Mathematical Concepts of Quantum Mechanics

The book gives a streamlined introduction to quantum mechanics while describing the basic mathematical structures underpinning this discipline.

Starting with an overview of key physical experiments illustrating the origin of the physical foundations, the book proceeds with a description of the basic notions of quantum mechanics and their mathematical content.

It then makes its way to topics of current interest, specifically those in which mathematics plays an important role. The more advanced topics presented include: many-body systems, modern perturbation theory, path integrals, the theory of resonances, adiabatic theory, geometrical phases, Aharonov-Bohm effect, density functional theory, open systems, the theory of radiation (non-relativistic quantum electrodynamics), and the renormalization group.

With different selections of chapters, the book can serve as a text for an introductory, intermediate, or advanced course in quantum mechanics. Some of the sections could be used for introductions to geometrical methods in Quantum Mechanics, to quantum information theory and to quantum electrodynamics and quantum field theory.

Table of Contents

1 Physical Background

2 Dynamics

3 Observables

4 Quantization

5 Uncertainty Principle and Stability of Atoms and Molecules

6 Spectrum and Dynamics

7 Special Cases and Extensions

8 Bound States and Variational Principle

9 Scattering States

10 Existence of Atoms and Molecules

11 Perturbation Theory: Feshbach-Schur Method

12 Born-Oppenheimer Approximation and Adiabatic Dynamics

13 General Theory of Many-particle Systems

14 Self-consistent Approximations

15 The Feynman Path Integral

16 Semi-classical Analysis

17 Resonances

18 Quantum Statistics

19 Open Quantum Systems

20 The Second Quantization

21 Quantum Electro-Magnetic Field – Photons

22 Standard Model of Non-relativistic Matter and Radiation

23 Theory of Radiation

24 Renormalization Group

25 Mathematical Supplement: Elements of Operator Theory

26 Mathematical Supplement: The Calculus of Variations

27 Comments on Literature, and Further Reading

About the Authors

Stephen J. Gustafson is Professor of Mathematics at the University of British Columbia. His research centres on various partial differential equations arising in physics.

Israel Michael Sigal is the Norman Stuart Robertson chair in Applied Mathematics at the University of Toronto. He works in several areas of mathematical physics and applied mathematics.