Mathematical Concepts of Quantum Mechanics

Preface to the second edition
Oneof the main goals motivating this new edition was to enhance the
elementary material. To this end, in addition to some rewriting and reorgani
zation, several new sections have been added (covering, for example, spin, and
conservation laws), resulting in a fairly complete coverage of elementary topics.
A second main goal was to address the key physical issues of stability of
atoms and molecules, and mean-field approximations of large particle systems.
This is reflected in new chapters covering the existence of atoms and molecules,
mean-field theory, and second quantization.
Our final goal was to update the advanced material with a view toward
reflecting current developments, and this led to a complete revision and reor
ganization of the material on the theory of radiation (non-relativistic quantum
electrodynamics), as well as the addition of a new chapter.
In this edition we have also added a number of proofs, which were omitted
in the previous editions. As a result, this book could be used for senior level
undergraduate, as well as graduate, courses in both mathematics and physics
departments.
Prerequisites for this book are introductory real analysis (notions of vec
tor space, scalar product, norm, convergence, Fourier transform) and com
plex analysis, the theory of Lebesgue integration, and elementary differential
equations. These topics are typically coveredbythethirdyearinmathematics
departments. The first and third topics are also familiar to physics undergrad
uates. However, even in dealing with mathematics students we have found it
useful, if not necessary, to review these notions, as needed for the course.
Hence, to make the book relatively self-contained, we briefly cover these sub
jects, with the exception of Lebesgue integration. Those unfamiliar with the
latter can think about Lebesgue integrals as if they were Riemann integrals.
This said, the pace of the book is not a leisurely one and requires, at least for
beginners, some amount of work.
Though, as in the previous two issues of the book, we tried to increase
the complexity of the material gradually, we were not always successful, and

irst in Chapter 12, and then in Chapter 18, and especially in Chapter 19,
there is a leap in the level of sophistication required from the reader. One
may say the book proceeds at three levels. The first one, covering Chapters 1
11, is elementary and could be used for senior level undergraduate, as well as
graduate, courses in both physics and mathematics departments; the second
one, covering Chapters 12- 17, is intermediate; and the last one, covering
Chapters 18- 22, advanced.
During the last few years since the enlarged second printing of this book,
there have appeared four books on Quantum Mechanics directed at mathe
maticians:
F. Strocchi, An Introduction to the Mathematical Structure of Quantum Me
chanics: a Short Course for Mathematicians. World Scientific, 2005.
L. Takhtajan, Quantum Mechanics for Mathematicians. AMS, 2008.
L.D. Faddeev, O.A. Yakubovskii, Lectures on Quantum Mechanics for Math
ematics Students. With an appendix by Leon Takhtajan. AMS, 2009.
J. Dimock, Quantum Mechanics and Quantum Field Theory. CambridgeUniv.
Press, 2011.
These elegant and valuable texts have considerably different aims and rather
limited overlap with the present book. In fact, they complement it nicely.
Acknowledgment: The authors are grateful to I. Anapolitanos, Th. Chen, J.
Faupin, Z. Gang, G.-M. Graf, M. Griesemer, L. Jonsson, M. Merkli, M. M¨ uck,
Yu. Ovchinnikov, A. Soffer, F. Ting, T. Tzaneteas, and especially J. Fr¨ohlich,
W. Hunziker and V. Buslaev for useful discussions, and to J. Feldman, G.-M.
Graf, I. Herbst, L. Jonsson, E. Lieb, B. Simon and F. Ting for reading parts
of the manuscript and making useful remarks.