Table Of ContentMean Field
Theory
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Mean Field
Theory
World Scientific
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British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
MEAN FIELD THEORY
Copyright © 2020 by World Scientific Publishing Co. Pte. Ltd.
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the publisher.
ISBN 978-981-121-177-5 (hardcover)
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TO
Natasha
Alexei
Benjamin
Aviva Ida
v
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Preface
The conceptofmeanfieldhasledtothe developmentofapproximatequantumand
semi-classicalapproacheswhichsuccessfullydescribethepropertiesofamany-body
interactingsystem. Inthisbookwepresentrecentdevelopmentsinmeanfieldtheory
and beyond. Although each section of the book is related to theoretical problems
in nuclear physics, with comparison to experimental data when available, there is
a logical extension of it to different aspects of other many-body physics, such as
atoms, condensed matter, plasma and other. The book is intended for graduate
studentsandactiveresearchers. Weassumebasicknowledgeofquantummechanics
and nuclear physics. An extensive list of references to original papers, reviews and
books is given, so that the reader can follow up on a subject of personal interest.
A significant part of the book is based on over three decades of collaborative
work by the authors which commenced when they visited the Niels Bohr Institute,
Copenhagen, Denmark. We have benefited from collaborative work with numerous
studentsandcolleagues,ascanbeseenfromthereferencesinthe text,particularly,
inthefiguresandtables. Theplanforthisbookoriginatedmorethanadecadeago.
Mybestfriendandco-author,Academician,ProfessorVladimirM.Kolomietz,died
in June 2018, a personal loss to his family, me and our collaborators. I hope the
completed book closely follows his vision.
Special thanks are due to Dr. Andrey I. Sanzhur for his help in preparing the
final version of the book for publication.
Shalom Shlomo
College Station, Texas
December 2019
vii
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Contents
Preface vii
Introduction xv
1. Self-consistent mean field approximations 1
1.1 Variational principle . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Effective interparticle interaction . . . . . . . . . . . . . . . . . . . 6
1.2.1 Nuclear matter . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.2 Finite nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2.3 Landau’s effective interaction . . . . . . . . . . . . . . . . . 13
1.3 Energy density functional . . . . . . . . . . . . . . . . . . . . . . . 15
1.3.1 Potential energy due to Skyrme forces . . . . . . . . . . . . 15
1.3.2 Spin-orbit potential energy . . . . . . . . . . . . . . . . . . 18
1.3.3 Coulomb energy . . . . . . . . . . . . . . . . . . . . . . . . 19
1.3.4 Energy density . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.4 Hartree-Fock equations for Skyrme forces . . . . . . . . . . . . . . 22
1.4.1 Nuclear matter properties in terms of Skyrme force
parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.4.2 Skyrme parameters in terms of nuclear matter properties . 26
1.4.3 Determining the Skyrme force parameters . . . . . . . . . . 28
2. Many-body problem in phase space 30
2.1 Weyl’s and Wigner’s representations . . . . . . . . . . . . . . . . . 30
2.1.1 Weyl representation . . . . . . . . . . . . . . . . . . . . . . 30
2.1.2 Wigner representation . . . . . . . . . . . . . . . . . . . . . 31
2.1.3 Composition formula . . . . . . . . . . . . . . . . . . . . . 31
2.2 Wigner distribution function. . . . . . . . . . . . . . . . . . . . . . 32
2.3 Application of the Wigner transformation . . . . . . . . . . . . . . 38
2.3.1 Kinetic equations . . . . . . . . . . . . . . . . . . . . . . . 39
2.3.2 Pauli blocking . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.3.3 Heavy-ion collisions . . . . . . . . . . . . . . . . . . . . . . 40
ix
