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" images in CMY (cyan, magenta, yellow) using a simple spectral construction. Method The process begins by specifying a radially symmetric amplitude profile in the spatial-frequency domain , for example: f(r) = A exp(-br) where r is the radial frequency (distance from the center in the Fourier plane), A is an amplitude scale, and b controls how quickly the spectrum decays with frequency. Next, assign a random phase to each Fourier coefficient, sampled uniformly in [0, 2π) , and form a complex spectrum F(k) = f(|k|) · exp(iφ(k)) where...