Geometry of Quantum Mechanics
``Somewhere in our doctrine is hidden a concept, unjustified by experience, which we must eliminate to open up the road.’’ – Max Born
Course information
Code: Fields Academy Shared Graduate Course
Instructor: Marco Gualtieri
Teaching Assistant: Joshua Lau (email)
Class schedule: Tuesday and Thursday 2-2:30 followed by 30m office hours
Evaluation: Five problem sets, attendance and in-class quizzes.
Assignments
You may discuss the problems, but avoid reading a written solution before you write your own, since these must be original. Also, do not share written solutions with anyone, even after the deadline.
Late assignments will not be accepted: please hand in what you have at the deadline.
Assignments are evaluated for correctness, but also clarity. Keep your solutions concise, and make sure the structure of your argument is clear. I recommend using a LaTeX template to improve the readability of your work.
Resources
Articles
- “On the law of distribution of energy in the normal spectrum” (1900), Max Planck
- “On a heuristic point of view concerning the production and transformation of light” (1905), A. Einstein.
- “On the quantum theory of radiation” (1917), A. Einstein.
- “On the quantum theory of line-spectra” (1918), N. Bohr.
- “The quantum-theoretical interpretation of the number of dispersion electrons” (1921), R. Ladenburg.
- “Quantum mechanics” (1924), M. Born.
- “Quantum-theoretical re-interpretation of kinematic and mechanical relations” (1925), W. Heisenberg.
- “On quantum mechanics” (1925), M. Born and P. Jordan.
- “The fundamental equations of quantum mechanics” (1925), P. A. M. Dirac.
- “On quantum mechanics II” (1925), M. Born, W. Heisenberg, P. Jordan.
- “On Unitary Representations of the Inhomogeneous Lorentz Group” (1939), E. Wigner.
- “The wave nature of the electron” (1929) L. de Broglie.
- “The statistical interpretation of quantum mechanics” (1954), M. Born.
- “Anharmonic Oscillator”, and Part II Bender and Wu.
- “Topological quantum field theory”, Atiyah.
- “Supersymmetry and Morse theory”, Witten.
Books
The course will draw from a variety of sources, including the recommended texts listed below.
- The Theory of Groups and Quantum Mechanics, H. Weyl.
- Group theory and physics, S. Sternberg
- Quantum Theory, Groups and Representations: An Introduction, P. Woit.
- Quantum Mechanics for Mathematicians, Takhtajan.
- Lectures on Quantum Mechanics for Mathematics Students, Faddeev and Yakubovskii.
- Quantum Field Theory: A Tourist Guide for Mathematicians, Folland.
- The Principles of Quantum Mechanics, Dirac.
- Quantum Mechanics and Integrals, Feynman & Hibbs.
- Quantum Many-Body Systems, Tasaki
Videos
- Double slit experiment - Hitachi Labs
- Dirac Lectures on Quantum Mechanics
- Feynman Lectures on Physics: Quantum mechanicsThe first part of the course will be an overview of the main ideas of quantum mechanics, phrased in a modern mathematical form. The second part will focus on the Exact WKB method, whereby the perturbative solutions to the Schrodinger equation have recently been shown to be Borel summable.
Main topics
The purpose of this course is to introduce the main phenomena of Quantum Mechanics and to introduce, in parallel, the mathematical structures and ideas which allow us to model and understand these phenomena.
We hope to cover most of the topics below:
- Groupoids and the Heisenberg approach to quantum mechanics
- Representations of the Lorentz group, elementary particles
- The Schrodinger equation on Riemannian manifolds
- The Dirac equation on Spin manifolds
- Heisenberg spin chains, the vacuum
- Penrose Spin networks
- Perturbation theory
- The Exact WKB method
- Supersymmetry