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.

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Contents:Sets, Mappings, GroupsMetric SpacesLinear Operators in Linear SpacesLinear Operators in Normed SpacesThe Extended Real LineMeasurable Sets and Measurable FunctionsMeasuresIntegrationLebesgue MeasureHilbert SpacesL2 Hilbert SpacesAdjoint OperatorsOrthogonal Projections and Projection Valued MeasuresIntegration with respect to a Projection Valued MeasureSpectral TheoremsOne-Parameter Unitary Groups and Stone's TheoremCommuting Operators and Reducing SubspacesTrace Class and Statistical OperatorsQuantum Mechanics in Hilbert SpacePosition 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.