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Although ideas from quantum physics play an important role in many parts of modern mathematics, there are few books about quantum mechanics aimed at mathematicians. This book introduces the main ideas of quantum mechanics in language familiar to mathematicians. Readers with little prior exposure to physics will enjoy the book's conversational tone as they delve into such topics as the Hilbert space approach to quantum theory; the Schrödinger equation in one space dimension; the Spectral Theorem for bounded and unbounded self-adjoint operators; the Stone–von Neumann Theorem; the Wentzel–Kramers–Brillouin approximation; the role of Lie groups and Lie algebras in quantum mechanics; and the path-integral approach to quantum mechanics.

The numerous exercises at the end of each chapter make the book suitable for both graduate courses and independent study. Most of the text is accessible to graduate students in mathematics who have had a first course in real analysis, covering the basics of L2 spaces and Hilbert spaces. The final chapters introduce readers who are familiar with the theory of manifolds to more advanced topics, including geometric quantization.

Inhaltsverzeichnis

Frontmatter

1. The Experimental Origins of Quantum Mechanics

Abstract
Quantum mechanics, with its controversial probabilistic nature and curious blending of waves and particles, is a very strange theory. It was not invented because anyone thought this is the way the world should behave, but because various experiments showed that this is the way the world does behave, like it or not. Craig Hogan, director of the Fermilab Particle Astrophysics Center, put it this way:
Brian C. Hall

2. A First Approach to Classical Mechanics

Abstract
We begin by considering the motion of a single particle in \({\mathbb{R}}^{1},\) which may be thought of as a particle sliding along a wire, or a particle with motion that just happens to lie in a line. We let x(t) denote the particle’s position as a function of time. The particle’s velocity is then
$$\displaystyle{v(t) :=\dot{ x}(t),}$$
where we use a dot over a symbol to denote the derivative of that quantity with respect to the time t.
Brian C. Hall

3. A First Approach to Quantum Mechanics

Abstract
In this chapter, we try to understand the main ideas of quantum mechanics. In quantum mechanics, the outcome of a measurement cannot—even in principle—be predicted beforehand; only the probabilities for the outcome of the measurement can be predicted.
Brian C. Hall

4. The Free Schrödinger Equation

Abstract
In this chapter, we consider various methods of solving the free Schrödinger equation in one space dimension. Here “free”means that there is no force acting on the particle, so that we may take the potential V to be identically zero
Brian C. Hall

5. A Particle in a Square Well

Abstract
It is difficult to solve the time-dependent Schrödinger equation explicitly, even in relatively simple cases. (Even for the free Schrödinger equation, we made do in Chap.4 with solutions that are either approximate or that involve an integral that is not explicitly evaluated.) Usually, then, one analyzes the time-independent Schrödinger equation (the eigenvector equation for \(\hat{H}\)) and then attempts to infer something about the time-dependent problem from the results. There are a number of problems, including the harmonic oscillator and the hydrogen atom, in which the time-independent Schrödinger equation can be solved explicitly.
Brian C. Hall

6. Perspectives on the Spectral Theorem

Abstract
Suppose A is a self-adjoint n × n matrix, meaning that \(A_{kj} = \overline{A_{jk}}\) for all 1≤ j, kn. Then a standard result in linear algebra asserts that there exist an orthonormal basis \(\{\mathbf{v}_{j}\}_{j=1}^{n}\) for \({\mathbb{C}}^{n}\) and real numbers λ 1,,λ n such that \(A\mathbf{v}_{j} =\lambda _{j}\mathbf{v}_{j}\). (See Theorem 18 in Chap. 8 of [24] and Exercise 4 in Chap. 7)
Brian C. Hall

7. The Spectral Theorem for Bounded Self-Adjoint Operators: Statements

Abstract
In the present chapter, we will consider the spectral theorem for bounded self-adjoint operators, leaving a discussion of unbounded operators to Chaps. 9 and 10. The proofs of the main theorems (two different versions of the spectral theorem) are moderately long and are deferred to Chap. 8.
Brian C. Hall

