Mathematical Foundations of Quantum Mechanics

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Princeton University Press, 1955 - Mathematics - 445 pages
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Mathematical Foundations of Quantum Mechanics was a revolutionary book that caused a sea change in theoretical physics. Here, John von Neumann, one of the leading mathematicians of the twentieth century, shows that great insights in quantum physics can be obtained by exploring the mathematical structure of quantum mechanics. He begins by presenting the theory of Hermitean operators and Hilbert spaces. These provide the framework for transformation theory, which von Neumann regards as the definitive form of quantum mechanics. Using this theory, he attacks with mathematical rigor some of the general problems of quantum theory, such as quantum statistical mechanics as well as measurement processes. Regarded as a tour de force at the time of publication, this book is still indispensable for those interested in the fundamental issues of quantum mechanics.

 

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Contents

Introductory Considerations
3
2 The Original Formulation of Quantum Mechanics
6
The Transformation Theory
17
Hilbert Space
28
Abstract Hilbert Space
34
2 The Geometry of Hilbert Space
46
3 Digression on the COnditions A E
59
4 Closed Linear Manifolds
73
2 The Statistical Interpretation
206
3 Simultaneous Measurability and Measurability in General
211
4 Uncertainty Relations
230
5 Projections as Propositions
247
6 Radiation Theory
254
Deductive Development of the Theory
295
2 Proof of the Statistical Formulas
313
3 Conclusions From Experiments
328

5 Operators in Hilbert Space
87
6 The Eigenvalue Problem
102
7 Continuation
107
8 Initial Considerations Concerning the Eigenvalue Problem
119
9 Digression of the Existence and Uniqueness of the Solutions of the Eigenvalue Problem
145
10 Commutative Operators
170
11 The Trace
178
The Quantum Statistics
196
General Considerations
347
2 Thermodynamical Considerations
358
3 Reversibility and Equilibrium Problems
379
4 The Macroscopic Measurement
398
The Measuring Process
417
2 Composite Systems
422
3 Discussion of the Measuring Process
437
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About the author (1955)

Von Neumann was one of the greatest mathematicians of the twentieth century and a pioneering figure in computer science. A native of Hungary who held professorships in Germany, he was appointed Professor of Mathematics at the Institute for Advanced Study (IAS) in 1933. Later he worked on the Manhattan Project, helped develop the IAS computer, and was a consultant to IBM.

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