March 27, 2015
Symmetry is your friend - overview
The power of thinking (version of March 27, 2015)
October 6
Lattice discretization
Lattice state (version of Oct 6, 2020)
October 6
Lattice derivative
Lattice derivative (version of Oct 6, 2020)
October 6
Shift operator
The generator of discrete translations (version of Oct 6, 2020)
October 6
Derivative
Derivative is a linear operator (version of Oct 6, 2020)
October 6
Lattice Laplacian
Lattice Laplacian (version of Oct 6, 2020)
October 6
Derivative
Derivative is a non-local operator (version of Oct 6, 2020)
October 8
Have symmetry?
Use it! (version of Oct 8, 2020)
October 8
Have symmetry?
Go to "eigen"subspace! Fourier decomposition of a 2-sites periodic lattice. (version of Oct 8, 2020)
October 8
Periodic lattices
Periodic lattices (version of Oct 8, 2020)
October 8
Fourier eigenvalues
Fourier eigenvalues (version of Oct 8, 2020)
October 8
Fourier representation
Discrete Fourier representation (version of Oct 8, 2020)
October 8
Laplacian
In Fourier representation (version of Oct 8, 2020)
October 8
Propagator
In Fourier representation (version of Oct 8, 2020)
October 8
A meta truth : We live in The Matrix
Fourier transformation is just a matrix (version of Oct 8, 2020)
October 8
Discussion
Shift matrix must have the periodic b.c.; Derivative being nonlocal is easiest to grasp on discrete lattice. It's so easy to make errors in the continuum formulation. (version of Oct 8, 2020)
October 8
Quantum Mechanics in a box
Sometimes it is simplest to impose the periodic b.c. on a localized solution, than relax it towards the correct (infinite extent) continuum solution. (version of Oct 13, 2020)
October 6
Discussion
Lattice discretization. What if geometry is not flat in all directions, but spherical? What about General Relativity? Life's persistent questions, skated around. (version of Oct 6, 2020)
October 6
Discussion
What is a derivative? Hypercubic lattice is a graph : nodes connected by links. Every graph has a notion of derivative associated with it; in particular a Laplacian. I was not allowed to say "Laplacian" here, as I have not gotten to defining it in my lecture at that point... (version of Oct 6, 2020)
October 6
Rocket science needs complex numbers
Why Fourier? Digital image processing! (version of Sep 22, 2020)