Mathematical Methods in Quantum Mechanics: With Applications to Schrodinger Operators

✍ Scribed by Gerald Teschl

• Publisher: American Mathematical Society
• Year: 2014
• Tongue: English
• Leaves: 375
• Series: Graduate Studies in Mathematics
• Edition: 2
• Category: Library

✦ Synopsis

Quantum mechanics and the theory of operators on Hilbert space have been deeply linked since their beginnings in the early twentieth century. States of a quantum system correspond to certain elements of the configuration space and observables correspond to certain operators on the space. This book is a brief, but self-contained, introduction to the mathematical methods of quantum mechanics, with a view towards applications to Schrodinger operators. Part 1 of the book is a concise introduction to the spectral theory of unbounded operators. Only those topics that will be needed for later applications are covered. The spectral theorem is a central topic in this approach and is introduced at an early stage. Part 2 starts with the free Schrodinger equation and computes the free resolvent and time evolution. Position, momentum, and angular momentum are discussed via algebraic methods. Various mathematical methods are developed, which are then used to compute the spectrum of the hydrogen atom. Further topics include the nondegeneracy of the ground state, spectra of atoms, and scattering theory. This book serves as a self-contained introduction to spectral theory of unbounded operators in Hilbert space with full proofs and minimal prerequisites: Only a solid knowledge of advanced calculus and a one-semester introduction to complex analysis are required. In particular, no functional analysis and no Lebesgue integration theory are assumed. It develops the mathematical tools necessary to prove some key results in nonrelativistic quantum mechanics. Mathematical Methods in Quantum Mechanics is intended for beginning graduate students in both mathematics and physics and provides a solid foundation for reading more advanced books and current research literature. This new edition has additions and improvements throughout the book to make the presentation more student friendly.

✦ Table of Contents

Contents

Preface xi

Part 0. Preliminaries
Chapter 0. A first look at Banach and Hilbert spaces 3
0.1. Warm up: Metric and topological spaces 3
0.2. The Banach space of continuous functions 14
0.3. The geometry of Hilbert spaces 21
0.4. Completeness 26
0.5. Bounded operators 27
0.6. Lebesgue $L^p$ spaces 30
0.7. Appendix: The uniform boundedness principle 38

Part 1. Mathematical Foundations of Quantum Mechanics
Chapter 1. Hilbert spaces 43
1.1. Hilbert spaces 43
1.2. Orthonormal bases 45
1.3. The projection theorem and the Riesz lemma 49
1.4. Orthogonal sums and tensor products 52
1.5. The $C^*$ algebra of bounded linear operators 54
1.6. Weak and strong convergence 55
1.7. Appendix: The Stone-Weierstraß theorem 59

Chapter 2. Self-adjointness and spectrum 63
2.1. Some quantum mechanics 63
2.2. Self-adjoint operators 66
2.3. Quadratic forms and the Friedrichs extension 76
2.4. Resolvents and spectra 83
2.5. Orthogonal sums of operators 89
2.6. Self-adjoint extensions 91
2.7. Appendix: Absolutely continuous functions 95

Chapter 3. The spectral theorem 99
3.1. The spectral theorem 99
3.2. More on Borel measures 112
3.3. Spectral types 118
3.4. Appendix: Herglotz-Nevanlinna functions 120

Chapter 4. Applications of the spectral theorem 131
4.1. Integral formulas 131
4.2. Commuting operators 135
4.3. Polar decomposition 138
4.4. The min-max theorem 140
4.5. Estimating eigenspaces 142
4.6. Tensor products of operators 143

Chapter 5. Quantum dynamics 145
5.1. The time evolution and Stone's theorem 145
5.2. The RAGE theorem 150
5.3. The Trotter product formula 155

Chapter 6. Perturbation theory for self-adjoint operators 157
6.1. Relatively bounded operators and the Kato-Rellich theorem 157
6.2. More on compact operators 160
6.3. Hilbert-Schmidt and trace class operators 163
6.4. Relatively compact operators and Weyl's theorem 170
6.5. Relatively form-bounded operators and the KLMN theorem 174
6.6. Strong and norm resolvent convergence 179

Part 2. Schrodinger Operators
Chapter 7. The free Schrödinger operator 187
7.1. The Fourier transform 187
7.2. Sobolev spaces 194
7.3. The free Schrodinger operator 197
7.4. The time evolution in the free case 199
7.5. The resolvent and Green's function 201

Chapter 8. Algebraic methods 207
8.1. Position and momentum 207
8.2. Angular momentum 209
8.3. The harmonic oscillator 212
8.4. Abstract commutation 214

Chapter 9. One-dimensional Schrodinger operators 217
9.1. Sturm-Liouville operators 217
9.2. Weyl's limit circle, limit point alternative 223
9.3. Spectral transformations I 231
9.4. Inverse spectral theory 238
9.5. Absolutely continuous spectrum 242
9.6. Spectral transformations II 245
9.7. The spectra of one-dimensional Schrodinger operators 250

Chapter 10. One-particle Schrodinger operators 257
10.1. Self-adjointness and spectrum 257
10.2. The hydrogen atom 258
10.3. Angular momentum 261
10.4. The eigenvalues of the hydrogen atom 265
10.5. Nondegeneracy of the ground state 272

Chapter 11. Atomic Schrodinger operators 275
11.1. Self-adjointness 275
11.2. The HVZ theorem 278

Chapter 12. Scattering theory 283
12.1. Abstract theory 283
12.2. Incoming and outgoing states 286
12.3. Schrodinger operators with short range potentials 289

Part 3. Appendix
Appendix A. Almost everything about Lebesgue integration 295
A.1. Borel measures in a nutshell 295
A.2. Extending a pre measure to a measure 303
A.3. Measurable functions 307
A.4. How wild are measurable objects? 309
A.5. Integration -- Sum me up, Henri 312
A.6. Product measures 319
A.7. Transformation of measures and integrals 322
A.8. Vague convergence of measures 328
A.9. Decomposition of measures 331
A.10. Derivatives of measures 334

Bibliographical notes 341
Bibliography 345
Glossary of notation 349
Index 353

✦ Subjects

Физика;Матметоды и моделирование в физике;