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## Über dieses Buch

The first fifteen chapters of these lectures (omitting four to six chapters each year) cover a one term course taken by a mixed group of senior undergraduate and junior graduate students specializing either in mathematics or physics. Typically, the mathematics students have some background in advanced anal­ ysis, while the physics students have had introductory quantum mechanics. To satisfy such a disparate audience, we decided to select material which is interesting from the viewpoint of modern theoretical physics, and which illustrates an interplay of ideas from various fields of mathematics such as operator theory, probability, differential equations, and differential geometry. Given our time constraint, we have often pursued mathematical content at the expense of rigor. However, wherever we have sacrificed the latter, we have tried to explain whether the result is an established fact, or, mathematically speaking, a conjecture, and in the former case, how a given argument can be made rigorous. The present book retains these features.

## Inhaltsverzeichnis

### 1. Physical Background

Abstract
In this introductory chapter, we present a very brief overview of the basic structure of quantum mechanics, and touch on the physical motivation for the theory. A detailed mathematical discussion of quantum mechanics is the focus of the subsequent chapters.
Stephen J. Gustafson, Israel Michael Sigal

### 2. Dynamics

Abstract
We recall that the evolution of the wave function, ψ, for a particle in a potential, V, is determined by the Schrödinger equation:
$$i\hbar \frac{{\partial \psi }} {{\partial t}} = H\psi$$
(2.1)
where
$$H = - \frac{{\hbar ^2 }} {{2m}}\Delta + V$$
is the appropriate Schrödinger operator. We supplement equation (2.1) with the initial condition
$$\psi |_{t = 0} = \psi _0$$
(2.2)
where ψ0L2. The problem of solving (2.1)– (2.2) is called an initial value problem or a Cauchy problem.
Stephen J. Gustafson, Israel Michael Sigal

### 3. Observables

Abstract
Observables are the quantities that can be experimentally measured in a given physical framework. In this chapter, we discuss the observables of quantum mechanics, as well as the notion of “quantizing” a classical theory.
Stephen J. Gustafson, Israel Michael Sigal

### 4. The Uncertainty Principle

Abstract
One of the fundamental implications of quantum theory is the uncertainty principle — that is, the fact that certain pairs of physical quantities cannot be measured simultaneously with arbitrary accuracy. In this chapter, we establish precise mathematical statements of the uncertainty principle.
Stephen J. Gustafson, Israel Michael Sigal

### 5. Spectral Theory

Abstract
Our next task is to classify the orbits (i.e. solutions) of the Schrödinger equation
$$i\hbar \frac{{\partial \psi }} {{\partial t}} = H\psi$$
with given initial condition
$$\psi |_{t = 0} = \psi _0$$
according to their behaviour in space-time. Naturally, we want to distinguish between states which are localized for all time, and those whose essential support moves off to infinity. Such a classification is made with the help of a very important notion — the spectrum of an operator. We begin by describing the general theory, and then we proceed to applications.
Stephen J. Gustafson, Israel Michael Sigal

### 6. Scattering States

Abstract
In this chapter we study scattering states in a little more det ail. As we saw in Section 5.4, scattering states are solutions of the time-dependent Schrödinger equation
$$i\hbar \frac{{\partial \psi }} {{\partial t}} = H\psi$$
(6.1)
with initial condit ion orthogonal to all eigenfunctions of H:
$$\psi |_{t = 0} = \psi _0 \in \mathcal{H}\frac{1} {b}$$
(6.2)
where H b := span eigenfunctions of H is the subspace of bound states of H.
Stephen J. Gustafson, Israel Michael Sigal

### 7. Special Cases

Abstract
In this chapter we will solve the Schrödinger eigenvalue equation in a few special cases (i.e., for a few particular potentials) which not only illustrate some of the general arguments presented above, but in fact form a basis for our intuition about quantum behaviour.
Stephen J. Gustafson, Israel Michael Sigal

### 8. Many-particle Systems

Abstract
In this chapter, we extend the concepts developed in the previous chapters to many-particle systems. Specifically, we consider a physical system consisting of N particles of masses m1,…,m N which interact pairwise via the potentials V ij (xi-x j ), where x j is the position of the j-th particle. Examples of such systems include atoms or molecules — i.e., systems consisting of electrons and nuclei interacting via Coulomb forces.
Stephen J. Gustafson, Israel Michael Sigal

