Math 528 – Mathematics of Quantum Theory (Spring 2019)

Section information

  • Class time and location: Tuesdays and Thursdays from 1:00PM to 2:30PM in Cupples I 218
  • Tentative office hours: Tuesdays 2:30PM-4:00PM (immediately after class); Wed 12:00PM-2:00PM

Subject

This is an introduction to Quantum Theory for graduate students in Math. (But all other students are welcome.) The emphasis will be on the mathematical foundations of the subject and applications to simple physical systems. No previous knowledge of physics beyond college mechanics will be assumed. On the other hand, familiarity with Measure Theory and the basics of Hilbert spaces will be helpful, but I will review background material as needed.

Text

I plan to use a few different texts for somewhat different purposes, the first of which (see below) will be our designated “official” textbook. Important! Do not buy these books. They are available in electronic form through the Olin Library web site.

  • Quantum Theory for Mathematicians by Brian C. Hall, Springer 2013. Library link here. Hall’s book will be the course’s “official” text. This means that I will try to follow its overall plan, and will define the content of the course by it. See below for more details on topics we intend to cover.

  • Mathematical Concepts of Quantum Mechanics by S.J. Gustafson, I.M. Sigal. Second Edition, Springer 2011. Library link here. This book covers more physical content, but the treatment is still congenial to people in math.)

  • A Mathematical Primer on Quantum Mechanics by Alessandro Teta. Springer 2018. Library link here. Some standard physical systems are developed in detail here, together with the theory of linear operators in Hilbert spaces. A possible source of topics for students’ presentations. (More on this below.)

Topics we hope to cover. (Chapters refer to B. Hall’s text.)

  • Generalities about classical mechanics - Chapter 2

  • A first approach to quantum mechanics - Chapter 3

  • Simple systems: Free particle, square well, the harmonic oscillator, etc. - Chapters 4, 5, 11

  • The spectral theorem, Stone-von Neumann - Chapters 6, 7, 8, 9, 10, 14

  • Quantum uncertainty (plus generalities on noncommutative probability) - Chapter 12

  • Quantization Schemes - Chapter 13

  • Further topics as time permits, from WKB, Lie group representations for angular momentum and spin, the hydrogen atom.

Coursework, exams and grades

Grades will be based on presence, homework, and a presentation at the end of the course. I expect approximately 7 to 8 homework assignments throughout the semester. Details will be given in class but, roughly, you are guaranteed a passing grade for good attendance; homeworks will count as 60% of the grade and the presentation 40%.

Homework assignments