Mathematical Concepts of Quantum Mechanics

The first fifteen chapters of these lectures (omitting four to six chapters each year) cover a one term course taken by a mixed group of senior undergraduate and junior graduate students specializing either in mathematics or physics. Typically, the mathematics students have some background in advanced anal­ ysis, while the physics students have had introductory quantum mechanics. To satisfy such a disparate audience, we decided to select material which is interesting from the viewpoint of modern theoretical physics, and which illustrates an interplay of ideas from various fields of mathematics such as operator theory, probability, differential equations, and differential geometry. Given our time constraint, we have often pursued mathematical content at the expense of rigor. However, wherever we have sacrificed the latter, we have tried to explain whether the result is an established fact, or, mathematically speaking, a conjecture, and in the former case, how a given argument can be made rigorous. The present book retains these features.

From the reviews:

"This is really a very beautiful book on the mathematical structure of quantum mechanics. It is very useful for mathematically oriented students because they can see functional analysis and spectral theory at work. The authors describe lesser-known ideas […]. The book will also be very useful for physicists. It shows which beautiful facets of mathematics are connected to the well-known parts of quantum mechanics. I must say that it has been difficult to find a good textbook containg both the physics and the mathematics of quantum mechanics. I think that this book solves the problem of how to teach quantum mechanics to mathematically oriented students in an optimal way. Congratulations must go to the authors."

EMS Newsletter December 2005

"I like this book. ... some mathematics books are written to be read; such books are as a light in the darkness to the wearied reader and a credit to the author(s). Mathematical concepts of quantum mechanics is one of these 'light in the darkness' books. Considering the technical nature of the material covered, a lot of the book is written in plain English - words that is, with short sentences and never a big word where a little one will do. The chapters too are digestibly short. The authors succeed in cutting through the mathematical technicalities of quantum mechanics by direct explanation and the ejection of unnecessary complications. ... It is a first class study book for graduates wishing to gain a very deep understanding of the mathematical theory that underlies quantum mechanics, but they will require more than a three-year mathematics degree to tackle all of it. As such a study book, it is first class - well done to Gustafson and Sigal."

D.Morris, The Mathematical Gazette, March 2005

"... A very strong point of this textbook is that in spite of the fact that it starts from scratch, it quickly arrives at a number of rather advanced topics thatare important in the modern literature. I expect that many physics students will feel a need for more physical explanations;but more mathematically inclined students will appreciate the book for its systematic approach."

D.Dieks, University of Utrecht, Contemporary Physics 2004, Vol. 45, Issue 4

"The present textbook is a very readable introduction to modern mathematical topics in quantum mechanics intended for students of mathematics or physics. However, even though much background material (both mathematical and physical) has been included to make the book relatively self-contained, it still proceeds at a fast pace and requires some amount of work, especially for beginners. On the other hand, it quickly reaches topics of current interest. …. I consider the book a valuable addition to the textbook literature in this field."

G. Teschl, Internationale Mathematische Nachrichten 196, p. 69, 2004

"... The book is well-organized and even though the presentation is mathematically advanced the authors do not get lost in mathematical technicalities. There is also a useful guide to the literature. Moreover, several of the results in the book are not normally included in introductory quantum theory textbooks. Therefore the book is particularly suitable for beginners who wish to become acquainted with essential mathematical results and structures underpinning quantum theory. The book may also serve as a useful guide for teachers of quantum theory. ..."

O. Rudolph, MathSciNet, Mathematical Reviews on the Web, AMS 2004

" ... This book is well-written, and the topics discussed have been well thought-out. It would provide a useful approach to quantum theory for the mathematician, and would also provide access for the physicist to some mathematically advanced methods and topics, but the physicist would definitely have to be prepared to work hard at the mathematics required."

J.Phys. A2004

"I like this book. … The authors succeed in cutting through the mathematical technicalities of quantum mechanics by direct explanation and ejection of unnecessary complications. … In addition, there is a large amount of clearly written mathematical tuition in the book. … The authors say they have aimed this book towards readers who have studied the third year of a mathematics degree and some undergraduate physics. … As such a study book, it is first class – well done to Gustafson and Sigal."

D.Morris, The Mathematical Gazette, March, 2005

"As the title suggests, the emphasis in this textbook is on the mathematical aspects of quantum theory. … A very strong point of this textbook is that in spite of the fact that it starts from scratch, it quickly arrives at a number of rather advanced topics that are important in the modern literature. … mathematically inclined students will appreciate the book for its systematic approach."

D. Dieks, Contemporary Physics, Vol. 45 (4), 2004

"This book is an introduction to the mathematics of quantum mechanics. … The strength of the book is where it shows how the mathematical treatment of quantum mechanics brings insights to physics. … There is much more in this book. It will be useful to the experienced reader as a guide to the impressive recent advances in mathematical quantum mechanics."

William G. Faris, SIAM Review, Vol. 47 (2), 2005

"The present textbook is a very readable introduction to modern mathematical topics in quantum mechanics intended for students of mathematics or physics. … I consider the book a valuable addition to the textbook literature in this field."

G.Teschl, Monatshefte für Mathematik, Vol. 143 (4), 2004

"This book presents a competent and interesting pedagogic account of certain topics in functional analysis and their applications to a selection of problemsin quantum mechanics, quantum field theory and quantum statistical mechanics. The typography and page layouts are, as one expects in a Springer publication, superb."

Chandra Shekhar Sharma, Zentralblatt MATH, Vol. 1033 (8), 2004

"The book under review contains a mathematical introduction to quantum theory. … The book is well organized and … the authors do not get lost in mathematical technicalities. There is also a useful guide to the literature. … the book is particularly suitable for beginners who wish to become acquainted with essential mathematical results and structures underpinning quantum theory. The book may also serve as a useful guide for teachers of quantum theory."

Oliver Rudolph, Mathematical Reviews, 2004 g

"This book is well-written, and the topics discussed have been well thought-out. It would provide a useful approach to quantum theory for the mathematician, and would also provide access for the physicist to some mathematically advanced methods and topics … ."

A Whitaker, Journal of Physics A: Mathematical and General, February, 2004