Table Of ContentReiner M. Dreizler
Tom Kirchner
Cora S. Lüdde
Quantum
Collision Theory
of Nonrelativistic
Particles
An introduction
Quantum Collision Theory of Nonrelativistic
Particles
Reiner M. Dreizler • Tom Kirchner (cid:129)
Cora S. Lüdde
Quantum Collision Theory
of Nonrelativistic Particles
An Introduction
ReinerM.Dreizler TomKirchner
InstitutfürTheoretischePhysik DepartmentofPhysicsandAstronomy
GoetheUniversität YorkUniversity
Frankfurt/Main,Germany Toronto,ON,Canada
CoraS.Lüdde
InstitutfürTheoretischePhysik
GoetheUniversität
Frankfurt/Main,Germany
ISBN978-3-662-65590-0 ISBN978-3-662-65591-7 (eBook)
https://doi.org/10.1007/978-3-662-65591-7
Thisbookisatranslation oftheoriginalGerman1stedition“Streutheorie indernichtrelativistischen
Quantenmechanik” by Reiner M. Dreizler, Tom Kirchner & Cora S. Lüdde, published by Springer-
Verlag GmbH Germany, part ofSpringer Nature in 2018. Thetranslation was done with the help of
artificial intelligence (machine translation by the service DeepL.com). The present version has been
revisedextensivelywithrespecttotechnicalandlinguisticaspectsbytheauthors.
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Preliminary Remarks
Scattering experiments are tools, which allow access to the quantum world. Our
knowledge of molecules, atoms, nuclei and elementary particles is largely based
on the analysis of such experiments. A classic example is the discovery of the
atomicnucleusbyelasticscatteringofα-particlespassingthroughgoldfoils.These
experiments were carried out between 1908 and 1913 by Geiger and Marsden,
following a suggestion of Rutherford. The central results were published in the
English journal Philosophical Magazine in 1913. This publication provides a
confirmationoftheRutherfordscatteringformula,whichwasderivedonthebasisof
classicalargumentsandpublishedinthe1911issueofPhil.Mag.1 Afirstexample
for the structural investigation of nuclei by scattering experimentsis the collision
ofprotons(denotedbyporalternativelyas1H)andlithiumnuclei(e.g.theisotope
1
7Li).Thefollowingprocessesarepossibleinthiscollision:
3
(cid:129) Elasticscattering,inwhichthedirectionofmotionoftheparticlesischangedbut
theirkineticenergyisnot
p+7Li −→ p+7Li.
3 3
(cid:129) In inelastic scattering, part of the collision energy is used to excite the lithium
nucleus. After the excitation, the Li nucleus returns to the ground state by
emitting a photon (γ). The excitation energy determines the frequency of the
photon
p +7Li −→ p(cid:3) + 7Li∗ −→ p(cid:3) +7Li + γ.
3 3 3
Thekineticenergyofthep-Lisystemisreducedafterthecollision.
(cid:129) Additional processes in which nuclear transformationsoccur. They are termed
reactionchannels.Thetotalenergyisalsoconservedintheseprocesses,butthe
initial energy is converted in various ways for the production of the reaction
products and their kinetic energy. One can recognise the validity of charge
1Thepublicationsreferredtoare:H.GeigerandE.Marsden,Phil.Mag.25,p.604(1913)andE.
Rutherford,Phil.Mag.21,p.669(1911).
v
vi PreliminaryRemarks
conservation(lowerindex)andtheconservationofnucleonnumber(upperindex)
bylookingattheequationscharacterisingthereactions
1p+7Li −→ 7Be+1n
1 3 4 0
−→ 6Li+2H
3 1
−→ 4He+4He
2 2
−→ 6He+1p+1p
2 1 1
−→ ...
Corresponding lists can be given for the scattering of ‘elementary particles’.
Thusonefinds,e.g.,forthecollisionofphotonsandprotonsbesidestheelastic
scattering
γ +p −→ γ +p
thegenerationofunchargedorchargedπ-mesons
γ +p −→ π0+p
−→ π++n.
These examples give a rough overview of the types of reactions possible
incollisionprocesses.Theyareexpressedinthefollowingform:Capitalletters
denotethetarget,andtheendproductofthereaction.Lowercaselettersrepresent
theincomingprojectileandtheoutgoingparticle(orparticles).
(cid:129) Elasticscattering
A(a,a)A
The particles are unchanged. Due to the conservation of the total energy, only
theirdirectionofmotionchanges.