8. The Spectral Theorem for Bounded Self-Adjoint Operators: Proofs

Abstract
In this chapter we give proofs of all versions of the spectral theorem stated in the previous chapter.
Brian C. Hall

9. Unbounded Self-Adjoint Operators

Abstract
Recall that most of the operators of quantum mechanics, including those representing position, momentum, and energy, are not defined on the entirety of the relevant Hilbert space, but only on a dense subspace thereof.
Brian C. Hall

10. The Spectral Theorem for Unbounded Self-Adjoint Operators

Abstract
This chapter gives statements and proofs of the spectral theorem for unbounded self-adjoint operators, in the same forms as in the bounded case, in terms of projection-valued measures, in terms of direct integrals, and in terms of multiplication operators.
Brian C. Hall

11. The Harmonic Oscillator

Abstract
The harmonic oscillator is an important model for various reasons. In solid-state physics, for example, a crystal is modeled as a large number of coupled harmonic oscillators. Using the notion of “normal modes,”this model is then transformed into independent one-dimensional harmonic oscillators with different frequencies. In the quantum mechanical setting, the excitations of the different normal modes are called phonons.
Brian C. Hall

12. The Uncertainty Principle

Abstract
In this chapter, we will continue our investigation of the consequences of the commutation relations among the position and momentum operators.
Brian C. Hall

13. Quantization Schemes for Euclidean Space

Abstract
One of the axioms of quantum mechanics states, “To each real-valued function f on the classical phase space there is associated a self-adjoint operator \(\hat{f}\) on the quantum Hilbert space.” The attentive reader will note that we have not, up to this point, given a general procedure for constructing \(\hat{f}\) from f.If we call \(\hat{f}\) the quantization of f,then we have only discussed the quantizations of a few very special classical observables, such as position, momentum, and energy.
Brian C. Hall

14. The Stone–von Neumann Theorem

Abstract
The Stone–von Neumann theorem is a uniqueness theorem for operators satisfying the canonical commutation relations. Suppose A and B are two self-adjoint operators on H satisfying \([A,B] = i\hslash I.\) Suppose also that A and B act irreducibly on H,meaning that the only closed subspaces of H invariant under A and B are {0} and H.Then provided that certain technical assumptions hold (the exponentiated commutation relations)
Brian C. Hall

15. The WKB Approximation

Abstract
The WKB method, named for Gregor Wentzel, Hendrik Kramers, and Léon Brillouin, gives an approximation to the eigenfunctions and eigenvalues of the Hamiltonian operator \(\hat{H}\) in one dimension.
Brian C. Hall

16. Lie Groups, Lie Algebras, and Representations

Abstract
An important concept in physics is that of symmetry, whether it be rotational symmetry for many physical systems or Lorentz symmetry in relativistic systems. In many cases, the group of symmetries of a system is a continuous group, that is, a group that is parameterized by one or more real parameters. More precisely, the symmetry group is often a Lie group, that is, a smooth manifold endowed with a group structure in such a way that operations of inversion and group multiplication are smooth. The tangent space at the identity in a Lie group has a natural “bracket” operation that makes the tangent space into a Lie algebra. The Lie algebra of a Lie group encodes many of the properties of the Lie group, and yet the Lie algebra is easier to work with because it is a linear space.
Brian C. Hall

17. Angular Momentum and Spin

Abstract
Classically, angular momentum may be thought of as the Hamiltonian generator of rotations (Proposition 2.30). Angular momentum is a particularly useful concept when a system has rotational symmetry, since in that case the angular momentum is a conserved quantity (Proposition 2.18). Quantum mechanically, angular momentum is still the “generator”of rotations, meaning that it is the infinitesimal generator of a one-parameter group of unitary rotation operators, in the sense of Stone’s theorem (Theorem 10.15).
Brian C. Hall

