Quantum theory for mathematicians

Preface

Contents

1 The Experimental Origins of Quantum Mechanics

1.1 Is Light a Wave or a Particle?
- 1.1.1 Newton Versus Huygens
- 1.1.2 The Ascendance of the Wave Theory of Light
- 1.1.3 Blackbody Radiation
- 1.1.4 The Photoelectric Effect
- 1.1.5 The Double-Slit Experiment, Revisited
1.2 Is an Electron a Wave or a Particle?
- 1.2.1 The Spectrum of Hydrogen
- 1.2.2 The Bohr–de Broglie Model of the Hydrogen Atom
- 1.2.3 Electron Diffraction
- 1.2.4 The Double-Slit Experiment with Electrons
1.3 Schrödinger and Heisenberg
1.4 A Matter of Interpretation
1.5 Exercises

2 A First Approach to Classical Mechanics

2.1 Motion in R1
- 2.1.1 Newton's law
- 2.1.2 Conservation of Energy
- 2.1.3 Systems with Damping
2.2 Motion in Rn
2.3 Systems of Particles
- 2.3.1 Conservation of Energy
- 2.3.2 Conservation of Momentum
- 2.3.3 Center of Mass
2.4 Angular Momentum
2.5 Poisson Brackets and Hamiltonian Mechanics
2.6 The Kepler Problem and the Runge–Lenz Vector
- 2.6.1 The Kepler Problem
- 2.6.2 Conservation of the Runge–Lenz Vector
- 2.6.3 Ellipses, Hyperbolas, and Parabolas
- 2.6.4 Special Properties of the Kepler Problem
2.7 Exercises

3 A First Approach to Quantum Mechanics

3.1 Waves, Particles, and Probabilities
3.2 A Few Words About Operators and Their Adjoints
3.3 Position and the Position Operator
3.4 Momentum and the Momentum Operator
3.5 The Position and Momentum Operators
3.6 Axioms of Quantum Mechanics: Operatorsand Measurements
3.7 Time-Evolution in Quantum Theory
- 3.7.1 The Schrödinger Equation
- 3.7.2 Solving the Schrödinger Equation by Exponentiation
- 3.7.3 Eigenvectors and the Time-Independent Schrödinger Equation
- 3.7.4 The Schrödinger Equation in R1
- 3.7.5 Time-Evolution of the Expected Position and Expected Momentum
3.8 The Heisenberg Picture
3.9 Example: A Particle in a Box
3.10 Quantum Mechanics for a Particle in Rn
3.11 Systems of Multiple Particles
3.12 Physics Notation
3.13 Exercises

4 The Free Schrödinger Equation

4.1 Solution by Means of the Fourier Transform
4.2 Solution as a Convolution
4.3 Propagation of the Wave Packet: First Approach
4.4 Propagation of the Wave Packet: Second Approach
4.5 Spread of the Wave Packet
4.6 Exercises

5 A Particle in a Square Well

5.1 The Time-Independent Schrödinger Equation
5.2 Domain Questions and the Matching Conditions
5.3 Finding Square-Integrable Solutions
5.4 Tunneling and the Classically Forbidden Region
5.5 Discrete and Continuous Spectrum
5.6 Exercises

6 Perspectives on the Spectral Theorem

6.1 The Difficulties with the Infinite-Dimensional Case
6.2 The Goals of Spectral Theory
6.3 A Guide to Reading
6.4 The Position Operator
6.5 Multiplication Operators
6.6 The Momentum Operator

7 The Spectral Theorem for Bounded Self-AdjointOperators: Statements

7.1 Elementary Properties of Bounded Operators
7.2 Spectral Theorem for Bounded Self-AdjointOperators, I
- 7.2.1 Spectral Subspaces
- 7.2.2 Projection-Valued Measures
- 7.2.3 The Spectral Theorem
7.3 Spectral Theorem for Bounded Self-AdjointOperators, II
7.4 Exercises

