I show how to calculate a Gaussian integral in arbitrary dimensions in the presence of a linear source term, as a first step toward applications in quantum field theory and statistical mechanics. Video. Notes.
Monthly Archives: November 2022
The adjoint operator
A quick look at the adjoint of a linear operator (aka the hermitian conjugate). An example with a differential operator and how to prove properties using the definition based on the inner product. Video. Notes.
A 1d Green’s function example two ways
A closer look at how to calculate a Green’s function in a 1D example, using a direct solution and a spectral representation. Video. Notes.
Diagonalizing differential operators on the lattice
How do you represent a function on a computer with discrete memory? What about its derivatives? A first look at stuff on a lattice, the matrix form of the first-derivative operator (forward, with periodic boundary conditions), and its eigenvectors and eigenvalues (Fourier modes). Video. Notes.
A very special unitary matrix. Part 2.
A second look at the matrix of the discrete Fourier transform. Proof that it is unitary. How to apply it to a function, and why it costs O(NlogN) vs O(N^2). Video. Notes.
A very special unitary matrix
Review of some properties of unitary matrices. Orthogonality and completeness. A first look at the matrix of the discrete Fourier transform. Diagonalization and efficiency. Video. Notes.
The Dirac delta
A first look at the Dirac delta and an application with Green’s functions. Video. Notes.
Eigensystems and Inverses. Part 2.
Recap of Part 1. Orthogonality and completeness relations. What do they look like in the infinite-dimensional case of part 1? Video. Notes.
Eigensystems and Inverses
Spectral representation of the inverse of a matrix. An example in infinite dimensions and… a first appearance of a Green’s function! Video. Notes.
Eigenvectors and Eigenvalues. Part 2.
Remider of part 1. Matrix diagonalization. Spectral representation. Similarity transformations. Two applications. Video. Notes.