18. Radial Potentials and the Hydrogen Atom

Abstract
If V is any radial function on \({\mathbb{R}}^{3}\), let \(\hat{H} = -({\hslash }^{2}/(2m))\Delta + V\) be the corresponding Hamiltonian operator, acting on \({L}^{2}({\mathbb{R}}^{3}).\) We will look for solutions to the time-independent Schrödinger equation \(\hat{H}\psi = E\psi\) of the form \(\psi (\mathbf{x}) = p(\mathbf{x})f(\left \vert \mathbf{x}\right\vert ),\) where f is a smooth function on (0,) and p is a harmonic polynomial on \({\mathbb{R}}^{3}\) that is homogeneous of degree l.
Brian C. Hall

19. Systems and Subsystems, Multiple Particles

Abstract
Up to this point, we have considered the state of a quantum system to be described by a unit vector in the corresponding Hilbert space, or more properly, an equivalence class of unit vectors under the equivalence relation \(\psi\)e i θ \(\psi\). We will see in this section that this notion of the state of a quantum system is too limited. We will introduce a more general notion of the state of a system, described by a density matrix. The special case in which the system can be described by a unit vector will be called a pure state.
Brian C. Hall

20. The Path Integral Formulation of Quantum Mechanics

Abstract
We turn now to a topic that is important already for ordinary quantum mechanics and essential in quantum field theory: the so-called path integral. In the setting of ordinary quantum mechanics (of the sort we have been considering in this book), the integrals in question are over spaces of “paths,” that is, maps of some interval [a, b] into \({\mathbb{R}}^{n}.\) In the setting of quantum field theory, the integrals are integrals over spaces of “fields,” that is, maps of some region inside \({\mathbb{R}}^{d}\) into \({\mathbb{R}}^{n}.\) Formal integrals of this sort abound in the physics literature, and it is typically difficult to make rigorous mathematical sense of them—although much effort has been expended in the attempt! In this chapter, we will develop a rigorous integral over spaces of paths by using the Wiener measure, resulting in the Feynman–Kac formula.
Brian C. Hall

21. Hamiltonian Mechanics on Manifolds

Abstract
In this chapter, we generalize the Hamiltonian approach to mechanics (introduced already in the Euclidean case in Sect. 2.5) to general manifolds. The chapter assumes familiarity with the basic notions of smooth manifolds, including tangent and cotangent spaces, vector fields, and differential forms. These notions are reviewed very briefly in Sect. 21.1, mainly in the interest of fixing the notation. See, for example, Chap. 2 of [40] for a concise treatment of manifolds and [29] for a detailed account. Throughout the chapter, we will use the summation convention, that repeated indices are always summed on.
Brian C. Hall

22. Geometric Quantization on Euclidean Space

Abstract
In this chapter, we consider the geometric quantization program in the setting of the symplectic manifold \({\mathbb{R}}^{2n},\) with the canonical 2-form ω = dp j dx j . We begin with the “prequantum” Hilbert space \({L}^{2}({\mathbb{R}}^{2n})\) and define “prequantum” operators Q pre(f). These operators satisfy
$$\displaystyle{Q_{\mathrm{pre}}(\{f,g\}) = \frac{1} {i\hslash }[Q_{\mathrm{pre}}(f),Q_{\mathrm{pre}}(g)]}$$
for all f and g. Nevertheless, there are several undesirable aspects to the prequantization map that make it physically unreasonable to interpret it as “quantization.” To obtain the quantum Hilbert space, we reduce the number of variables from 2n to n. Depending on how we do this reduction, we will obtain either the position Hilbert space, the momentum Hilbert space, or the Segal–Bargmann space. Each of these subspaces is preserved by the prequantized position and momentum operators, and by certain other operators of the form Q pre(f).
Brian C. Hall

23. Geometric Quantization on Manifolds

Abstract
Geometric quantization is a type of quantization, which is a general term for a procedure that associates a quantum system with a given classical system. In practical terms, if one is trying to deduce what sort of quantum system should model a given physical phenomenon, one often begins by observing the classical limit of the system. Electromagnetic radiation, for example, is describable on a macroscopic scale by Maxwell’s equations. On a finer scale, quantum effects (photons) become important. How should one determine the correct quantum theory of electromagnetism? It seems that the only reasonable way to proceed is to “quantize” Maxwell’s equations—and then to compare the resulting quantum system to experiment.
Brian C. Hall

Backmatter

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