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Topics in Mathematical Physics: Classical and Quantum Mechanics (Winter 2019-20)

Ma 148aWinter 2019-20Caltech

Classical and quantum mechanics, and field theory — a mathematical primer.

Overview

This class gives an introduction to the basic principles of Classical Mechanics, Quantum Mechanics, Special Relativity and elementary Classical Field Theory from a mathematician’s point of view. The goal of this class is to introduce students with a math background to topics in modern physics. No physics background, other than an introductory course in physics, is assumed. The course assumes knowledge of elementary differential geometry, elementary functional analysis, algebra, and differential equations.

Meeting time and location

  • Monday, Wednesday and Friday, 10:00 – 10:55 am
  • 187 Linde Hall

Instructor

  • Sarthak Parikh (he/him) · 202 Kellogg/Linde Hall · Office hours by appointment

Grading

  • This class is offered Pass/Fail.
  • Student presentations: 60%
  • Class participation: 40%

Each student gives two 15-minute presentations (30% of the grade each). The first is on a relevant topic from classical mechanics, around midterms week; the second is on a relevant topic from quantum mechanics, around finals week. Topics must be approved by the instructor. For classical mechanics, consider presenting a section/appendix from Arnold’s book.

Lecture outline

  1. Overview of topics to be covered.
  2. Maxwell’s equations in different forms, Hamilton’s equations and Poisson bracket.
  3. Exterior forms, exterior products review.
  4. Differentiable manifolds, differential forms review — I.
  5. Differentiable manifolds, differential forms review — II.
  6. Symplectic manifolds.
  7. Hamiltonian vector fields.
  8. Poisson brackets.
  9. Lagrangian mechanics.
  10. Classical field theory.
  11. Special relativity.
  12. Electrodynamics in tensor notation, differential form notation; Yang-Mills theory.
  13. Student presentations on classical mechanics.
  14. Quantum mechanics.
  23. Student presentations on quantum mechanics.

Suggested books and reference material

  • Vladimir Arnold, Mathematical Methods of Classical Mechanics
  • Herbert Goldstein, Classical Mechanics
  • Gerald Folland, Quantum Field Theory: A Tourist Guide for Mathematicians
  • Jonathan Dimock, Quantum Mechanics and Quantum Field Theory: A Mathematical Primer
  • Peskin and Schroeder, An Introduction to Quantum Field Theory (chapter 15)
  • Pierre Deligne et al., Quantum Fields and Strings: A Course for Mathematicians (Vols. 1 and 2)