Hilbert Space and Quantum Mechanics

The topics of this book are the mathematical foundations of non-relativistic quantum mechanics and the mathematical theory they require. The main characteristic of the book is that the mathematics is developed assuming familiarity with elementary analysis only. Moreover, all the proofs are carried out in detail. These features make the book easily accessible to readers with only the mathematical training offered by undergraduate education in mathematics or in physics, and also ideal for individual study. The principles of quantum mechanics are discussed with complete mathematical accuracy and an effort is made to always trace them back to the experimental reality that lies at their root. The treatment of quantum mechanics is axiomatic, with definitions followed by propositions proved in a mathematical fashion. No previous knowledge of quantum mechanics is required. This book is designed so that parts of it can be easily used for various courses in mathematics and mathematical physics, as suggested in the Preface.

The book is of interest to researchers and graduate students in functional analysis, who can see how closely an important part of their chosen field is linked with quantum mechanics, and also to physicists, who can see how the abstract language of functional analysis brings unity to the apparently distinct approaches employed in quantum theory.

Contents:

• Sets, Mappings, Groups
• Metric Spaces
• Linear Operators in Linear Spaces
• Linear Operators in Normed Spaces
• The Extended Real Line
• Measurable Sets and Measurable Functions
• Measures
• Integration
• Lebesgue Measure
• Hilbert Spaces
• L2 Hilbert Spaces
• Adjoint Operators
• Orthogonal Projections and Projection Valued Measures
• Integration with respect to a Projection Valued Measure
• Spectral Theorems
• One-Parameter Unitary Groups and Stone's Theorem
• Commuting Operators and Reducing Subspaces
• Trace Class and Statistical Operators
• Quantum Mechanics in Hilbert Space
• Position and Momentum in Non-Relativistic Quantum Mechanics

Readership: Advanced undergraduate and graduate students in mathematics or in physics as well as researchers in functional analysis or in fundamental quantum mechanics.
Key Features:

• Self-contained, assuming only basic mathematic knowledge