Many-Body Theory Exposed! Propagator description of quantum mechanics in many-body systems Many-Body Theory Exposed! Propagator description of quantum mechanics in many-body systems Willem H Dickhoff Department of Physics, Washington University in St. Louis Dimitri Van Neck Laboratory of Theoretical Physics, Ghent University World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONGKONG • TAIPEI • CHENNAI Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. MANY-BODY THEORY EXPOSED! Propagator Description of Quantum Mechanics in Many-Body Systems Copyright © 2005 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN 981-256-294-X Printed by Fulsland Offset Printing (S) Pte Ltd, Singapore to J to Lut, Ida, and Cor Preface Surveying the available textbooks that deal with the quantum mechanics of many-particle systems, one might easily arrive at the incorrect conclusion that few new developments have taken place in the last couple of decades. We only mention the recent discovery of Bose-Einstein condensation of di- lute vapors of atoms at low temperature to make the point that this is not the case. In addition, coincidence experiments involving electron beams have clarified in wonderful detail the properties of electrons in atoms and protons in nuclei, since the majority of textbooks have been written. Also, most of them do not provide a satisfactory transition from the typical single- particle treatment of quantum mechanics to the more advanced material. Our experience suggests that exposure to the properties and intricacies of many-body systems outside the narrow scope of one's own research can be tremendously beneficial for practitioners as well as students, as does a unified presentation. It usually takes quite some time before a student of this material masters the subject sufficiently so that new research can be initiated. Any reduction of that time facilitated by a student-friendly textbook therefore appears welcome. For these reasons we have made an attempt at a systematic development of the quantum mechanics of nonrel- ativistic many-boson and many-fermion systems. Some material originated as notes that were made available to students taking an advanced graduate course on this subject. These students typi- cally take a one-year course in graduate quantum mechanics without actu- ally seeing many of the topics that deal with the many-body problem. We note that motivated undergraduate students with one semester of upper- level quantum mechanics are also able to absorb the material, if they are willing to fill some small gaps in their knowledge. As indicated above, an important goal of the presentation is to provide vii viii Many-body theory exposed! a unified perspective on different fields of physics. Although details differ greatly when one studies atoms, molecules, electrons in solids, quantum liquids, nuclei, nuclear/neutron matter, Bose-Einstein or fermion conden- sates, it is helpful to use the same theoretical framework to develop physi- cally relevant approximation schemes. We therefore emphasize the Green's function or propagator method from quantum field theory, which provides this flexibility, and in addition, is formulated in terms of quantities that can often be studied experimentally. Indeed, from the comparison of the calcu- lation of these quantities with data, it is often possible to identify missing ingredients of the applied approximation, suggesting further improvements. The propagator method is applied to rederive essential features of one- and two-particle quantum mechanics, including eigenvalue equations (dis- crete spectrum) and results relevant for scattering problems (continuum problem). Employing the occupation number representation (second quan- tization), the propagator method is then developed for the many-body sys- tem. We use the language of Feynman diagrams, but also present the equa- tion of motion method. The important concept of self-consistency is empha- sized which treats all the particles in the system on an equal footing, even though the self-energy and the Dyson equation single out one of the parti- cles. Atomic systems, the electron gas, strongly correlated liquids including nuclear matter, neutron matter, and helium systems, as well as finite nuclei illustrate various levels of sophistication needed in the description of these systems. We introduce the mean-field (Hartree-Fock) method, random phase approximation (ring diagram summation), summation of ladder dia- grams, and further extensions. A detailed presentation of the many-boson problem is provided, containing a discussion of the Gross-Pitaevskii equa- tion relevant for Bose-Einstein condensation of atomic gases. Spectacular features of many-particle quantum mechanics in the form of Bose-Einstein condensation, superfluidity, and superconductivity are also discussed. Results of these methods are, where possible, confronted with experi- mental data in the form of excitation spectra and transition probabilities or cross sections. Examples of actual theoretical calculations that rely on numerical calculations are included to illustrate some of the recent applica- tions of the propagator method. We have relied in some cases on our own research to present this material for the sole reason that we are familiar with it. References to different approaches to the many-body problem are sometimes included but are certainly not comprehensive. The book offers several options for use as an advanced course in quantum mechanics. The first six chapters contain introductory material and can Preface ix be omitted when it was covered in the standard sequence on quantum me- chanics. Starting from Ch. 7 canonical material is developed supplemented by topics that have not been treated in other textbooks. It is possible to tailor the material to the specific needs of the instructor by emphasizing or omitting sections related to Bose-Einstein condensation, atoms, nuclei, nu- clear matter, electron gas, etc. In addition to standard problems, we also introduce a few computer exercises to pursue interesting and illustrative calculations. We have attempted a more or less self-contained presenta- tion, but include a sizable list of references for further study. By providing detailed steps we have tried to reduce the level of frustration many students encounter when first confronting this challenging material. We hope that the book will also be useful to researchers in different fields. As usual with a text of this kind, it is impossible to cover all available material. We have refrained from discussing important topics in solid state physics, confident that these are more than adequately covered in appropri- ate textbooks. We have also omitted the finite-temperature formalism of many-body perturbation theory, since it is well documented in other texts. It is a pleasure to thank the many colleagues, students, and others who have contributed to the material in this book, in particular those who have collaborated on the research reported here and those from the Department of Subatomic and Radiation Physics at the University of Ghent. Without their scholarship and interest we would not have been motivated to complete this lengthy project. A special thanks goes to our colleagues who have provided us with data and information that allowed us to construct many of the figures in the text. We anticipate unavoidable corrections to the text. Readers can track these at http://wuphys.wustl.edu/~wimd. Willem H. Dickhoff, St. Louis [email protected] Dimitri Van Neck, Ghent [email protected]