Table Of ContentVIEWEG 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
