Math 444 – The Mathematics of Quantum Theory

Renato Feres

Fall 2022

Section information

  • Class time and location: Tuesday and Thursday, 2:30PM to 3:50PM, classrom: Cupplies I, 215
  • Office hours: Monday-Wednesday from 4:00PM to 5:00PM, Friday from 11:00AM to 1:00PM (shared)

General description of the course

This is an introduction to the mathematical foundations of quantum theory aimed at advanced undergraduate/beginning graduate students in Mathematics and Engineering, although students from other disciplines are equally welcome to attend.

Topics may include: the mathematical postulates of quantum theory from a quantum probability perspective, multilinear algebra on finite dimensional Hilbert spaces, rudiments of Lie groups and Lie algebras, spectral theory of self-adjoint operators, simple physical systems in finite and infinite dimensions, quantum probability and quantum information.

It is expected that you have a good grounding in linear algebra and multivariate calculus, as well as familiarity with probability theory, although I will make every effort to make the course as self-contained as possible.

Coursework will consist of weekly homework and reading assignments, and a final paper on a project related to the topics of the course. The final grade will be calculated on the basis of the weekly assignments and final project, with special emphasis on the former.

Text

There is, unfortunately, no single text that approaches the subject matter of this course quite at the level and from the perspective we plan to cover it. My intention is to make available to you a variety of sources throughout the course. These sources will include typed course notes, occasional chapters from various texts and expository papers. I should have a fair amount of this material collated earlier on in the semester and available in Canvas.

All the material needed for the homework assignments will be contained in the typed course notes.

Topics we hope to cover.

The course will be divided into two parts: Part I covers the foundations of Quantum Theory in a finite dimensional setting from a perspective that emphasizes non-commutative probability theory. It should be accessible to students with a strong background in linear algebra (although I will review the necessary facts). Part II is dedicated to systems governed by differential (Schrödinger) operators.

The following is a very tentative list of topics. Some will be developed in detail while others may be surveyed more briefly.

  1. Mathematical postulates of quantum probability spaces
    • Hilbert spaces, operators, the spectral theorem;
    • Gleason’s theorem and Born’s rule, the uncertainty relation;
    • Dynamics in finite dimensional QT;
    • Quantum trajectories, strong and weak measurements;
    • The unitary group;
    • The qubit and the Bloch sphere.
  2. Composite systems and entanglement
    • Tensor products of Hilbert spaces;
    • Symmetric and antisymmetric tensor products (indistinguishability, bosons and fermions);
    • Fock spaces;
    • Tensor diagrams and quantum circuits;
    • von Neumann entropy, Schmidt decomposition and rank; entanglement;
    • EPR states, Bell’s inequalities, superdense coding, quantum teleportation.
  3. Infinite dimensional systems
    • Self-adjointness, Schrödinger operators;
    • Wave packets and the Fourier transform;
    • Examples of quantum systems (mostly in dimension 1):
      • Particle in a box;
      • Harmonic oscillator;
      • Quantum graphs;
  4. Open systems and basic concepts in quantum stochatic processes

Coursework and grades

Coursework will consist mainly of weekly homework assignments (I plan around 9 or 10 assignments), and a final project paper. Final grades are determined as follows: homework 80%, final project 20%. Your lowest homework grade will be dropped. Assignments will be handled through Gradescope and grades will be posted on Canvas.

Grades may be curved, but will not be less than the following scale:

A+ A A- B+ B B- C+ C C- D F
TBA 90+ [85,90) [80,85) [75,80) [70,75) [65,70) [60,65) [55,60) [50,55) [0,50)

If you are taking this class Pass/Fail, you need a C- to pass. There is no requirement for auditing.

In writing the tests (mid-term and final), I will rely strongly on the homework assignments. To feel confident, you should master the topics and exercises covered by the assignments. Test questions will not involve R in any form.

Recording

I plan to record the in-person lectures using Zoom and make them available in Canvas. I expect, and strongly urge, that students will attend classes in person, but the option of attending online is available if you cannot be there for health related reasons.

Academic integrity

I will follow the University’s academic integrity policy, which you can read here. Work submitted under your name is expected to be your own. You are permitted, and encouraged, to collaborate on assignments. You may research broadly over the internet and in books when working on assignments.

If you have any concerns or questions about this policy or academic integrity in class, please contact me.

Email

Please include Math 444 in the subject line of any email message that pertains to this course.

Homework assignments

Weekly homework assignments will be posted below on this syllabus. Solutions will be posted in Canvas (under Pages).