8 The Spectral Theorem for Bounded Self-AdjointOperators: Proofs

8.1 Proof of the Spectral Theorem, First Version
- 8.1.1 Stage 1: The Continuous Functional Calculus
- 8.1.2 Stage 2: An Operator-Valued Riesz Representation Theorem
8.2 Proof of the Spectral Theorem, Second Version
8.3 Exercises

9 Unbounded Self-Adjoint Operators

9.1 Introduction
9.2 Adjoint and Closure of an Unbounded Operator
9.3 Elementary Properties of Adjoints and ClosedOperators
9.4 The Spectrum of an Unbounded Operator
9.5 Conditions for Self-Adjointness and EssentialSelf-Adjointness
9.6 A Counterexample
9.7 An Example
9.8 The Basic Operators of Quantum Mechanics
9.9 Sums of Self-Adjoint Operators
9.10 Another Counterexample
9.11 Exercises

10 The Spectral Theorem for Unbounded Self-AdjointOperators

10.1 Statements of the Spectral Theorem
10.2 Stone's Theorem and One-Parameter Unitary Groups
10.3 The Spectral Theorem for Bounded NormalOperators
10.4 Proof of the Spectral Theorem for UnboundedSelf-Adjoint Operators
10.5 Exercises

11 The Harmonic Oscillator

11.1 The Role of the Harmonic Oscillator
11.2 The Algebraic Approach
11.3 The Analytic Approach
11.4 Domain Conditions and Completeness
11.5 Exercises

12 The Uncertainty Principle

12.1 Uncertainty Principle, First Version
12.2 A Counterexample
12.3 Uncertainty Principle, Second Version
12.4 Minimum Uncertainty States
12.5 Exercises

13 Quantization Schemes for Euclidean Space

13.1 Ordering Ambiguities
13.2 Some Common Quantization Schemes
13.3 The Weyl Quantization for R2n
- 13.3.1 Heuristics
- 13.3.2 The L2 Theory
- 13.3.3 The Composition Formula
- 13.3.4 Commutation Relations
13.4 The ``No Go'' Theorem of Groenewold
13.5 Exercises

14 The Stone–von Neumann Theorem

14.1 A Heuristic Argument
14.2 The Exponentiated Commutation Relations
14.3 The Theorem
14.4 The Segal–Bargmann Space
- 14.4.1 The Raising and Lowering Operators
- 14.4.2 The Exponentiated Commutation Relations
- 14.4.3 The Reproducing Kernel
- 14.4.4 The Segal–Bargmann Transform
14.5 Exercises

15 The WKB Approximation

15.1 Introduction
15.2 The Old Quantum Theory and the Bohr–SommerfeldCondition
15.3 Classical and Semiclassical Approximations
15.4 The WKB Approximation Away from the TurningPoints
- 15.4.1 The Classically Allowed Region
- 15.4.2 The Classically Forbidden Region
15.5 The Airy Function and the Connection Formulas
15.6 A Rigorous Error Estimate
- 15.6.1 Preliminaries
- 15.6.2 The Regions Near the Turning Points
- 15.6.3 The Classically Allowed and Classically Forbidden Regions
- 15.6.4 The Transition Regions
- 15.6.5 Proof of the Main Theorem
15.7 Other Approaches
15.8 Exercises

16 Lie Groups, Lie Algebras, and Representations

16.1 Summary
16.2 Matrix Lie Groups
16.3 Lie Algebras
16.4 The Matrix Exponential
16.5 The Lie Algebra of a Matrix Lie Group
16.6 Relationships Between Lie Groups and Lie Algebras
16.7 Finite-Dimensional Representations of Lie Groupsand Lie Algebras
- 16.7.1 Finite-Dimensional Representations
- 16.7.2 Unitary Representations
- 16.7.3 Projective Unitary Representations
16.8 New Representations from Old
16.9 Infinite-Dimensional Unitary Representations
- 16.9.1 Ordinary Unitary Representations
- 16.9.2 Projective Unitary Representations
16.10 Exercises

