The Quantum Mechanical Three-Body Problem. Vieweg Tracts in Pure and Applied Physics PDF
Preview The Quantum Mechanical Three-Body Problem. Vieweg Tracts in Pure and Applied Physics
VIEWEG TRACTS IN PURE AND APP IED PHYSICS Volume 2 Advisory Board Prof. Dr. H. Stumpf University of Tübingen, Germany Prof. Dr. K. Wildermuth University of Tübingen, Germany Erich W. Schmid Horst Ziegelmann The Quantum Mechanical Three-Body Problem edited by H. Stumpf Pergamon Press Oxford · Edinburgh · New York · Toronto · Sydney Friedr. Vieweg + Sohn Braunschweig Prof. Dr. Erich W. Schmidt Dr. Horst Ziegelmann Institute for Theoretical Physics University of Tübingen 1974 All rights reserved Copyright ©1974 by Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig No part of this publication may be reproduced, stored in a retrieval system or transmitted mechanically, by photocopies, recordings or other means, without prior permission of the Copyright holder. Set by Vieweg, Braunschweig Printed by E. Hunold, Braunschweig Bookbinder: W. Langeliiddecke, Braunschweig Printed in Germany-West ISBN 3 528 08337 9 Vieweg ISBN 0 08 0182402 Pergamon Preface One of the outstanding problems of classical physics was the three-body problem, namely, to solve Isaac Newton's equations of motion for three interacting bodies. There is a similar problem in modern quantum mechanics, namely, to describe the motion of three interacting atomic or subatomic particles by rigorous integral equa- tions. L. D. Faddeev, in 1960, published the first work on this problem, Scattering Theory for a Three-Particle System, and since then a great deal of work has been done, the results of which are mainly available in original articles and reviews. The present book originated from a course given by the authors in the summer semester of 1972. It was felt that the student who wants to acquaint himself with the field should no longer have to work his way through the review articles and the original literature. Therefore, the course lectures were revised and expanded into this volume. It is not the aim of this book to cover the field completely. For example, there are different formulations of the abstract theory, any of which can equally well serve as a mathematical basis for the theory. Those of S. Weinberg, L. Rosenberg, J. V. Noble, R. G. Newton, R. Sugar and R. Blankenbecler, and P. A. Kazaks and K. R. Greider, however, have been omitted, while that of Faddeev has been dealt with extensively. Emphasis, instead, has been placed on the various ways of getting from the abstract operator equations to numerical solutions and to data which can be compared with experimental results. We hope that this book will make the field of three-particle physics more readily accessible to the interested student. We also hope that it will help the experiment- alist to understand better how the theoretical data are derived. We thank Dr. J. Schwager and Dr. F. Sohre for their critical reading of the manuscript and Mrs. Ch. Stiller for the typing. We also thank the authors of original papers for giving us permission to reproduce their figures. Erich W. Schmid / Horst Ziegelmann - 1 - 1. Introduction The quantum mechanical three-body problem deals with the motion of three particles under the influence of an interaction potential. Its fundament is the Schrödinger equation for three particles, ^11^ = ^^. with where V(i,k) is a two-body potential; also a three-body poten- tial V(i,k,j) is thinkable, but it is used very seldom. In atomic-, nuclear-, and elementary-particle physics, there exists a large number of states and processes which demand to be treated as three-particle systems. Here are a few examples: a) the scattering of electrons from hydrogen b) the nuclei 3H and \e c) the nucleon-deuteron scattering d) the photodisintegration of ^H and ^He e) the scattering of mesons from deuterons f) the scattering of a meson from a nucléon, where the latter is treated as a meson- nucleon bound state g) multiparticle systems, which can be treated approximately as three-particle systems as, for instance, the scattering of deuterons h from He with breakup of the deuteron and 11 the scattering of protons from B with breakup into three a-particles. Particular difficulties arise when a reaction leads to a final state with three free particles, as in the case of proton- deuteron scattering at a center-of-mass energy of more than 2.22 MeV, - 2 - p + d P + d <^ ^p + p + n Because of such processes, the three-particle problem becomes especially interesting. The three-particle scattering system is still simple enough to yield exact solutions, but it is also complicated enough to have channels other than the elastic one, namely, rearrangement and breakup channels. We now shall discuss the difficulties of three-particle breakup and examine the experimental situation. 