Quantum Mechanics for Mathematicians
American Mathematical Soc., 1 janv. 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.
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Basic Principles of Quantum Mechanics
4 Notes and references
2 Onedimensional Schrodinger equation
3 Angular momentum and SO3
5 Hydrogen atom and SO4
6 Semiclassical asymptotics I
Spin and Identical Particles
2.3 of Chapter angular momentum anticommuting asymptotics boundary conditions bounded operator called canonical classical mechanics commutation relations continuous spectrum corresponding defined definition degrees of freedom denote differential equation differential operator eigenfunctions eigenvalues fermion Feynman path integral finite finite-dimensional follows formula Gaussian given graded vector space Grassmann algebra Hamiltonian operator harmonic oscillator Hilbert space inner product integrals of motion irreducible isomorphism kernel Lagrangian system Lemma Lie algebra Lie group linear mapping matrix metric multiplication non-degenerate obtain one-dimensional operator H orthogonal orthonormal basis phase space Poisson bracket Poisson manifold polynomials Problem projection-valued measure Proof Proposition Prove quantization quantum mechanics quantum particle Remark Riemannian satisfies Schrodinger equation Schrodinger operator Section 2.3 self-adjoint operator solution spin subset subspace superalgebra supermanifold symmetric symplectic form symplectic manifold Theorem theory transform unitary operators variables vector field wave function Weyl Wick symbol Wiener measure