Mathematical methods in quantum mechanics : with applications to Schrödinger operators
著者
書誌事項
Mathematical methods in quantum mechanics : with applications to Schrödinger operators
（Graduate studies in mathematics, v. 157）
American Mathematical Society, c2014
2nd ed
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注記
Includes bibliographical references (p. 345-348) and index
内容説明・目次
内容説明
Quantum mechanics and the theory of operators on Hilbert space have been deeply linked since their beginnings in the early twentieth century. States of a quantum system correspond to certain elements of the configuration space and observables correspond to certain operators on the space. This book is a brief, but self-contained, introduction to the mathematical methods of quantum mechanics, with a view towards applications to Schroedinger operators.
Part 1 of the book is a concise introduction to the spectral theory of unbounded operators. Only those topics that will be needed for later applications are covered. The spectral theorem is a central topic in this approach and is introduced at an early stage. Part 2 starts with the free Schroedinger equation and computes the free resolvent and time evolution. Position, momentum, and angular momentum are discussed via algebraic methods. Various mathematical methods are developed, which are then used to compute the spectrum of the hydrogen atom. Further topics include the nondegeneracy of the ground state, spectra of atoms, and scattering theory.
This book serves as a self-contained introduction to spectral theory of unbounded operators in Hilbert space with full proofs and minimal prerequisites: Only a solid knowledge of advanced calculus and a one-semester introduction to complex analysis are required. In particular, no functional analysis and no Lebesgue integration theory are assumed. It develops the mathematical tools necessary to prove some key results in nonrelativistic quantum mechanics.
Mathematical Methods in Quantum Mechanics is intended for beginning graduate students in both mathematics and physics and provides a solid foundation for reading more advanced books and current research literature.
This new edition has additions and improvements throughout the book to make the presentation more student friendly.
目次
Preface
Part 0. Preliminaries
Chapter 0. A first look at Banach and Hilbert spaces
Part 1. Mathematical foundations of quantum mechanics
Chapter 1. Hilbert spaces
Chapter 2. Self-adjointness and spectrum
Chapter 3. The spectral theorem
Chapter 4. Applications of the spectral theorem
Chapter 5. Quantum dynamics
Chapter 6. Perturbation theory for self-adjoint operators
Part 2. Schroedinger operators
Chapter 7. The free Schroedinger operator
Chapter 8. Algebraic methods
Chapter 9. One-dimensional Schroedinger operators
Chapter 10. One-particle Schroedinger operators
Chapter 11. Atomic Schroedinger operators
Chapter 12. Scattering theory
Part 3. Appendix
Chapter 13. Almost everything about Lebesgue integration
Bibliographical notes
Bibliography
Glossary of notation
Index
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