Not long ago I had occasion to review Quantum Mechanics for Mathematicians, by Leon Takhtajan. The present book is mentioned very favorably by Takhtajan in the Preface to his book: “[The] excellent lectures notes for undergraduate students by L. D. Faddeev and O. A. Yakubovskiĭ seems to be the only book on quantum mechanics completely accessible to mathematicians” — at least that how it was before the appearance of Takhtajan’s contribution, which, however, is aimed at a more mathematically mature audience. À propos, there is at least one more player in this game, namely Teschl’s brand-new Mathematical Methods in Quantum Mechanics, also reviewed in the present column.
In any event, it is an established fact that QM in the style of the physicists, who after all have custody of it, so to speak, is painful for mathematicians to have to deal with, pace such mathematically pure expositors as Von Neumann, Mackey, Barry Simon, and Varadarajan. But it’s the business of learning QM that’s at issue at the moment and the indicated works of the authors just mentioned are pitched higher. So it appears that at the level of a well-prepared undergraduate the present book by Faddeev-Yakubovskiĭ is a sine qua non.
In point of fact, Lectures on Quantum Mechanics for Mathematics Students, while accurately aimed at novices, depends heavily on the two books by Von Neumann and by Mackey, both interestingly bearing the same title: Mathematical Foundations of Quantum Mechanics. (The books have ostensibly different thrusts, however: Von Neumann’s functional analysis vs. Mackey’s unitary representation theory — same coin, different sides?) However, the book under review is more elementary than either of these. Also, its pace is good: not too fast, not too slow. And the coverage is thorough, modulo its pre-delineated scope; nonetheless, in less than 250 pages, the authors take us all the way to scattering theory.
A thorough course in QM for mathematicians might start with Faddeev-Yakubovskiĭ and proceed on to Takhtajan’s book, given that he in fact took the indicated course at Math-Mech (Leningrad) and has in fact contributed an appendix on the formalism of classical mechanics to the present Lectures. Additionally, Faddeev states in the Preface to the English Edition that “[he] had hoped to modify the book [under review,] making it more informative. However, the recent book by Leon Takhtajan… [has] made such modifications unnecessary.” And the arc might be completed by Teschl’s book…
Finally, Lectures on Quantum Mechanics for Mathematics Students is obviously a most valuable contribution: it presents to the English speaking mathematical world. i.e. just about every one (including the French), the marvelous opportunity to learn QM in the style of mathematicians properly so-called.
Post Scriptum: O. A. Yakubovskiĭ passed away at the time the decision was made to launch this English edition. May he rest in eternal peace.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.