Mathematical Methods in Quantum Mechanics: With Applications to Schrödinger Operators
About this Title
Gerald Teschl, University of Vienna, Vienna, Austria
Publication: Graduate Studies in Mathematics
Publication Year 2009: Volume 99
ISBNs: 978-0-8218-4660-5 (print); 978-1-4704-1838-0 (online)
MathSciNet review: MR2499016
MSC: Primary 81-01; Secondary 47N50, 81Qxx
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 Schrödinger 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 Schrödinger 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. It is well suited for self-study and includes numerous exercises (many with hints).
Graduate students and research mathematicians interested in mathematical physics and quantum mechanics.
Table of Contents
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. Schrödinger operators
- Chapter 7. The free Schrödinger operator
- Chapter 8. Algebraic methods
- Chapter 9. One dimensional Schrödinger operators
- Chapter 10. One-particle Schrödinger operators
- Chapter 11. Atomic Schrödinger operators
- Chapter 12. Scattering theory
Part 3. Appendix
- Appendix A. Almost everything about Lebesgue integration