Gustaf Ullman

Exploring Nodal Lines in Random De Broglie Waves Exploring Nodal Lines in Random De Broglie Waves By Gustaf Ullman Introduction Nodal lines are a fascinating feature of random wave superpositions in quantum chaos. These lines, where either the real or imaginary parts of the wavefunction equal zero, reveal intricate patterns that merge randomness and structure. Using equation 1 from the paper "Nodal lines of random wavefunctions: perimeter corrections, statistics and scaling" by Saichev et al, we generate nodal lines for a superposition of De Broglie waves with random amplitudes, directions, and phases. Method The wavefunction is constructed as: ψ(r) = Σ a j exp(i(k j ·r + φ j )) Here, a j represents random amplitudes, k j is the wavevector with random directions, and φ j is the random phase. Nodal lines are identified where Re(ψ) = 0 or Im(ψ) = 0 . Results The generated nodal lines for R...

Creating Digital Art Using 2D Fourier Transforms Creating Digital Art Using 2D Fourier Transforms By Gustaf Ullman Introduction Digital art can explore concepts and techniques rooted in mathematics. One such method involves the use of two-dimensional Fourier transforms to create intricate and randomized patterns. This article outlines how to generate "lumpy" spatial frequency images in CMY (cyan, magenta, yellow) using a simple mathematical framework. Method The process begins with a radially symmetric function, such as: f(r) = A exp(-br) where r is the radial distance from the center of the image, A is the amplitude, and b controls the frequency distribution. This function is then assigned a random phase for each pixel, sampled uniformly between 0 and 2π. After applying a 2D Fourier transform to the resulting function, the output image exhibits a "lumpy" pattern, with spatial frequencies distri...