17 Angular Momentum and Spin

17.1 The Role of Angular Momentumin Quantum Mechanics
17.2 The Angular Momentum Operators in R3
17.3 Angular Momentum from the Lie Algebra Pointof View
17.4 The Irreducible Representations of so(3)
17.5 The Irreducible Representations of SO(3)
17.6 Realizing the Representations Inside L2(S2)
17.7 Realizing the Representations Inside L2(R3)
17.8 Spin
17.9 Tensor Products of Representations: “Addition ofAngular Momentum”
17.10 Vectors and Vector Operators
17.11 Exercises

18 Radial Potentials and the Hydrogen Atom

18.1 Radial Potentials
18.2 The Hydrogen Atom: Preliminaries
18.3 The Bound States of the Hydrogen Atom
18.4 The Runge–Lenz Vector in the Quantum KeplerProblem
- 18.4.1 Some Notation
- 18.4.2 The Classical Runge–Lenz Vector, Revisited
- 18.4.3 The Quantum Runge–Lenz Vector
- 18.4.4 Representations of so(4)
18.5 The Role of Spin
18.6 Runge–Lenz Calculations
18.7 Exercises

19 Systems and Subsystems, Multiple Particles

19.1 Introduction
19.2 Trace-Class and Hilbert–Schmidt Operators
19.3 Density Matrices: The General Notionof the State of a Quantum System
19.4 Modified Axioms for Quantum Mechanics
19.5 Composite Systems and the Tensor Product
19.6 Multiple Particles: Bosons and Fermions
19.7 “Statistics” and the Pauli Exclusion Principle
19.8 Exercises

20 The Path Integral Formulation of Quantum Mechanics

20.1 Trotter Product Formula
20.2 Formal Derivation of the Feynman Path Integral
20.3 The Imaginary-Time Calculation
20.4 The Wiener Measure
20.5 The Feynman–Kac Formula
20.6 Path Integrals in Quantum Field Theory
20.7 Exercises

21 Hamiltonian Mechanics on Manifolds

21.1 Calculus on Manifolds
- 21.1.1 Tangent Spaces, Vector Fields, and Flows
- 21.1.2 Differential Forms
21.2 Mechanics on Symplectic Manifolds
- 21.2.1 Symplectic Manifolds
- 21.2.2 Poisson Brackets and Hamiltonian Vector Fields
- 21.2.3 Hamiltonian Flows and Conserved Quantities
- 21.2.4 The Liouville Form
21.3 Exercises

22 Geometric Quantization on Euclidean Space

22.1 Introduction
22.2 Prequantization
22.3 Problems with Prequantization
22.4 Quantization
22.5 Quantization of Observables
22.6 Exercises

23 Geometric Quantization on Manifolds

23.1 Introduction
23.2 Line Bundles and Connections
23.3 Prequantization
23.4 Polarizations
23.5 Quantization Without Half-Forms
- 23.5.1 The General Case
- 23.5.2 The Real Case
- 23.5.3 The Complex Case
23.6 Quantization with Half-Forms: The Real Case
- 23.6.1 The Space of Leaves
- 23.6.2 The Canonical Bundle
- 23.6.3 Square Roots of the Canonical Bundle
- 23.6.4 The Half-Form Hilbert Space
- 23.6.5 Quantization of Observables
23.7 Quantization with Half-Forms: The Complex Case
23.8 Pairing Maps
23.9 Exercises

Appendix A Review of Basic Material

A.1 Tensor Products of Vector Spaces
A.2 Measure Theory
A.3 Elementary Functional Analysis
- A.3.1 The Stone–Weierstrass Theorem
- A.3.2 The Fourier Transform
- A.3.3 Distributions
- A.3.4 Banach Spaces
A.4 Hilbert Spaces and Operators on Them
- A.4.1 Inner Product Spaces and Hilbert Spaces
- A.4.2 Orthogonality
- A.4.3 The Riesz Theorem and Adjoints
- A.4.4 Quadratic Forms
- A.4.5 Tensor Products of Hilbert Spaces

References

Index