Green’s Functions in Quantum Physics

324 Pages·1983·4.622 MB·English

by  Professor Eleftherios N. Economou PhD (auth.)

7
Springer Series in Solid-State Sciences
Edited by Peter Fulde
Springer Series in Solid-State Sciences
Editors: M. Cardona P. Fulde H.-J. Queisser
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Volume 42 Crystal Optics with Spatial Dispersion and the Theory of Excitations
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Volume 43 Resonant Nonlinear Interactions of Light with Matter
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Volume 44 Elastic Media with Microstmcture II Three-Dimensional Models
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Volumes 1 - 39 are listed on the back inside cover
E. N. Economou
Green's Functions
in Quantum Physics
Second Corrected and Updated Edition
With 52 Figures
Springer-Verlag Berlin Heidelberg GmbH
Professor Ele/therios N. Economou, PhD
Department of Physics, University of Crete,
Heraklion, Crete, Greece
Series Editors:
Professor Dr. Manuel Cardona
Professor Dr. Peter Fulde
Professor Dr. Hans-Joachim Queisser
Max-Planck-Institut fUr Festkărperforschung, HeisenbergstraBe 1
D-7000 Stuttgart 80, Fed. Rep. of Germany
ISBN 978-3-540-12266-1 ISBN 978-3-662-02369-3 (eBook)
DOI 10.1007/978-3-662-02369-3
Library of Congress Cataloging in Publication Data. Economou, E. N., 1940-. Green's functions in
quantum physics. (Springer series in solid-state sciences ; v. 7). Bibliography: p. Inc1udes index. 1. Green's
functions. 2. Quantum theory. 1. Title. II. Series. QC174.17.G68E25 1983 530.1'43 83-4739
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© by Springer-Verlag Berlin Heidelberg 1979 and 1983
Urspriinglich erschienen bei Springer-Verlag Berlin Heidelberg New York 1983
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to Sophia
Preface to the Second Edition
In this edition the second and main part of the book has been considerably
expanded as to cover important applications of the formalism.
In Chap.5 a section was added outlining the extensive role of the tight
binding (or equivalently the linear combination of atomic-like orbitals)
approach to many branches of solid-state physics. Some additional informa
tion (including a table of numerical values) regarding square and cubic
lattice Green's functions were incorporated.
In Chap.6 the difficult subjects of superconductivity and the Kondo
effect are examined by employing an appealingly simple connection to the
question of the existence of a bound state in a very shallow potential well.
The existence of such a bound state depends entirely on the form of the un
perturbed density of states near the end of the spectrum: if the density of
states blows up there is always at least one bound state. If the density of
states approaches zero continuously, a critical depth (and/or width) of the
well must be reached in order to have a bound state. The borderline case of
a finite discontinuity (which is very important to superconductivity and the
Kondo effect) always produces a bound state with an exponentially small
binding energy.
Chapter 7 has been expanded to cover details of the new and fast developing
field of wave propagation in disordered media. The coherent potential ap
proximation (a simple but powerful method) is presented with an extensive
list of references to the current literature. Then the electrical conductivity
is examined both because it is an interesting quantity in its own right and
because it plays a central role in demonstrating how disorder can create a
qualitatively different behavior. Since the first edition of this book, very
significant advances in the field of random media have taken place. An effort
has been made to present in a simple way the essential pOints (for the reader
with a casual interest in this subject) and to review the current literature
(for the sake of the reader whose research activities are or will be related
to the field of disordered systems).
VII
In this edition each chapter is preceded by a short outline of the material
to be covered and it is concluded by a summary containing the most important
equations numbered as in the main text.
I would like to thank A. Andriotis and A. Fertis for pointing out to me
several misprints in the first edition. I would also like to express my
gratitude to Exxon Research and Engineering Company for its hospitality
during the conclusion of this work.
Heraklion, Crete E.N. Economou
January, 1983
VIII
Preface to the First Edition
This text grew out of a series of lectures addressed to solid-state experi
mentalists and graduate students beginning their research career in solid
state physics.
The first part consisting of Chaps.1 and 2 is a rather extensive mathe
matical introduction which covers the material about Green's functions
usually included in a graduate course on mathematical physics. Emphasis is
given to those topics which are of significance in quantum physics. On the
other hand, little attention is given to the important question of deter
mining the Green's functions associated with boundary conditions on sur
faces at finite distance from the source.
The second and main part of the book is, in my opinion, a first effort
to collect in a systematic but concise way various topics of quantum physics,
where the Green's functions (as defined in Part I) can be successfully
applied. Chapter 3 is a direct application of the formalism developed in
Part I. In Chap.4 the perturbation theory for Green's functions is present
ed and applied to scattering and to the question of formation of bound
states. Next, the Green's functions for the so-called tight-binding Hamil
tonian (TBH) are calculated. The TBH is of central importance for solid
state physics because it is the simplest example of wave propagation in
periodic structures. It is also important for quantum physics in general,
because it is rich in physical phenomena (e.g., negative effective mass,
creation of a bound state by a repulsive perturbation) and, at the same
time, simple in its mathematical treatment. Thus one can derive simple,
exact expressions for scattering cross sections and for bound and resonance
levels. The multiple scattering formalism is presented within the frame-
work of the TBH, and it is applied to questions related with the behavior
of disordered systems (such as amorphous semiconductors). The material of
Part II is of interest not only to solid-state physicists but to students
of a graduate course in quantum mechanics (or scattering theory) as well.
IX
In Part III, with the help of the second quantization formalism, the
many-body Green's functions are introduced and utilized in extracting physical
information about interacting many particle systems. Many excellent books
have been devoted to the material of Part III (e.g., FETTER and WALECKA:
Quantum Theory of Many-Particle Systems [4.1]). Thus the present treatment
must be viewed as a brief introduction to the subject; this introduction may
help the solid-state theorist approach the existing thorough treatments of
the subject and the solid-state experimentalist to become acquainted with the
formalism.
I would like to thank Nuclear Research Center "Demokritos" and the Greek
Atomic Energy Commission for their hospitality during the writing of the
second half of this book.
Athens, Greece, November 1978 E.N. Economou
x
Contents
Part I: Green's Functions in Mathematical Physics
1. Time-Independent Green's Functions •...•.••......•...••........•... 3
1 . 1 Forma 1 ism ...................................................... 3
1.2 Examples ....................................................... 10
1.2.1 Three-Dimensional Case .................................. 10
1.2.2 Two-Dimensional Case .................................... 12
1.2.3 One-Dimensional Case .................................... 14
1.2.4 Finite Domain Il ......................................... 14
1.3 Summary ........................................................ 15
2. Time-Dependent Green's Functions .••............................... 19
2.1 First-Order Case ............................................... 19
2.2 Examples ....................................................... 23
2.3 Second-Order Case .............................................. 24
2.4 Examp 1e s ....................................................... 29
2.5 Summary ........................................................ 33
Part 1/: Green's Functions in One-Body Quantum Problems
3. Physical Significance of G. Application to the Free-Particle Case ..... 39
3.1 General Relations .............................................. 39
3.2 The Free-Particle (Ho =p2/2m) Case ............................. 42
3.3 The Free-Particle Klein-Gordon Case ...........................• 46
3.4 Summary ........................................................ 48
4. Green's Functions and Perturbation Theory ••.......••.•.•..••..••.. 50
4.1 Formalism ...................................................... 50
4.1.1 Time-Independent Case ................................... 50
XI