Mathematical Foundations of Quantum Mechanics

by John von Neumann

Overview

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.

Product Details

ISBN-13:	9780691028934
Publisher:	Princeton University Press
Publication date:	11/17/1996
Series:	Princeton Landmarks in Mathematics and Physics , #16
Edition description:	Reprint
Pages:	460
Product dimensions:	6.00(w) x 9.25(h) x (d)

Table of Contents

Translator’s Preface vii

Preface to This New Edition ix

Foreword xi

Introduction 1

I Introductory Considerations

1 The Origin of the Transformation Theory 5

2 The Original Formulations of Quantum Mechanics 7

3 The Equivalence of the Two Theories: The Transformation Theory 13

4 The Equivalence of the Two Theories: Hilbert Space 21

II Abstract Hilbert Space

1 The Definition of Hilbert Space 25

2 The Geometry of Hilbert Space 32

3 Digression on the Conditions A-E 40

4 Closed Linear Manifolds 48

5 Operators in Hilbert Space 57

6 The Eigenvalue Problem 66

7 Continuation 69

8 Initial Considerations Concerning the Eigenvalue Problem 77

9 Digression on the Existence and Uniqueness of the Solutions of the Eigenvalue Problem 93

10 Commutative Operators 109

11 The Trace 114

III The Quantum Statistics

1 The Statistical Assertions of Quantum Mechanics 127

2 The Statistical Interpretation 134

3 Simultaneous Measurability and Measurability in General 136

4 Uncertainty Relations 148

5 Projections as Propositions 159

6 Radiation Theory 164

IV Deductive Development of the Theory

1 The Fundamental Basis of the Statistical Theory 193

2 Proof of the Statistical Formulas 205

3 Conclusions from Experiments 214

V General Considerations

1 Measurement and Reversibility 227

2 Thermodynamic Considerations 234

3 Reversibility and Equilibrium Problems 247

4 The Macroscopic Measurement 259

VI The Measuring Process

1 Formulation of the Problem 271

2 Composite Systems 274

3 Discussion of the Measuring Process 283

Name Index 289

Subject Index 291

Locations of Flagged Propositions 297

Articles Cited: Details 299

Locations of the Footnotes 303

Editorial Reviews

The translator and publisher have performed a service in making the 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.

Quarterly of Applied Mathematics - A. F. Stevenson

It remains indispensable to those who desire a rigorous presentation of the foundations of the subject.

Quarterly of Applied Mathematics

The translator and publisher have performed a service in making the 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.

Quarterly of Applied Mathematics - A.F. Stevenson

"[Von Neumann's] book was the first exposition of quantum mechanics that made the theory mathematically respectable. The concepts were rigorously defined and the consequences rigorously deduced. . . . From that book I learned most of what I know about quantum mechanics."—-Freeman Dyson, Notices of the American Mathematical Society

"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 Publisher