Spectral Theory and Quantum Mechanics

verfasst von: Mathieu Lewin

Verlag: Springer International Publishing

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

Inhaltsverzeichnis

Über dieses Buch

This textbook presents the spectral theory of self-adjoint operators on Hilbert space and its applications in quantum mechanics. Based on a course taught by the author in Paris, the book not only covers the mathematical theory but also provides its physical interpretation, offering an accessible introduction to quantum mechanics for students with a background in mathematics. The presentation incorporates numerous physical examples to illustrate the abstract theory. The final two chapters present recent findings on Schrödinger’s equation for systems of particles.

While primarily designed for graduate courses, the book can also serve as a valuable introduction to the subject for more advanced readers. It requires no prior knowledge of physics, assuming only a graduate-level understanding of mathematical analysis from the reader.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction to Quantum Mechanics: The Hydrogen Atom

Abstract

This chapter is an introduction to the mathematical formalism of quantum mechanics. It presents the modeling of a quantum particle such as the electron in the hydrogen atom. The approach is purely variational and the role played by spectral theory is only hinted at.

Mathieu Lewin

Chapter 2. Self-adjointness

Abstract

We introduce the concept of self-adjoint operators in infinite dimension. Several examples are presented, including the momentum and the Laplacian on an interval of the real line, or on the whole space in any dimension.

Mathieu Lewin

Chapter 3. Self-adjointness Criteria: Rellich, Kato and Friedrichs

Abstract

We first present a result of Rellich and Kato, that provides the self-adjointness of operators in the form \(A+B\) on the same domain as that of A, under simple assumptions on the perturbation B. This is followed by a discussion of Friedrich’s theory of quadratic forms, that allows one to construct self-adjoint extensions that are natural from a physical point of view.

Mathieu Lewin

Chapter 4. Spectral Theorem and Functional Calculus

Abstract

This chapter contains the spectral theorem (diagonalization of any self-adjoint operator) and the construction of functional calculus. Numerous consequences are discussed: spectral projectors, powers, Stone’s theorem, Schrödinger’s equation, heat equation, wave equation, etc.

Mathieu Lewin

Chapter 5. Spectrum of Self-adjoint Operators

Abstract

This chapter contains multiple tools for studying the spectrum of a self-adjoint operator. First, we discuss how the spectrum is modified when a small perturbation is added to a given operator. Next, we define the different types of spectrum (point, continuous, essential, discrete) and present criteria due to Weyl in order to identify them. Compact and compact-resolvent operators are treated as examples. The Courant-Fischer formula and the Lieb-Thirring inequality are then used to quantify the number and size of eigenvalues below the essential spectrum. The chapter concludes with a short introduction to semi-classical analysis.

Mathieu Lewin

Chapter 6. N-particle Systems, Atoms, Molecules

Abstract

We discuss the modeling of a system containing several quantum particles interacting with each other, such as the electrons in a molecule or an atom. Recent results are described for the case of large atoms.

Mathieu Lewin

Chapter 7. Periodic Schrödinger Operators, Electronic Properties of Materials

Abstract

We study Schrödinger operators comprising a periodic potential. Bloch-Floquet theory is used to explain differences in the electrical behavior of solids, depending on the shape of the spectrum.

Mathieu Lewin

Backmatter

Titel: Spectral Theory and Quantum Mechanics
verfasst von: Mathieu Lewin
Verlag: Springer International Publishing
Electronic ISBN: 978-3-031-66878-4
Print ISBN: 978-3-031-66877-7
DOI: https://doi.org/10.1007/978-3-031-66878-4