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Mechanics for. Leon A. Graduate Studies. Volume American Mathematical Society. Providence, Rhode Island. Editorial Board. David Cox Chair. Steven G. Rafe Mazzeo. Martin Scharlemann. Primary For additional information and updates on this book, visit. Library of Congress Cataloging-in-Publication Data. Takhtadzhian, L. Leon Armenovich. Includes bibliographical references and index.
ISBN alk. Quantum theory. Mathematical physics. Copying and reprinting. Individual readers of this publication, and nonprofit libraries. Permission is granted to quote brief passages from this publication in. Republication, systematic copying, or multiple reproduction of any material in this publication.
Requests for such. Requests can also be made by. All rights reserved. The American Mathematical Society retains all rights. Printed in the United States of America. The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. To my teacher Ludwig Dmitrievich Faddeev with admiration and gratitude. Part 1. Classical Mechanics. Lagrangian Mechanics. Generalized coordinates. The principle of the least action. Examples of Lagrangian systems.
Symmetries and Noether's theorem. One-dimensional motion. The motion in a central field and the Kepler problem. Legendre transform.
Hamiltonian Mechanics. Hamilton's equations. The action functional in the phase space. The action as a function of coordinates. Classical observables and Poisson bracket. Canonical transformations and generating functions. Symplectic manifolds. Poisson manifolds. Hamilton's and Liouville's representations. Notes and references. Basic Principles of Quantum Mechanics. Observables, states, and dynamics.
Mathematical formulation. Heisenberg's uncertainty relations. Heisenberg commutation relations. Coordinate and momentum representations. Free quantum particle. Examples of quantum systems. Old quantum mechanics. Harmonic oscillator. Holomorphic representation and Wick s y m b o l s Weyl relations. Stone-von Neumann theorem. Invariant formulation. Weyl quantization. Deformation quantization. Chapter 3. Schrodinger Equation. General properties. Characterization of the spectrum.
The virial theorem. One-dimensional Schrodinger equation. Jost functions and transition coefficients. Eigenfunction expansion. S-matrix and scattering theory. Other boundary conditions. Angular momentum and SO 3. Angular momentum operators. Representation theory of 50 3. Two-body problem.
Separation of the center of m a s s. Three-dimensional scattering theory. Particle in a central potential.
Hydrogen atom and S0 4. Discrete spectrum. Continuous spectrum. Hidden S0 4 symmetry. Semi-classical asymptotics - I. Time-dependent asymptotics. Time independent asymptotics. Bohr-Wilson-Sommerfeld quantization rules. Spin and Identical Particles. Spin operators. Spin and representation theory of SU 2 Charged spin particle in the magnetic field.
Pauli Hamiltonian. Particle in a uniform magnetic field. System of identical particles. The symmetrization postulate. Young diagrams and representation theory of Symnr.
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What does an analytic number theorist do when he is caught in the tractor-beam of non-commutative geometry with its end-of-the-rainbow promises of magical new number theory methods, and methods with roots in quantum field theory at that? What does he do when he hears rumors of future wild and reckless new attacks on the fortresses of the primes where the zeta function of Riemann stands guard, still seemingly invincible after all these years? Well, the keenly excited but dizzy and disoriented analytic number theorist starts frantically to look for a royal road to quantum mechanics and thereafter to quantum field theory, things he has shunned ever since his youthful all-consuming discovery of for instance modular forms and theta functions. No more, however: those days are gone…].
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Quantum Mechanics for Mathematicians
Toggle navigation. Quantum Mechanics for Mathematicians Leon A. Takhtajan This book provides a comprehensive treatment of quantum mechanics from a mathematics perspective and is accessible to mathematicians starting with second-year graduate students. In 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 quantum mechanics. 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.