About the Book:
This book emphasizes the close connection between the shorter classical periodic orbits and the partially resolved quantum fluctuations in the level density and the response of an autonomous finite quantum system. Particular care is taken to present a detailed derivation of Gutzwiller's trace formula and its extensions to continuous symmetries, zeta function techniques, and diffractive orbits. Use is made of simple model examples for illustrating the formalism. The self-consistent mean-field approach to the many-body problem is used, and the extended Thomas-Fermi model posited for the average properties of finite fermion systems. Strutinsky's energy theorem is exploited to bring out the quantum effects in interacting systems. Experimental manifestations of quantum shell structure, and their understanding in terms of a few classical orbits, are illustrated in atomic nuclei, metal clusters, and mesoscopic devices. Chapters 1, 2, and 8 are meant for the general reader interested in semiclassical physics and a survey of relevant experiments. The other five chapters give a detailed, but elementary exposition of the theory, aimed at the second-year graduate student level.
Hard cover, 444 pages, 141 figures
ISBN 0-201-48351-3 (First printing: March 1997)
About the authors:
Matthias Brack was born in Basel, Switzerland, and studied there. He spent post-doctorate years at Copenhagen (NBI), Stony Brook (SUNY) and Grenoble (ILL), and is currently a professor of physics at the University of Regensburg, Germany. His present main research efforts are directed toward unraveling the quantum behavior of many-fermion systems in nuclear, atomic and condensed matter physics by semiclassical methods.
Rajat K. Bhaduri was born in Raipur, India. He studied in Calcutta, Bombay, and McMaster University in Hamilton, Canada. He spent post-doctorate years at Oxford, Bombay and McMaster, where he is an emeritus professor of physics. His recent research is in the areas of low-dimensional quantum systems and quantum statistics.
Reviews:
David Brink, former codirector of the European Centre for Theoretical Studies in Nuclear Physics and Related Areas (ECT*), Trento:
To my knowledge there is no other book which studies the field on such a broad base. One of the confusing aspects for the beginner is to understand the differences between Gutzwiller's theory for isolated periodic orbits, the extension for degenerate orbits, the work of Strutinsky, and the theory of Berry and collaborators. These things are clearly explained in the present book as is the comparison between the various methods for calculating the average density of states.
T. Dittrich, Physikalische Blätter 54 (1998) Nr. 4:
Auch vor den trockenen Gefilden der Physik macht die Postmoderne nicht halt: Semiklassik ist in. Was als verspätetes Eindringen des Zeitgeistes in die exakte Naturwissenschaft interpretiert werden könnte, hat ganz pragmatische Gründe. Die fortschreitende Miniaturisierung in der Halbleitertechnologie, die Hinwendung zu grösseren und komplexeren Systemen in der Atomphysik und der Quantenchemie und andere Tendenzen lenken die Aufmerksamkeit zunehmend auf Skalen, wo klassische Dynamik und quantenmechanische Kohärenz zugleich die Physik beherrschen. In diesem Grenzland sind Näherungen kurzer Wellenlänge das Mittel der Wahl. In den vergangenen Jahren hat sich ein vielseitiges Repertoire an "semiklassischen" Methoden entwickelt. Sie sind in einigen klassischen Übersichtsartikeln von Sir Michael Berry und, in Teilaspekten, in mehreren Monographien zugänglich. Eine gut lesbare, elementare Einführung in Buchform fehlte bisher. Diese Lücke soll der 462 Seiten starke Band von Brack und Bhaduri füllen.
Die Autoren verzichten weitgehend auf eine allgemeine Herleitung von Näherungen kurzer Wellenlänge und eine darauf aufsetzende hierarchische Gliederung des Stoffs. Stattdessen (auch darin postmodern ...) handeln sie eine Reihe aktueller Themen (u. a. Quantisierung integrabler vs. nichtintegrabler Systeme, Einteilchen-Zustandsdichte, Spurformel, Schalenstruktur in Mehrteilchensystemen) in relativ unabhängig nebeneinanderstehenden Kapiteln ab. Die Themenauswahl spiegelt durchaus die Vorlieben der Autoren wieder: Anwendungen auf Mehrfermionensysteme der Kern- und der Festkörperphysik wird viel Raum gegeben. Semiklassische Aspekte irregulären Streuens oder dissipativer Dynamik fehlen hingegen fast bzw. ganz. Einleuchtend ist es, wenn im Umkreis der Spurformel die Betonung auf Schalenstrukturen auf gröberen Energieskalen liegt, die sich auf wenige kurze, periodische Bahnen zurückführen lassen.
