I introduce the Fourier transform as a generalization of the Fourier series from a finite interval to the whole real line. I give two basic examples that start to show the mathematics underlying Heisenberg’s uncertainty principle in quantum mechanics.Video. Notes.
Author Archives: joaquindrut
Green’s functions of the Laplacian: eigenfunction expansion.
Using the cartesian and spherical eigenfunctions of the Laplacian discussed in previous videos, we build the corresponding Green’s functions. What happens when we remove the boundaries? A step toward Fourier transforms and quantum field theory.Video. Notes.
Fourier series: generalizations & applications. Part 3.
Last part! I give a few more details on how to find the radial part of the eigenfunctions of the Laplacian with vanishing boundary conditions on a sphere. This radial part is given by the spherical Bessel functions and their roots, and the latter quantize the eigenvalues of the full problem. A brief example is …
Continue reading “Fourier series: generalizations & applications. Part 3.”
Fourier series: generalizations & applications. Part 2.
More details on solving the diffusion equation in finite interval in 1D. Generalization to 3D in cartesian and spherical coordinates. Further details on the latter in part 3!Video. Notes.
Fourier series: generalizations & applications. Part 1.
Generalized Fourier series. A few comments and two examples: spherical harmonics and Chebyshev polynomials. An application to PDEs: diffusion in a finite interval in one spatial dimension. More in part 2! Video. Notes.
Calculating roots: the bisection algorithm
A first look at numerical methods to calculate roots: the bisection algorithm. Steps, precision bounds, and an important caveat. Video. Notes.
The geometric sum and series
A quick reminder of the derivation of the geometric sum and its limit as a series. Video. Notes.
The binomial theorem
A simple proof, by induction, of the binomial theorem. Video. Notes.
Fourier series: [sin(x)]^n
Using the binomial theorem, I derive the Fourier series for [Sin(x)]^n in exponential form, without doing any integrals. Video. Notes.
Fourier series: exponential form
From the trigonometric (sine/cosine) form to the complex exponential form of the Fourier series, using Euler’s identity.Video. Notes.