A quick discussion of exp(-x^2) cos(3x), followed by an explicit derivation of the Fourier series of sin^3(x) without doing integrals.Video. Notes.
Author Archives: joaquindrut
Fourier series: example 1
A first example of calculating a Fourier series explicitly. More examples in upcoming videos! Video. Notes.
The quantum free particle
Using background on complex numbers and Fourier modes, here is a first discussion of the most essential quantum mechanical system. Video. Notes.
Fourier series: first concepts
A first look at the definition and concept of Fourier series as a global expansion in a finite interval, compared with a Taylor series as a local expansion. More in upcoming videos! Video. Notes.
Euler’s formula
A quick look at the derivation of Euler’s formula and its uses and connections to trigonometric and hyperbolic sine and cosine functions of complex variables. Video. Notes.
Complex numbers basics
Definition and basic properties of complex numbers. Addition, multiplication, division, conjugation. Polar representation. Trigonometric identities. Differentiation & Cauchy-Riemann conditions. Simple examples. Video. Notes.
More about Gaussian integrals
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.
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.