Das Buch besticht durch eine Fülle klug gewählter und z. T. bis ins Detail ausgearbeiteter Beispiele. Zusammen mit den jedes Kapitel abschließenden Übungsaufgaben und den zahlreichen Abbildungen machen sie es zu einer ausgezeichneten Grundlage für Spezialvorlesungen. Ebensogut ist es dafür geeignet, sich selbständig in das Gebiet einzuarbeiten. Schade, dass das Layout (Standard-LaTeX) etwas fad geraten ist und auf ein professionelles Korrekturlesen offenbar verzichtet wurde. Das ist nicht den Autoren anzulasten, sondern einem Verfall der Kultur des (Physik-)Büchermachens. Eine liebevollere Betreuung durch den Verlag wäre dem Werk zu gönnen.
D. Sen, Current Science 75 (1998) No. 5:
Ever since its development in the 1920s, quantum mechanics has proved to be spectacularly successful in describing the microscopic world of atomic and nuclear physics. Although quantum mechanics represents a radical departure from the concepts of classical mechanics, there is often a close relationship between the behaviour of a quantum mechanical system and its classical limit. Indeed, this is the main idea behind the Bohr-Sommerfeld quantization condition which played a major role in the early history of quantum mechanics before its mathematical structure was fully developed. The connection between classical and quantum mechanics becomes more transparent in the path integral fromalism which was introduced by Feynman and developed further by Van Vleck, Gutzwiller and others. The path integral technique clearly shows when and how the concept of classical trajectories of particles can be useful in quantum mechanics. Very briefly, the path integral method consists of identifying the classical trajectories in a systematic expansion in powers of Planck's constant. This is called a semiclassical approach and it is the main subject of the book under review.
For a small number of particles or degrees of freedom, the path integral method can be a more convenient and geometrically appealing way of studying a system than solving the Schrödinger equation. This is particularly true if the classical system is not integrable. In that case, the Schrödinger equation governing the quantum mechanical system is generally not exactly solvable by analytical methods. While the equation can certainly be solved numerically up to any desired accuracy, many features of the numerical solutions may be difficult to understand in a physically intuitive way. This is where a path integral approach may prove to be fruitful. It turns out that the classical periodic orbits, which generally exist even in nonintegrable systems, play an important role in determining certain features of the quantum mechanical solutions, such as `scars' in the wave functions of a single particle in a confined region, and shell structures in the energy spectra of multiparticle systems like atomic nuclei, metal clusters, and quantum dots. The high point of the periodic orbit theory is the Gutzwiller trace formula which expresses the quantum mechanical amplitude for a particle to go from one space-time point to another in terms of the appropriate classical trajectories and the fluctutations around them. The discussions of the trace formula and its applications form a major part of the book. To this end, the authors examine a variety of classically integrable and nonintegrable systems, and the different methods which are required to study them quantum mechanically.
The authors also consider a related semiclassical technique called the Thomas-Fermi method. This method and its various extensions are particularly useful for studying systems with a large number of particles, either at zero or at finite temperature. The Thomas-Fermi and path integral methods are presented in most books as unrelated topics. However they can be connected to each other through the single-particle density of states, as is demonstrated in this book.
The introductory chapter of the book lists a number of experimental results which can be understood in a fairly simple way using semiclassical ideas. The examples include wave packets of an atomic electron with a high Rydberg number, the spectra of atoms in a strong magnetic field, the electrical resistance of mesoscopic systems particularly in the presence of a magnetic field, and the energy levels of quantum dots in semiconductors. This chapter provides a strong motivation for learning the theoretical tools explained in the rest of the book.
The authors have presented lots of worked out problems for the reader to solve at the end or each chapter. The problems often lead to more advanced topics for which ample references are provided. The lucid style of writing and the large number of figures make the book a pleasure to read. There are brief notes on a number of topics such as chaotic motion, the Riemann zeta function, the quantum theory of scattering, and Hartree-Fock and density functional theories which make the book extremely self-contained. The entire book should be understandable by someone who knows elementary quantum mechanics. It is indispensable for anyone who plans to begin work in this area and is highly recommended for those who are just curious about the subject. This book is the latest addition to the Frontiers in Physics series which has over the years covered almost the entire spectrum of modern physics. Libraries of science would certainly benefit by acquiring this book written by two experts in the field.