Mathematical Foundation of Quantum Mechanics

This is abrief introduction to the mathematical foundations of quan- tum mechanics based on lectures given by the author to Ph.D.students at the Delhi Centre of the Indian Statistical Institute during the years 1980- 1985 in order to initiate active research in the emerging field of quantum prob ability.

A student's first course on quantum mechanics provides the foundation essential for much of his or her future work in physics, be it in atomic, elementary particle, or solid state physics. This introductory textbook contains not only the foundations and many applications of quantum mechanics, but also new aspects and their experimental verification. It has as its particular virtues clarity and conciseness of presentation, while at the same time being self-contained. Comprehensibility is further guaranteed by the inclusion of all the mathematical steps required for a complete understanding. Carefully chosen problems help to consolidate the student's knowledge. In the introductory chapters, starting from the historical evolution of the subject, the fundamental postulates are developed inductively by means of an interference experiment. Thereafter the structure is purely deductive, covering all of nonrelativistic quantum mechanics, plus the quantization of the radiation field in the context of optical transitions. As well as the quantum mechanical "essentials", for instance, detailed treatments of scattering theory, time dependent phenomena, and the density matrix, such topics as the theory of quantum mechanical measurement and the Bell inequality are dis- cussed. A separate chapter is devoted to supersymmetric quantum mechanics, an area which to date has been accessible only in the research literature.

This book provides an in-depth and accessible description of special relativity and quantum mechanics which together form the foundation of 21st century physics. A novel aspect is that symmetry is given its rightful prominence as an integral part of this foundation. The book offers not only a conceptual understanding of symmetry, but also the mathematical tools necessary for quantitative analysis. As such, it provides a valuable precursor to more focused, advanced books on special relativity or quantum mechanics.

The spectral theory of linear operators plays a key role in the mathematical formulation of quantum theory. Furthermore, such a rigorous mathematical foundation leads to a more profound insight into the nature of quantum mechanics. This textbook provides a concise and comprehensible introduction to the spectral theory of (unbounded) self-adjoint operators and its application in quantum dynamics.

This volume provides a broad perspective on the state of the art in the philosophy and conceptual foundations of quantum mechanics. Its essays take their starting point in the work and influence of Itamar Pitowsky, who has greatly influenced our understanding of what is characteristically non-classical about quantum probabilities and quantum logic, and this serves as a vantage point from which they reflect on key ongoing debates in the field.

Advanced Quantum Mechanics, the second volume on quantum mechanics by Franz Schwabl, discusses nonrelativistic multi-particle systems, relativistic wave equations and relativistic quantum fields. Characteristic of the author´s work are the comprehensive mathematical discussions in which all intermediate steps are derived and where numerous examples of application and exercises help the reader gain a thorough working knowledge of the subject. The topics treated in the book lay the foundation for advanced studies in solid-state physics, nuclear and elementary particle physics. This text both extends and complements Schwabl´s introductory Quantum Mechanics, which covers nonrelativistic quantum mechanics and offers a short treatment of the quantization of the radiation field. The fourth edition has been thoroughly revised with new material having been added. Furthermore, the layout of the figures has been unified, which should facilitate comprehension.