(cid:129) Inelasticscattering
A(a,a∗)A∗
The kinetic energy of the projectile is changed, and it may also be excited but
retainsitsidentity.Thetargetsystemisexcitedandcanreturntothegroundstate
byemittingphotonsordisintegrateintofragments.
(cid:129) Particletransfer
A(a,b)B
PreliminaryRemarks vii
The particles before the collision differ from the particles after the collision.
Therebycan
– Partsoftheprojectilebetransferredtothetarget(strippingreaction)
– Partsofthetargetbetransferredtotheprojectile(pickuporcapturereaction)
– Particles, which are formed during the collision process, can disintegrate
(breakuporfragmentationreaction)
In each case, the entrance channel consists of two particles that are initially far
enoughapartandmovefreelytowardseachother.Whentheyaresufficientlyclose,
theyinteractwitheachother.Thetaskoftheexperimentsistodetectandinvestigate
thepossibleexitchannelsbesideselasticscattering.Thisinvolvestheidentification
of the exiting particles and the determination of their respective momentum and
energyasafunctionoftheinitialconditions.Theresultsobtainedinthiswaycanbe
usedtofindcluesabouttheinteractionbetweentheparticles,toexploretheinternal
structureoftheparticlesand,ifpossible,todiscoverparticlesthatarenotyetknown.
Thepresenceofalargenumberofreactionchannelsindicatesthatoneisconfronted
withaverycomplicatedsituation.Thecomplexitydependsonthecollisionenergy
andtheinternalstructureofthecollisionpartners,whichtakepartintheprocess.For
theinvestigationoftheprocesses,scatteringtheorybasedonquantummechanicsis
used.Thisbookisanintroductiontothescatteringtheoryofnonrelativisticcollision
systems. It is intended for students, who are familiar with the basic elements of
quantummechanics.
The notion that the two collision partners move towards each other (or one of
thecollisionpartnershitsastationaryparticle)impliesatime-dependenttreatment
ofthecollisionprocesses.Themotionofthecollisionpartnersmustberepresented
by wave packets whose time evolution is determined by an initial value problem
basedonthetime-dependentSchrödingerequationinthe nonrelativisticcase.The
fact that in practice not the scattering of one particle by another one is observed,
but a particle beam hits a (stationary) target (like the gold foil in the Rutherford
experiment),allowsastationaryformulationofthescatteringtheory.Thetaskinthis
approachistosolvethestationarySchrödingerequationwithboundaryconditions
thatreflect the scatteringsituation. One can show that the time-dependentand the
stationary formulations give comparable results under suitable conditions. As an
approachtothescatteringtheory,weproposethefollowingprogram:
(cid:129) Foundationandextensionofthestationaryscatteringtheoryforelasticscattering,
thesimplesttypeofreaction.Adirectapproachisthestationarytreatmentofthe
problemusingtheSchrödingerequationwithboundaryconditionscharacterising
theincidentparticlebeamandtheoutgoingscatteredparticles(Chap.1).
(cid:129) An alternative treatment is offered by a formulation of the theory with the aid
of integral equations, in which the boundaryconditions are explicitly included
via the use of Green’s functions (Chap.2). Central equations are in this case
the Lippmann–Schwinger equations for the scattering states and an operator
equationfortheT-matrix.
viii PreliminaryRemarks
(cid:129) A third approach, which addresses the actual time development of a single
collision,isthetime-dependentformulationoftheproblem(Chap.3).Itcanbe
shown that a mathematically appropriate treatment of the asymptotic states in
time (and in space) yields the same experimentallyverifiable results (the cross
sections)asthestationaryformulation.Thequantitiesthatplayaspecialrolein
thischapteraretheMølleroperatorsandtheoperatoroftheS-matrix,anoperator
closelyrelatedtotheT-matrix.
(cid:129) TheconsiderationofS-matrixelementsallowsaunifieddiscussionofselection
rulesfortheresultsofthecollisionprocessesduetoparticularsymmetriesofa
givenproblem,ase.g.translationalorrotationalsymmetry(Chap.4).
(cid:129) The study of the analytic structure of the S-matrix elements as a function of
complex wave numbers provides further insights into the potential scattering
problem, such as a connection of the scattering solutions of the Schrödinger
equationwiththeboundstatesofthecorrespondingenergyspectrum(Chap.5).
(cid:129) It is necessary to include a discussion of partially spin-polarised beams and
targetsbymeansofaformulationusingdensitymatricesinordertoanalysethe
resultsofmoreextensiveexperiments(Chap.6).
