Customer Reviews

Mathematical Foundations of Quantum Mechanics

5 star:		(4)
4 star:		(2)
3 star:		(0)
2 star:		(1)
1 star:		(0)

 

 

First the strengths:
(1) The author gives a rigorous proof of the incompatibility of the two theories of quantum mechanics, that is, the evolution of the state vector as determined by Schrodinger's equation and the collapse of the state vector by means of an observation. He does this by the concept of entropy.
(2) He gives a thorough treatment of the "consciousness school" of interpretation, which makes for an interesting,if not entirely convincing read.

The weaknesses:
(1) the font is a pain to read; it looks like it came off an old typewriter. I wonder why the publishers couldn't put it into a more modern readable form.
(2) Von Neumann writes this book , in part, with the intention to dispel the mathematical nonsense, as he perceives it, of the Dirac delta function. Therefore he casts everything into the unwieldy formalism required to do without the distribution. Indoubtedly he was trying to change the dirac formalism in use in quantum mechanics at the time but was fortunately unsucessful in persuading physicists to use his alternative language.

Summary: I recommend this book for anyone wishing to deepen his or her understanding of the foundations, conceptual and mathematical of quantum theory.

This is not the kind of book I would recommend to a novice person in the area, but it does give a very interesting view of how was quantum mechanics born. It begins with a thorough discussion about the mathematics of Hilbert spaces and operator theory to later merge Heisenberg's and Schrodinger's theories in one rigorous mathematical theory. It makes some remarks that allows the reader to see how was the 'new' quantum theory born and developed, since it briefly discusses the theories of Heisenberg and Schrodinger in the way they originally stated them. Maybe the most disturbing issue would be the notation since in 1932 dirac had still not developed the bracket formalism.

The ultimate source of Hilbert Space applied to Quantum Mechanics. John von Neumann was the first to systematically formulate QM in such a powerful and elegant vector space. If this is the Bible of QM in HS, Hughes is the missioner! Get the book as well -- The Structure and Interpretation of Quantum Mechanics.

As an undergrad, I am sorry that I cannot share the perspectives of professionals as expressed below. After initial introductory courses, I got fascinated by certain untold conceptual issues. And one of the textbooks (probably Griffiths) suggested von Neumann had tried to prove mathematically that the classical formulation is just the furthest the formalism can go and we don't have to worry about underlying complexities. Later, Bohm created a formalism which von Neumann "proves" to be mathematically impossible in this book. I bought this book just to find out how the proof goes. But I got stuck with some tedious proofs on Hilbert space (which he calls a "digression"). This part isn't essential but as the braket notation is not used you need to consult this part. I think at least a strong background in linear algebra is required. Definitly not an introductory textbook. Most useful for those who study history of physics.

I strongly recommend perusing a printed copy of this text before deciding upon purchase. A large body of the text consists of mathematical proofs of mostly elementary properties of Hilbert space. Despite the historic significance of the book in establishing a rigorous foundation for the math underlying QM, it is not a good text for developing either a conceptual or functional understanding of modern mathematical techniques in QM.

This is a beautifully written book, more for mathematicians than for physicists, since von Neumann does not really discuss the physics that goes into the subject. There is also an emphasis on "foundation", so you will not have too many examples of hydrogen atoms work out. The style is also unmistakably mathematical. Every mathematician with an interest in quantum mechanics should have this book.

von Neumann's text is still the standard, some 40 yrs after its pub date, for the definitive mathematical treatment of Quantum Mechanics. This comes to me from an expert in the field at the U of Chicago. (He also reccomended Dirac's seminal text to those wishing knowledge of the inner workings of Q. M.) Read the Quantum Bible and learn from the very same source from which the experts cut their teeth. Available from Amazon dot product com. and the Seminary Co-op in Chicago 773-752-4381. Mark Witucke Bookseller Chicago