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Solving the quantum scattering problem for systems of two and three charged particles
Stockholm University, Faculty of Science, Department of Physics. (Molecular Physics Division)
2011 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

A rigorous formalism for solving the Coulomb scattering problem is presented in this thesis. The approach is based on splitting the interaction potential into a finite-range part and a long-range tail part. In this representation the scattering problem can be reformulated to one which is suitable for applying exterior complex scaling. The scaled problem has zero boundary conditions at infinity and can be implemented numerically for finding scattering amplitudes. The systems under consideration may consist of two or three charged particles.

The technique presented in this thesis is first developed for the case of a two body single channel Coulomb scattering problem. The method is mathematically validated for the partial wave formulation of the scattering problem. Integral and local representations for the partial wave scattering amplitudes have been derived. The partial wave results are summed up to obtain the scattering amplitude for the three dimensional scattering problem. The approach is generalized to allow the two body multichannel scattering problem to be solved. The theoretical results are illustrated with numerical calculations for a number of models.

Finally, the potential splitting technique is further developed and validated for the three body Coulomb scattering problem. It is shown that only a part of the total interaction potential should be split to obtain the inhomogeneous equation required such that the method of exterior complex scaling can be applied. The final six-dimensional equation is reduced to a system of three dimensional equations using the full angular momentum representation. Such a system can be numerically implemented using the existing full angular momentum complex exterior scaling code (FAMCES). The code has been updated to solve the three body scattering problem.

Place, publisher, year, edition, pages
Stockholm: Department of Physics, Stockholm University , 2011. , 64 p.
Keyword [en]
Scattering theory, three body scattering
National Category
Atom and Molecular Physics and Optics Other Physics Topics
Research subject
Physics
Identifiers
URN: urn:nbn:se:su:diva-54832ISBN: 978-91-7447-213-4OAI: oai:DiVA.org:su-54832DiVA: diva2:398414
Public defence
2011-03-23, FA32, AlbaNova universitetscentrum, Roslagstullsbacken 21, Stockholm, 13:00 (English)
Opponent
Supervisors
Funder
Swedish Research Council
Note
At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 4: Submitted. Paper 5: Manuscript.Available from: 2011-03-01 Created: 2011-02-17 Last updated: 2011-02-22Bibliographically approved
List of papers
1. Solving the Coulomb scattering problem using the complex-scaling method
Open this publication in new window or tab >>Solving the Coulomb scattering problem using the complex-scaling method
2009 (English)In: Europhysics letters, ISSN 0295-5075, E-ISSN 1286-4854, Vol. 85, no 3, 30001- p.Article in journal (Refereed) Published
Abstract [en]

We present a rigorous formalism for solving the scattering problem for long-rangeinteractions without using exact asymptotic boundary conditions. The long-range interaction maycontain both Coulomb and short-range potentials. The exterior complex-scaling method, appliedto a specially constructed inhomogeneous Schr¨odinger equation, transforms the scattering probleminto a boundary problem with zero boundary conditions. The local and integral representations forthe scattering amplitudes have been derived. The formalism is illustrated with numerical examples.

Place, publisher, year, edition, pages
Bristol, UK: IOP Publishing Ltd., 2009
Keyword
Scattering theory, Elastic scattering
National Category
Atom and Molecular Physics and Optics
Research subject
Physics
Identifiers
urn:nbn:se:su:diva-54826 (URN)10.1209/0295-5075/85/30001 (DOI)000263693200001 ()
Funder
Swedish Research Council
Available from: 2011-02-17 Created: 2011-02-17 Last updated: 2011-02-18Bibliographically approved
2. Quantum Scattering with the Driven Schrödinger Approachand Complex Scaling
Open this publication in new window or tab >>Quantum Scattering with the Driven Schrödinger Approachand Complex Scaling
Show others...
2009 (English)In: Few-body systems, ISSN 0177-7963, E-ISSN 1432-5411, Vol. 45, no 2-4, 197-201 p.Article in journal (Refereed) Published
Abstract [en]

Quantum scattering calculations of two and three-body systems with Coulomb interaction using thedriven Schrödinger equation combined with exterior complex scaling are discussed. A rigorous formulationfor two-body scattering is reported, and its generalization to three-body scattering is considered.