1. Scattering Experiments with Three-Particle Breakup The three particles which participate in the reaction have 3 times 3 = 9 degrees of freedom in configuration space. The reaction also can be described in momentum space, and here we also have 9 degrees of freedom. Asymptotically, the momenta are sharp, i.e., they are measurable for every single event. In a so-called complete experiment, all 9 momenta are de- termined. The difficulty arises from the multiplicity of data to be measured simultaneously and from the small number of events for each combination of data. Let us look at the reaction p + d > p + p + n Four out of nine momentum components of the final state are determined by the initial state via momentum and energy con- servation. Conservation of momentum means that the sum of momenta of the outgoing particles in the laboratory system must be equal to the momentum of the incoming proton. Conser- vation of energy means that the sum of the kinetic energies of the outgoing particles must be equal to the kinetic energy of the incoming proton minus the binding energy of the deuteron. In the notation of Fig. 1, this means - 3 - C,-E^ , Λ, = 1 L* 5 |l Λ,» 3 There remain 9 minus 4 = 5 momentum components of the outgoing particles to be measured. One can proceed in the following way. Two of the outgoing particles are detected, the third one remains undetected. G> Fig. 1. Notation of a three-particle breakup reaction detector v particle 3J> beam target detector particle 4 Fig. 2. Experimental setup - 4 - Each counter determines three parameters (see Fig. 2), namely energy and direction of momentum of one outgoing nucléon. The two counters together measure six parameters. One of them is redundant and the data, therefore, are not independent. They have to fulfill a kinematical condition. When, for a given position of the counters, all events are plotted in an E , Ε^ diagram, they will lie, theoretically, on a certain curve, as shown in Fig. 3· E 4 Fig. 3. Kinematically allowed curve in a three-particle breakup reaction In practice, one uses scintillation counters of finite aper- ture, finite energy resolution and multichannel analyzers. The numbers of events are represented by more or less pronounced dots. When the background is suppressed by a complicated coinci- dence technique, a band appears along the kinematical curve (Fig. 4). Its width shows the finite resolution of the experi- mental setup. The picture changes for every location of the two counters. Usually, the intensity of the band along the kine- matical curve is plotted against a curve parameter (Fig. 5). The angles Θ-, and Qu can be chosen so as to pick out interesting details of the scattering process, as in the process where two particles have very low relative velocity. The described experiment is called c o m p l e te because a maximum of information is determined (except for polarization). It is hard to perform since the counting rates are very low. - 5 - Fig. 4. Result of a coincidence experiment (H. Brückmann et al. [ 1 ] ) d3d [ mb dSdn d0 [MeVsr2 3 4 d + p—*p*p*n Ed-52MeV θ -42.0° 3 θ -25.3° 4 10 15 20 22 22 20 15 10 5 E[MeV] 4 Fig. 5. Three-particle breakup cross section plotted versus curve parameter S (H. Brückmann et al. [ 1 ] ) - 6 - In the majority of scattering events, the outgoing particles miss either one or both of the counters. The rare events where both counters respond in coincidence are distributed over a whole kinematical curve. The statistic, therefore, is very bad. Earlier experiments always were incomplete because one had to achieve a reasonable intensity by measuring less than five parameters. One used, for instance, only one proton counter and received a cross section which represented an integral of the complete differential cross section over the undetermined parameters. 2. Difficulties of the Theory When we ask for difficulties of the three-particle problem, it is advantageous to ask first why the two-particle problem is so simple. a) Scattering of Two Elementary Particles In the initial state, a wave packet - which describes the projectile - approaches a wave which describes the target in the laboratory system. In the center-of-mass system, a wave packet approaches the origin where the interaction takes place (Fig. 6). Fig. 6. Scattering of two particles in the center-of-mass system The initial condition or boundary condition is, basically, a condition in t i m e. One can now go over to the limit
The list of books you might like
