)
∂x ∂x ∂y ∂y
∂u ∂v ∂v ∂u
⇒ = and =− (1.8)
∂x ∂y ∂x ∂y
(theCauchy–Riemannequations)