Mathematical Foundations of Quantum Mechanics

John von Neumann (1903–57) was one of the most important mathematicians of the twentieth century. His work included fundamental contributions to mathematics, physics, economics, and the development of the atomic bomb and the computer. He was a founding member of the Institute for Advanced Study in Princeton. Nicholas A. Wheeler is a mathematical physicist and professor emeritus of physics at Reed College.

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