روش‌های ریاضی در مکانیک کوانتومی: با کاربردهایی در اپراتورهای شرودینگر (مطالعات تحصیلات تکمیلی در ریاضیات)

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. 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 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 Schrödinger 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 Schrödinger 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. The book is written in a very clear and compact style. It is well suited for self-study and includes numerous exercises (many with hints). —Zentralblatt MATH The author presents this material in a very clear and detailed way and supplements it by numerous exercises. This makes the book a nice introduction to this exciting field of mathematics. —Mathematical Reviews