Curriculum Vitae of Matthew S. Foster

“Once men turned their thinking over to machines in the hope that this would set them free. But that only permitted other men with machines to enslave them.” —Frank Herbert, Dune

0. Recent Talks

1. Simulating fractal surface states with twisted moiré graphene
(MPIPKS Dresden, 09-2024)

2. Success and failure: The Dirac equation and topological materials
(Rice University, 09-2024)

3. Suppression of shot noise in a dirty marginal Fermi liquid
(APS March Meeting, 03-2025)

4. Non-Linear Sigma Model for Disordered and Ergodic Systems: Basic Principles and Applications.
Part 4: Measurement-induced (entanglement) phase transitions
(APS March Meeting tutorial, 03-2026)

1. Papers on arXiv

2. Lecture notes

• Introduction to quantum mechanics I [Course notes]

Course notes for a junior-level quantum class taught from Fall 2021-2025 at Rice. Starts with an introduction to math and notation ala Shankar, and proceeds to the quantum physics of rotation via spin 1.

I. Mathematical formalism

1. Introduction, wave equation and eigenmodes
2. Linear vector spaces, Dirac notation, inner product
3. Operators, matrix elements, adjoint operation
4. Hermitian operators, eigenvalue problem
5. Diagonalising Hermitian operators, degenerate eigenspaces, unitary basis change
6. Functions of operators, Hilbert space, Dirac delta function
7. Position and wavenumber operators, boundary conditions, Fourier transforms
7A. Summary: Position, wavenumber operators on function space
8. Commutator, inhomogeneous classical string

II. Basic quantum mechanics in a nutshell

9. Uncertainty, postulates of quantum, Schroedinger eigenvalue problem

III. Eigenstates of rotation, quantum magnetic moments, spin-1 and precession

10. Generators of rotations, eigenstates of rotation, SO(3)
10A. Infinitesimal rotation
11. Quantum spin 1, magnetic moment, Hamiltonian
12. Quantum time evolution (SE), solution methods, spin dynamics, operator EOM

IV. Spin-1/2 (single qubit), Rabi oscillations, entangled pairs, measurements

13. Spin 1/2, Bloch sphere, rotating frame
14. Paramagnetic resonance and Rabi oscillations, projective measurements
15. Direct product spaces, particle in the plane, 2 spin 1/2s
16. Singlet, triplet states, entanglement, Stern-Gerlach apparatus

V. Particles in 1D redux: Continuum states, scattering, SHO

17. Probability current, Gaussian wave packet
18. Potential scattering in 1D
19. SHO 1: Intro, position-basis solution
20. SHO 2: Creation and annihilation operators

VI. General theory of angular momentum, spherical harmonics, 3D central force and hydrogen atom

21. Orbital angular momentum in 3D, general representation theory
22. Orbital angular momentum in position basis, spherical harmonics
23. Rotationally invariant Hamiltonians in 3D
24. Hydrogen atom

• Topological Materials Physics [Guest lecture]

Slides for a guest lecture in undergraduate solid state physics being taught at Rice in Spring 2017. This is an introduction to topological materials, mainly by way of the dimerized chain (Su-Schrieffer-Heeger) model.

Topological Materials Physics (Spring 2017)

• "Modern" mathematical physics I: Mainly Lie algebra representation theory [Course notes]

These are course notes for a class being taught in Fall 2016 at Rice on Lie algebras and their representations. Most of the material will be standard, but the plan is for the presentation to be unabashedly applied, emphasizing visualization and algorithms at the expense of rigor and generality. I am following roughly the presentation in Robert Cahn's excellent Semi-simple Lie Algebras and Their Representations, available on the web here. The main difference is that I will work many examples and elaborate on various topics. All lecture notes will be posted here. These are intended to be complete enough for self study, and I hope they will prove generally useful for physics students that wish to learn this material. As the current course is being taught at the graduate level, I hope to cover some advanced topics by the end (some subset of Riemannian symmetric spaces and random matrix theory; classification of topological phases; affine Lie algberas, WZNW models, quantum equivalence; quantum groups, anyons, fusion and braiding rules).

Syllabus PHYS 600 (Fall 2016)

1. Rotations, so(3) and su(2). [v2.1]
2. Lie groups as manifolds. SU(2) and the 3-sphere. [v1.5]
2A. SU(n), SO(n), and Sp(2n) Lie groups. [v1.5]
3. Introduction to su(3). [v1.7]
4. Killing form and commutation relations. [v1.6]
5. Roots and weights. [v2.0]
6. Cartan classification of Hamiltonians: The 10-fold way. [v1.2]
7. The classical and exceptional Lie algebras. [v1.5]
8. Highest-weight representations. [v2.5]
8b. Informal lecture--some applications. [v1.0]
9. Casimir, characters, dimension and strange formulae. [v1.8]
10. Spinor representations of SO(N). [rough notes]
11. Affine Lie algebra representation theory I. [rough notes]