Quantum Theory - Analysis and PDE

M393C in Fall 2013

Topics in Applied Mathematics # 57705


Instructor: Thomas Chen
Office: RLM 12.138.
Office hours: TBA.
Email: t c A_T m a t h . u t e x a s . e d u
Lectures: MWF, 10:00 - 10:50 AM
Location: RLM 10.176.


Syllabus and Course Information


The purpose of this graduate course is to provide an introduction to Quantum Mechanics and Quantum Field Theory for mathematicians, with an emphasis on methods of Functional Analysis, Harmonic Analysis, and PDE's. One of the key goals will be to make a selection of fundamental topics in this research area accessible to graduate students specializing in Analysis, Mathematical Physics, PDE's, or Applied Mathematics. No background in physics is required.

Updated course information will be posted here and on Blackboard.

This website will be frequently updated.

Tentative list of topics:
  1. Basics of classical Hamiltonian dynamics. Principle of least action. Symmetries and conserved quantities. Legendre transform and Hamiltonian formalism. Symplectic form, Poisson bracket. Liouville theorem, Liouville equation.
    Integrable Hamiltonian systems, Arnol'd-Jost theorem, action-angle variables.
  2. Quantum mechanics. Link to classical mechanics via path integrals and Wigner transform. Derivation of classical Liouville equation. Ehrenfest theorem.
    Spectral theory, selfadjointness. Point spectrum, bound states, Birman-Schwinger principle, Lieb-Thirring estimates.
    Continuous spectrum, scattering states, wave operators, asymptotic completeness, scattering operator.
  3. Manybody systems, bosons, fermions. Kinetic energy estimates. Electrostatic inequalities. Stability of matter of 1st and 2nd kind.
  4. Fock space and second quantization. Weyl operators, coherent states, Bogoliubov transforms, Van Hove Hamiltonian.
    Classical and Quantum Electrodynamics. Field quantization. Renormalization and spectral theory. Feynman graphs.
  5. Bose gases. Electron gases. Mean field limits, macroscopic scaling limits. Derivation of nonlinear Schrodinger, Boltzmann, and Vlasov equations. Hamiltonian structure. Positive temperature formalism.
Recommended texts:
  1. V.I. Arnol'd, Mathematical Methods of Classical Mechanics, Springer.
  2. R. Abraham, J.E. Marsden, Foundations of Mechanics, AMS.
  3. T. Cazenave, Semilinear Schrodinger Equations (Courant Lecture Notes), AMS.
  4. L.C. Evans, Partial differential equations, AMS.
  5. S. Gustafson, I.M. Sigal, Mathematical concepts of Quantum Mechanics, Springer.
  6. J. Glimm, A. Jaffe, Quantum Physics from a functional integral point of view, Springer.
  7. R. Haag, Local quantum physics: Fields, particles, algebras. Springer.
  8. T. Kato, Perturbation Theory for Linear Operators, Springer.
  9. L.D. Landau, E.M. Lifshitz, Course of Theoretical Physics, Elsevier.
  10. E.H. Lieb, M. Loss, Analysis, AMS.
  11. E.H. Lieb, R. Seiringer, The stability of matter in Quantum Mechanics, Cambridge.
  12. J. Moser, E.J. Zehnder, Notes on Dynamical Systems (Courant Lecture Notes), AMS.
  13. C. Muscalu, W. Schlag, Classical and multilinear harmonic analysis, Vol. 1, Cambridge.
  14. M. Reed, B. Simon, Methods of Modern Mathematical Physics, Vols. 1 - 4, Academic Press.
  15. E. Stein, Harmonic Analysis, Princeton University Press.
  16. T. Tao, Nonlinear dispersive equations, AMS.
  17. G. Teschl, Mathematical Methods in Quantum Mechanics, AMS.
  18. T. Wolff, Lectures on harmonic analysis, AMS.
Online resources:
  1. Lecture notes by T. Arbogast and J. Bona for Methods of Applied Mathematics.
  2. Lecture notes by S. Klainerman, which include an introduction to Harmonic Analysis.
  3. Lecture notes by W. Schlag on Harmonic Analysis.
  4. G. Teschl's book is available for download here.
  5. Lecture notes by T. Wolff on Harmonic Analysis.
If you would like to prepare ahead over the summer break, the following will be useful: Topics in Functional Analysis, including distributions, Hilbert spaces, spectral theory of selfadjoint operators (book by Teschl, lecture notes by Arbogast-Bona). Methods of Harmonic Analysis (lecture notes by Schlag, Wolff, and Klainerman).
The University of Texas at Austin provides upon request appropriate academic accommodations for qualified students with disabilities.
For more information, contact the Office of the Dean of Students at 471-6259, 471-4641 TTY.