### 9. Density Matrices

Abstract
In this chapter we extend the basic notions of quantum mechanics to the situation of open systems and positive temperatures. This topic is called quantum statistical mechanics. Since the notion of temperature pertains to systems with an infinite number of degrees of freedom — infinite systems — which are in states of thermal equilibrium, we should explain what the first sentence really means. By a quantum system at a positive temperature, we mean a quantum system coupled to an infinite system which is initially in a state of thermal equilibrium at a given temperature. In what follows, we develop a mathematical framework with which to handle such situations. We will see that we will have to replace the notion of wave function (i.e. a square integrable function of the particle coordinates — an element of L2(ℝ3)) with the notion of density matrix, a positive, trace class operator on the state space L2(ℝ3). The notions of ground state and ground state energy go over to the notions of Gibbs state and (Helmholtz) free energy. We consider some element ary examples and formulate some key problems. The notions of trace and trace class operators will be defined in the “mathematical supplement” at the end of this chapter.
Stephen J. Gustafson, Israel Michael Sigal

### 10. The Feynman Path Integral

Abstract
In this chapter, we derive a convenient representation for the integral kernel of the Schrödinger evolution operator, eitH/ħ. This representation, the “Feynman path integral”, will provide us with a heuristic but effective tool for investigating the connection between quantum and classical mechanics. This investigation will be undertaken in the next section.
Stephen J. Gustafson, Israel Michael Sigal

### 11. Quasi-classical Analysis

Abstract
In this chapter we investigate the connection between quantum and classical mechanics. More precisely, taking advantage of the fact that the Planck constant provides us with a small parameter, we compute some key quantum quantities — such as quantum energy levels — in terms of relevant classical quantities. This is called quasi-classical (or semi-classical) analysis. To do this, we use the Feynman path integral representation of the evolution operator (propagator) eiHt/ħ. This representation provides a non-rigorous but highly effective tool, as the path integral is expressed directly in terms of the key classical quantity — the classical action.
Stephen J. Gustafson, Israel Michael Sigal

### 12. Mathematical Supplement: the Calculus of Variations

Abstract
The calculus of variations, an extensive mathematical theory in its own right, plays a fundamental role throughout physics. This chapter contains an overview of some of the basic aspects of the variational calculus. This material in used in Chapters 11 and 13, in conjunction with the path integral introduced in the previous chapter, to obtain useful quantitative results about quantum systems.
Stephen J. Gustafson, Israel Michael Sigal

### 13. Resonances

Abstract
The notion of a resonance is a key notion in quantum physics. It refers to metastable states — i.e., to states which behave like stationary states for long time intervals, but which eventually break up. In other words, these are states of the continuous spectrum (i.e. scattering states), which for a long time behave as if they were bound states. In fact, the notion of a bound state is an idealization: most of the states which are (taken to be) bound states in certain models, turn out to be resonance states in a more realistic description of the system.
Stephen J. Gustafson, Israel Michael Sigal

### 14. Introduction to Quantum Field Theory

Abstract
The goal of quantum field theory (which we will often abbreviate as QFT) is to describ e element ary particles and their interactions. QFT has deep connections with a variety of disciplines, including statistical mechani cs and condensed matter physics, probability theory (i.e. sto chastic PDEs), and nonlinear PDEs.
Stephen J. Gustafson, Israel Michael Sigal

### 15. Quantum Electrodynamics of Non-relativistic Particles: the Theory of Radiation

Abstract
We conclude this book by outlining the theory of the phenomenon of emission and absorption of electromagnetic radiation by systems of non-relativistic particles such as atoms and molecules. Attempts to understand this phenomenon led, at the beginning of the twentieth century, to the birth of quantum physics. Only by treating the matter and the radiation as quantum mechanical can one give a consistent description of the phenomenon in question. Thus, our starting point should be a Schrodinger operator describing quantum particles interacting amongst themselves, and with quantum radiation. In mathematical terms, the question we address is how the bound state structure of the particle system is modified by the interaction with radiation. One expects that the ground state of the particle system survives, while the excited states turn into resonances. The real parts of the resonance eigenvalues — the resonance energies — produce the Lamb shift, first experimentally measured by Lamb and Retherford (Lamb was awarded the Nobel prize for this discovery). The imaginary parts of the resonance eigenvalues — the decay probabilities — are given by the Fermi Golden Rule (see, eg, [HuS]). This picture was established rigorously, under somewhat restrictive conditions, in [BFS1]- [BFS4], whose results we describe here. The method in these papers also provides an effective computational technique to any order in the electron charge, something the conventional perturbation theory fails to do.
Stephen J. Gustafson, Israel Michael Sigal

### 16. Supplement: Renormalization Group

Abstract
In this chapter we describe an operator version of the renormalization group method, due to [BFS1]-[BFS4]_(see also [GaW, Weg, KM]). We demonstrate how this method works by applying it to the problem of radiation described in Chapter 15. In particular, we continue our study of the Hamiltonian H (ε) which describes quantum particles coupled to the quantized EM field. We outline a proof of part (i) of Theorem 15.2, which states the existence of the ground state of the operator H(ε) for sufficiently small |ε|. The problems of instability of the excited states and existence of the resonances — statements (ii) and (iii) of Theorem 15.2 — can be treated in the same way.
Stephen J. Gustafson, Israel Michael Sigal