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Product Code:  GSM/157 
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Electronic ISBN:  9781470418885 
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Book DetailsGraduate Studies in MathematicsVolume: 157; 2014; 356 ppMSC: Primary 81; 46; 34; 47;
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 selfcontained, 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 selfcontained 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 onesemester 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.
Solutions are available electronically for instructors only. Please send email to textbooks@ams.org for more information. Updates and corrections are available for the book; please see errata.The book is written in a very clear and compact style. It is well suited for selfstudy and includes numerous exercises (many with hints).
—Zentralblatt MATH
The author presents this material in a very clear and detailed way and supplements it by numerous exercises. This makes the book a nice introduction to this exciting field of mathematics.
—Mathematical Reviews
ReadershipGraduate students and research mathematicians interested in spectral theory and quantum mechanics, with an emphasis on Schrödinger operators.

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. Selfadjointness and spectrum

Chapter 3. The spectral theorem

Chapter 4. Applications of the spectral theorem

Chapter 5. Quantum dynamics

Chapter 6. Perturbation theory for selfadjoint operators

Part 2. Schrödinger operators

Chapter 7. The free Schrödinger operator

Chapter 8. Algebraic methods

Chapter 9. Onedimensional Schrödinger operators

Chapter 10. Oneparticle Schrödinger operators

Chapter 11. Atomic Schrödinger operators

Chapter 12. Scattering theory

Part 3. Appendix

Appendix A. Almost everything about Lebesgue integration


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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 selfcontained, 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 selfcontained 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 onesemester 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.
Solutions are available electronically for instructors only. Please send email to textbooks@ams.org for more information. Updates and corrections are available for the book; please see errata.
The book is written in a very clear and compact style. It is well suited for selfstudy and includes numerous exercises (many with hints).
—Zentralblatt MATH
The author presents this material in a very clear and detailed way and supplements it by numerous exercises. This makes the book a nice introduction to this exciting field of mathematics.
—Mathematical Reviews
Graduate students and research mathematicians interested in spectral theory and quantum mechanics, with an emphasis on Schrödinger operators.

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. Selfadjointness and spectrum

Chapter 3. The spectral theorem

Chapter 4. Applications of the spectral theorem

Chapter 5. Quantum dynamics

Chapter 6. Perturbation theory for selfadjoint operators

Part 2. Schrödinger operators

Chapter 7. The free Schrödinger operator

Chapter 8. Algebraic methods

Chapter 9. Onedimensional Schrödinger operators

Chapter 10. Oneparticle Schrödinger operators

Chapter 11. Atomic Schrödinger operators

Chapter 12. Scattering theory

Part 3. Appendix

Appendix A. Almost everything about Lebesgue integration