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Solving the Quantum Scattering Problem for Systems of Two and Three Charged Particles Solving the quantum scattering problem for systems of two and three charged particles Solving the quantum scattering problem for systems of two and three charged particles Mikhail Volkov c Mikhail Volkov, Stockholm 2011 ISBN 978-91-7447-213-4 Printed in Sweden by Universitetsservice US-AB, Stockholm 2011 Distributor: Department of Physics, Stockholm University In memory of Professor Valentin Ostrovsky Abstract 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 mathe- matically 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 scat- tering 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 mo- mentum complex exterior scaling code (FAMCES). The code has been up- dated to solve the three body scattering problem. 7 List of Papers This thesis is based on the following papers, which are referred to in the text by their Roman numerals. I Solving the Coulomb scattering problem using the complex scal- ing method M. V. Volkov, N. Elander, E. Yarevsky and S .L. Yakovlev Euro Phys. Lett. 85, 30001 (2009). II Quantum scattering with the driven Schrödinger approach and complex scaling N. Elander, M. V. Volkov, Å. Larson, M. Stenrup, J. Z. Mezei, E. Yarevsky and S. L. Yakovlev Few-Body Systems 45, 197 (2009). III The impact of sharp screening on the Coulomb scattering problem in three dimensions S. L. Yakovlev, M. V. Volkov, E. Yarevsky and N. Elander J. Phys. A: Math. Theor. 43, 245302 (2010). IV Splitting potential approach to multichannel Coulomb scattering: driven Schrödinger equation formulation M. V. Volkov, S .L. Yakovlev, E. Yarevsky and N. Elander submitted to Phys. Rev. A V A potential splitting approach to multichannel Coulomb scattering: the driven Schrödinger equation formulation II. Comparing an Adiabatic versus a Diabatic representation. Application to the fundamental low-energy mutual neutralization + reaction H + H− H2∗ H(1)+ H(n) M. V. Volkov, S .L.→ Yakovlev,→ E. Yarevsky and N. Elander preliminary manuscript Reprints were made with permission from the publishers. Contributions by the author I I wrote the computercodeandperformedall the calculations for the paper and participated equally with the other authors in ob- taining the theoretical results and writing the paper. 9 II InthispaperIparticipatedin writing thesectiondedicated to the two-body problem for Coulomb plus short-range potentials. III In this paper I took part in writing the section dedicated to the scattering problem for the potential tail and the section dedicated to the driven Schrödinger equation for the Coulomb scattering problem. IV I wrote the computercodeandperformedall the calculations for the paper and participated equally with the other authors in ob- taining the theoretical results and writing the paper. V I wrote the computercodeandperformedall the calculations for the paper and participated equally with the other authors in ob- taining the theoretical results and writing the paper. Publications not included in the thesis 1. Exact results for survival probability in the multistate Landau-Zener model M.V. Volkov and V.N. Ostrovsky J. Phys. B: At. Mol. Opt. Phys. 37, 4069 (2004). 2. No-go theorem for bands of potential curves in multistate Landau-Zener model M.V. Volkov and V.N. Ostrovsky J. Phys. B: At. Mol. Opt. Phys. 38, 907 (2005). 3. Reply to Comment on Exact results for survival probability in the multistate Landau-Zener model M.V. Volkov and V.N. Ostrovsky J. Phys. B: At. Mol. Opt. Phys. 39, 1261-1267 (2006). 4. Landau-Zener transitions in a system of interacting spins: Exact results for demagnetization probability V.N. Ostrovsky and M.V. Volkov Phys. Rev. B 73, 060405(R) (2006). 5. Four-state (two-spin) non-stationary models V.N. Ostrovsky, M.V. Volkov, J.P. Hansen and S. Selstø Phys. Rev. B 75, 014441 (2007). 6. Analytical results for state-to-state transition probabilities in the multistate Landau-Zener model by nonstationary perturbation theory M.V. Volkov and V.N. Ostrovsky Phys. Rev. A 75, 022105 (2007). 10 Contents 1 Introduction .......................................... 13 2 Solving the two body single channel Coulomb scattering problem . 19 2.1 Formulation of the two body single channel scattering problem .......... 19 2.2 Partial wave decomposition. One-dimensional scattering problem ........ 21 2.3 Complex scaling approach to scattering theory ..................... 23 2.4 Coulomb scattering problem .................................. 26 2.5 Potential splitting approach to solving the Coulomb scattering problem ..... 28 3 Solving the two body multichannel Coulomb scattering problem . 33 3.1 Formulation of the two body multichannel scattering problem. ........... 33 3.2 Potential splitting approach to the multichannel scattering problem. ....... 35 4 Solving the three body Coulomb scattering problem ............. 37 4.1 The three body scattering problem. Jacobi coordinates ................ 37 4.2 The potential splitting approach to the Coulomb scattering problem ....... 42 4.3 Separation of the rotation. Full angular momentum representation ........ 45 5 Overview of papers ..................................... 49 5.1 Paper I ................................................. 49 5.2 Paper II ................................................ 49 5.3 Paper III ................................................ 50 5.4 Paper IV ............................................... 50 5.5 Paper V ................................................ 51 6 Conclusions .......................................... 53 A Special functions ....................................... 55 A.1 Coulomb functions ........................................ 55 A.2 Riccati-Bessel and Riccati-Hankel functions ....................... 56 A.3 Wigner D-functions ........................................ 57 Bibliography ............................................. 63 1. Introduction Investigation of few body quantum systems is very important in physics. Knowledge about the processes in few body systems may give qualitative, and in many cases quantitative, information for the problems generated by a large number of interacting particles. Problems appearing in few body systems can in principle be divided into two classes: bound state problems and scattering problems. The bound state problems are usually related to the spectroscopy of such systems while scattering problems describe their reactions. The main focus in the present thesis is the scattering problem for systems consisting of two or three particles. Two body scattering theory is described in many monographs, textbooks, and reviews [1–5]. The simplest case of two body scattering appears when the particles A and B are structureless. Only elastic scattering A + B A + B → is possible in this case. In the multichannel two body scattering problem the structure of the colliding particles can be partially taken into account. Thus, this problem is a model to describe any reactions of the type A + B C + D → The application of such models to real processes is often an open question. The Coulomb scattering problem requires special considerations in both of the single channel and multichannel cases. The reason for this is the slow de- crease of the Coulomb potential as the inter-particle distance increases. One cannot neglect the Coulomb interaction at any inter-particle distance [6]. For- tunately, the scattering problem with a pure Coulomb interaction has an ana- lytical solution in the two body case. This solution can be expressed in terms of so called Coulomb functions (see Appendix A). The Coulomb functions can be used as a basis set for solving scattering problems with a Coulomb plus short range interaction. Another approach to the two body Coulomb scattering problem, namely potential splitting combined with an exterior complex scaling transformation, is the subject of this thesis. In this approach the scattering problem is reduced to a boundary problem with zero boundary condition at infinity. Moreover, it is possible to avoid using the Coulomb functions in practical calculations. The history of the applications of such a complex
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