Place, publisher, year, edition, pages
Heidelberg: Springer Verlag, 2009
Keyword
Scattering theory
National Category
Atom and Molecular Physics and Optics
Research subject
Physics
Identifiers
urn:nbn:se:su:diva-54828 (URN)10.1007/s00601-009-0046-3 (DOI)000265704400028 ()
Funder
Swedish Research Council
Available from: 2011-02-17 Created: 2011-02-17 Last updated: 2011-02-17Bibliographically approved
3. The impact of sharp screening on the Coulomb scattering problem in three dimensions
Open this publication in new window or tab >>The impact of sharp screening on the Coulomb scattering problem in three dimensions
2010 (English)In: Journal of Physics A: Mathematical and General, ISSN 0305-4470, Vol. 43, no 24, 245302- p.Article in journal (Refereed) Published
Abstract [en]

The scattering problem for two particles interacting via theCoulomb potential isexamined for the case where the potential has a sharp cut-off at some distance.The problem is solved for two complementary situations, firstly, when theinterior part of the Coulomb potential is left in the Hamiltonian and, secondly,when the long-range tail is considered as the potential. The partial wave resultsare summed up to obtain the wavefunction in three dimensions. It is shownthat in the domains where the wavefunction is expected to be proportionalto the known solutions, the proportionality is given by an operator actingon the angular part of the wavefunction. The explicit representation for thisoperator is obtained in the basis of Legendre polynomials. We proposed adriven Schr¨odinger equation including an inhomogeneous term of the finiterange with purely outgoing asymptotics for its solution in the case of thethree-dimensional scattering problem with long-range potentials.

Place, publisher, year, edition, pages
Bristol, UK: IOP Publishing Ltd., 2010
Keyword
Coulomb scattering
National Category
Atom and Molecular Physics and Optics Other Physics Topics
Research subject
Physics
Identifiers
urn:nbn:se:su:diva-54829 (URN)10.1088/1751-8113/43/24/245302 (DOI)000278026000011 ()
Funder
Swedish Research Council
Available from: 2011-02-17 Created: 2011-02-17 Last updated: 2011-02-18Bibliographically approved
4. Potential splitting approach to multichannel Coulomb scattering: the driven Schrödinger equation formulation
Open this publication in new window or tab >>Potential splitting approach to multichannel Coulomb scattering: the driven Schrödinger equation formulation
(English)Article in journal (Refereed) Submitted
Abstract [en]

In this paper we suggest a new approach for the multichannel Coulomb scattering problem. TheSchr¨odinger equation for the problem is reformulated in the form of a set of inhomogeneous equationswith a finite-range driving term. The boundary conditions at infinity for this set of equations havebeen proven to be purely outgoing waves. The formulation presented here is based on splittingthe interaction potential into a finite range core part and a long range tail part. The conventionalmatching procedure coupled with the integral Lippmann-Schwinger equations technique are usedin the formal theoretical basis of this approach. The reformulated scattering problem is suitablefor application in the exterior complex scaling technique: the practical advantage is that after thecomplex scaling the problem is reduced to a boundary problem with zero boundary conditions. TheCoulomb wave functions are used only at a single point: if this point is chosen to be at a sufficientlylarge distance, on using the asymptotic expansion of Coulomb functions, one may completely avoidthe Coulomb functions in the calculations. The theoretical results are illustrated with numericalcalculations for two models.

Keyword
Scattering theory
National Category
Atom and Molecular Physics and Optics
Research subject
Physics
Identifiers
urn:nbn:se:su:diva-54830 (URN)
Funder
Swedish Research Council
Available from: 2011-02-17 Created: 2011-02-17 Last updated: 2011-02-18Bibliographically approved
5. A Potential-splitting approach to multichannel Coulomb scattering: the driven Schrödinger equation formulation II. Comparing an Adiabatic versus a Diabaticrepresentation. Application to the fundamental low-energy mutual neutralisationreaction H+ + H− ! H2 ! H(1) + H(n)
Open this publication in new window or tab >>A Potential-splitting approach to multichannel Coulomb scattering: the driven Schrödinger equation formulation II. Comparing an Adiabatic versus a Diabaticrepresentation. Application to the fundamental low-energy mutual neutralisationreaction H+ + H− ! H2 ! H(1) + H(n)
(English)Manuscript (preprint) (Other academic)
Abstract [en]

In this paper it is demonstrated that the split-potential driven Schrödinger approach to two-body Coulomb multichannel quantum scattering in a diabatic framework presented by us in a previouspaper [XXXXX] also can be formulated within an adiabatic framework. The new formulation of thetheory is numerically realised using finite element discrete variable representation. The method isapplied to a realistic model of the fundamental mutual neutralisation reaction H+ + H−! H2 !H(1) + H(n) described in terms of the seven lowest 1+g electronic states of the H2 molecule. Theobtained cross sections are in good agreement with other methods applied to the same model.

Keyword
Scattering theory
National Category
Atom and Molecular Physics and Optics
Research subject
Physics
Identifiers
urn:nbn:se:su:diva-54831 (URN)
Funder
Swedish Research Council
Available from: 2011-02-17 Created: 2011-02-17 Last updated: 2011-02-18Bibliographically approved

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