Quantum Mechanics for MathematiciansAmerican Mathematical Soc., 2008 - 387 pages This book provides a comprehensive treatment of quantum mechanics from a mathematics perspective and is accessible to mathematicians starting with second-year graduate students. It addition to traditional topics, like classical mechanics, mathematical foundations of quantum mechanics, quantization, and the Schrodinger equation, this book gives a mathematical treatment of systems of identical particles with spin, and it introduces the reader to functional methods in quantummechanics. This includes the Feynman path integral approach to quantum mechanics, integration in functional spaces, the relation between Feynman and Wiener integrals, Gaussian integration and regularized determinants of differential operators, fermion systems and integration over anticommuting (Grassmann)variables, supersymmetry and localization in loop spaces, and supersymmetric derivation of the Atiyah-Singer formula for the index of the Dirac operator. Prior to this book, mathematicians could find these topics only in physics textbooks and in specialized literature.This book is written in a concise style with careful attention to precise mathematics formulation of methods and results. Numerous problems, from routine to advanced, help the reader to master the subject. In addition to providing a fundamental knowledge of quantum mechanics, this book could also serve as a bridge for studying more advanced topics in quantum physics, among them quantum field theory.Prerequisites include standard first-year graduate courses covering linear and abstract algebra, topology and geometry, and real and complex analysis. |
Table des matières
| 3 | |
Basic Principles of Quantum Mechanics | 63 |
Schrödinger Equation | 149 |
Spin and Identical Particles | 217 |
Functional Methods and Supersymmetry | 237 |
Integration in Functional Spaces | 289 |
Fermion Systems | 307 |
Supersymmetry | 343 |
| 373 | |
| 383 | |
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Expressions et termes fréquents
2.3 of Chapter angular momentum anticommuting asymptotics boundary conditions called canonical classical mechanics commutation relations corresponding defined definition degrees of freedom denote det(A differential equation eigenfunctions eigenvalues fermion Feynman path integral first follows formula Gaussian given graded vector space Grassmann algebra Hamiltonian operator harmonic oscillator Heisenberg Hilbert space integrals of motion irreducible isomorphism kernel Lagrangian system Lemma Lie algebra linear mapping matrix multiplication non-degenerate obtain operator H phase space Poisson bracket Poisson manifold Problem projection-valued measure Proof Proposition Prove quantization quantum mechanics quantum particle Remark Riemannian satisfies satisfy Schrodinger Schrödinger equation Schrödinger operator Section 2.3 self-adjoint operator solution spin subspace superalgebra supermanifold symmetric SymN symplectic form symplectic manifold Theorem theory transform unitary operators variables vector field vector space wave function Weyl Wick symbol Wiener measure