Mathematical Foundations of Quantum Mechanics 1st Edition
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"That von Neumann has been ‘par excellence' the mathematician of quantum mechanics is as obvious to every physicist now as it was a quarter of a century ago. Quantum mechanics was very fortunate indeed to attract, in the very first years after its discovery in 1925, the interest of a mathematical genius of von Neumann’s stature. As a result, the mathematical framework of the theory was developed and the formal aspects of its entirely novel rules of interpretation were analyzed by one single man in two years."---Léon Van Hove, Bulletin of the American Mathematical Society
"After almost a quarter of a century this book is still a valuable addition to any library. There are few references in which quantum statistics, including the concept of the density matrix and a discussion of the theory of measurement, is treated in as masterful a manner. . . . [The translator] is to be congratulated on an excellent piece of work."---M. E. Rose, Physics Today
"The translator and publisher have performed a service in making this classic available to a wider circle of English-speaking readers. It remains indispensable to those who desire a rigorous presentation of the foundations of the subject."---A. F. Stevenson, Quarterly of Applied Mathematics
"[W]hat a delight. . . . [I]nsightful, motivated, intuitive and understandable. . . . [A] classic."---S. Gudder, Mathematical Reviews
From the Back Cover
- Publisher : Princeton University Press; 1st edition (January 1, 1955)
- Language : English
- Paperback : 464 pages
- ISBN-10 : 0691028931
- ISBN-13 : 978-0691028934
- Item Weight : 1.44 pounds
- Dimensions : 6 x 1.05 x 9.25 inches
- Best Sellers Rank: #1,547,207 in Books (See Top 100 in Books)
- Customer Reviews:
Top reviews from the United States
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(1) Perhaps the most captivating aspects deal with causality (for instance, pages 302, 305, 323, 326, 328). Let us read: "As we see, the attempt to interpret causality as an equality definition led to a question of fact which can and must be answered, and which might conceivably be answered negatively." Chapter four will explain that statement !
(2) The structure of this treatise is interesting: First, physical (chapter one) then, mathematical (chapter two), then an amalgamation of the physical and the mathematical (chapter three), following which, statistical (chapter four) and, finally, measurements (chapters five and six; these two chapters are the pages reprinted in the 1983 Wheeler & Zurek compendium).
(3) Now, if you have John von Neumann, you need to study: Dirac, Kramers, Heisenberg, Pauli, and Schrodinger. It is my opinion that in order to appreciate all of these historical documents, you need to compare and contrast each. Previous to commencing, read two articles: London and Baer "The Theory of Observation in Quantum Mechanics (1939)," and Bryce dewitt "Quantum Mechanics and Reality" (1970, Physics Today 23(9):30-35). All are well worth the effort !
(4) Not everything herein is difficult (that was another mistaken bias I possessed): For example, von Neumann explicitly shows how he justifies Dirac's "delta function" with (instead) his "function sequences." (see: footnote #84, page 128). Another is where we learn that the "trace is invariant" (footnote #113, page 179). So you will see, the more involved details are relegated to the footnotes. That is, the "rigor" which is here does not detract from the lucid exposition.
(5) Something which is explicated herein: the interplay between discontinuity and continuity. That is, between discrete and continuous. Or, when the total energy is "known" the time-dependent schrodinger wave-equation is "continuous and causal," otherwise, confronted with discontinuous, instantaneous, and non-causal. Read: "the chief weakness of quantum mechanics is that it presupposes a simultaneity concept" but, "what we really need is not that the change of t (time) be small, but only that it have little effect in the calculation of probabilities." (page 354).
(6) Probability and Born: "the first statistical statements on the behavior of a system in the 'state- theta' originated with Max Born. Also, "although we believe that after specifying the 'state' we know the state completely, nevertheless, only statistical statements can be made of the physical quantities involved." (page 207).
(7) Uncertainty Relations: "it will not be clear to common-sense without a further discussion why the position and velocity (coordinate and momentum) of a material body cannot both be measured with arbitrarily high accuracy.
Therefore it is necessary to elucidate by an exact analysis... that this is not the case." (page 238). This, he proceeds to do. This is chapter three, a fine chapter entitled "the quantum statistics."
(8) I conclude my review. In so doing, I apologize to John von Neumann and readers of my review. There is simply too much here that is fascinating and well-written. I have spent many more hours studying the texts and papers of Dirac, Kramers, Heisenberg and Pauli and Schrodinger than anything von Neumann ever wrote. That is a mistake for which I intend to make amends. Get a copy of this fascinating treatise, and study the entirety of it !
This book represents that immensity. Covering the development of the Transformation Theory and its origin to the Measuring Process, von Neumann is capable of providing the mathematical rigor as well as detailed and easy to understand commentary throughout this important work.
The Notes in this work stand out, especially. They are informative and compliment the main text explicitly, expanding it and making it more informative. They often go beyond a simple reference to operate as a subtext of the main text, not to be ignored.
Further adding to this point is the fact that von Neumann, throughout this work, continues to give personal commentary: *reasons* for and historical references to, the many mathematical pronouncements and derivations. For instance, on page 196 he begins to develop the statistical assertions of quantum mechanics. By page 198 we have been shown "one of the first and simplest examples by means of which the statistical character of quantum mechanics was recognized." Not only was the derivation clear and concise, the reader is provided the historical context as well.
Often a book of this sort is more a historical document than active reference (unless you are capable of the math). As such, books on quantum mechanics authored by the early founders (and, in this case, a later superb contributor and inventor of notions like "quantum logic") offer an insider look at the mindset of both the classically trained physicist against what the new physical theory asked of that mind.
For instance, this is reflected in a superb Preface, wherein the author states the object of this book ("to present the new quantum mechanics in a unified representation which, so far as it is possible and useful, is mathematically rigorous...what is presumably a definitive form: the so-called "transformation theory."). We also learn in this Preface that "we shall as a rule omit any discussion of the application of quantum mechanical methods to particular problems, as well as any discussion of special theories derived from the general theory - at least so far as this is possible without endangering the understanding of the general relationships."
He goes on to point out that his mathematical treatment in this work "deviates considerably from that of Dirac." Thus, he takes Dirac's "elegant" theory for it "in no way satisfies the requirements of mathematical rigor - not even if these are reduced in a natural and proper fashion to the extent common elsewhere in theoretical physics. For example, the method adheres to the fiction that each self-adjoint operator can be put in diagonal form."
Von Neumann's solution is to start from the beginning with Hermitean operators and Hilbert spaces which "provide the framework" for the Transformation Theory. This book is that story in the authors own voice.
I suggest you purchase a copy for your library today.
So very satisfied !
Top reviews from other countries
You get the feeling that von Neumann just steam-rollerred the whole infant subject into submission.
Given who the author is, this is to be expected. It is undoubtedly a magnificent achievement.
But the book is clearly a photocopy of a manuscript knocked up on an old fashioned typewriter. The layout is very poor. It even contains hand-written special characters. It is a very poor production for a vast amount of money.
I particularly wanted to see his proof that the actual point of measurement is when that measurement sinks into the consciousness. Found it. The logic is impeccable, but I don't know that I believe it.
I had also heard that the concept of "the collapse of the wave function" is due to von Neumann. I did not find that confirmed in this book. But as a wave function is, for von Neumann, only another chance to get straight back into Hilbert space, perhaps he did not invent this concept, after all.
This is a historical document - not a book for students or (like me) the casual observer.