Mathematical Methods in Quantum Mechanics (2nd ed.)

Description

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 Schrodinger 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 Schrodinger 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.

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MR3243083 This book presents an introduction to the theory of operators onHilbert spaces and the applications of this theory to the study ofSchrödinger operators. Beginning graduatestudents are the intended readership, and the mathematical tools are developedin a detailed andself-contained way, keeping the prerequisites at a minimallevel.
The book is divided into two parts, the first one covering the abstracttheory of unbounded operators in Hilbert spaces and the second oneconcentrating on the spectral theory of Schrödinger operators. Thecontents of the first edition of the book were reviewed in detail[G. Teschl, Mathematical methods in quantum mechanics, Grad. Stud.Math., 99, Amer. Math. Soc., Providence, RI, 2009; MR2499016]. In the second edition the book has grown by about 50pages. In particular, the chapter on the spectral theorem for unboundedoperators has been reworked, the section on inverse spectral theory ofSchrödinger operators has been improved and the appendix on measuretheory has been expanded.
The author admirably selects the main results from the theory andpresents them in a clear and detailed way. The material correspondsapproximately to a two-semester course, but because of the numerousexercises, the book is also suitable for self-study. It provides anice introduction to this exciting field of mathematics. Reviewed by Rupert L. Frank