(cid:129) In Chap.7 the multichannel situation (elastic scattering in competition with
reactions such as excitation, particle transfer, etc.) is outlined. Examples of
such problems are the treatment of the three-bodyproblem on the basis of the
Faddeev equations, as well as the consideration of the Distorted Wave Born
Approximation(DWBA)fornucleontransferreactions.
Inallchapters,aseparationoftheargumentsbasedonphysicsandoncomputational
aspects is attempted. For example, in the first chapter, the basic elements of
stationarypotentialscatteringtheoryarecoveredinfoursections.Thefifthsection
contains additionalinformation on some of the concepts as well as more detailed
calculationsinordertocomplementtheprecedingfoursections.Itisrecommended
tocheckthesecalculationsasanexercise.
Attheendofthisvolumeonefindsalistofrelevantliteratureoncollisiontheory.It
contains
(cid:129) Suggestionsforfurtherreading.
(cid:129) Alistoftheliteraturecitedineachofthechapters.
(cid:129) Mathematicalbackgroundmaterial, thatis used and referredto throughoutthis
book,arethe‘SpecialFunctionsofMathematicalPhysics’.Astandardworkon
thistopicis
M. Abramowitz and I. Stegun: Handbook of Mathematical Functions. Dover
Publications,NewYork(1974).
Material from this source is cited in the form ‘Abramowitz/Stegun, p. yyy’.
Alternativeeditionsexistinadditiontotheeditionoftheyear1974,forinstance
alsointheformofseveralopensourceeditions.
(cid:129) Additionalsourcesdealingwith‘SpecialFunctions’,aree.g.
W.Magnus,F.Oberhettinger,R.P.Soni:FormulaeandTheoremsfortheSpecial
FunctionofMathematicalPhysics.SpringerVerlag,NewYork(1966).
PreliminaryRemarks ix
(cid:129) Alternatively,onecanconsultbooksspecialisingononeparticularsetofSpecial
Functionsas,e.g.,
L. J.Slater:ConfluentHypergeometricFunctions.CambridgeUniversityPress,
Cambridge(1960),
G. N. Watson: A Treatise on the Theory of Bessel Functions. Cambridge
UniversityPress,Cambridge(firstedition1922,lastedition2018).
(cid:129) Someaspectsofclassicalmechanicsarecitedfromthebook
R. M. Dreizler, C. S. Lüdde: Theoretical Physics 1, Theoretical Mechanics.
SpringerVerlag,Heidelberg(2002and2008)
intheformDreizler/Lüdde,Vol.1,Sect.orChap.xxx.
Contents
1 Elastic Scattering:StationaryFormulation—Differential
Equations ..................................................................... 1
1.1 SomeBasicConcepts .................................................. 1
1.1.1 ElasticScatteringExperiments ................................ 1
1.1.2 FormulationoftheTwo-ParticleScatteringProblem ......... 3
1.1.3 ConnectionBetweenTheoryandExperiment ................ 7
1.1.4 The DifferentialCross Sectionin the Laboratory
SystemandintheCentreofMassSystem .................... 8
1.2 ThePartialWaveExpansion ........................................... 10
1.2.1 PartialWaveExpansionintheCaseofCylindrical
Symmetry ....................................................... 12
1.2.2 ThePhaseShift ................................................. 15
1.2.3 AnExample:ScatteringbyaSphericalPotentialWell ....... 17
1.3 ScatteringbyaCoulombPotential .................................... 26
1.3.1 PartialWaveExpansion ........................................ 27
1.3.2 SolutioninParabolicCoordinates ............................. 30
1.4 PotentialScatteringofParticleswithSpin ............................ 33
1.4.1 DifferentialCrossSections .................................... 35
1.4.2 SelectionRules ................................................. 36
1.4.3 RoleofParticleStatistics ...................................... 37
1.5 DetailedCalculationsforChap.1 ...................................... 40
1.5.1 TheDifferentialCrossSection ................................ 40
1.5.2 TheOpticalTheorem .......................................... 41
1.5.3 CentreofMassandLaboratorySystems ...................... 42
1.5.4 VariousBesselFunctions ...................................... 46
1.5.5 PartialWaves ................................................... 48
1.5.6 TheConfluentHypergeometricFunction ..................... 50
1.5.7 PropertiesoftheCoulombWaveFunction ................... 51
1.5.8 The CoulombProblemin ParabolicCoordinates:
AsymptoticLimit .............................................. 53
LiteratureQuotedinthePreliminaryRemarksandinChap.1 ............. 54
xi
