Ulysses' trip

Apr 11, 2025

Eigenfunctions of the Laplacian on an annulus with homogeneous Neumann boundary condition

Several features of the eigenfunctions of the Laplacian on an annulus with homogeneous Neumann boundary condition are discussed. The distribution of the eigenvalues is discussed in detail, making use of a phase angle function called $\tht$ . The limiting cases of a disk and a circle are discussed.

Categories: math
Tags: mathematical physics, pde, ode, long paper

Mar 3, 2025

The role of particle indistinguishability in statistical mechanics

Indistinguishability plays an important role in enumerative problems in combinatorics. This article explains the concept and significance of particle indistinguishability in statistical mechanics.

Categories: physics
Tags: mathematical physics, statistical mechanics, probability, long paper, combinatorics, quantum mechanics

Jan 9, 2025

Why it is a bad idea to use GitHub for submiting school homework

It is common for school courses to require students to submit their homework using GitHub. However, this is usually a bad idea for several reasons.

Categories: misc
Tags: programming, github, school

Nov 15, 2024

How to use JavaScript in GitHub Actions without committing `node_modules`

I was creating my first GitHub Action. According to the official documentation, I should commit the node_modules to the repo or generate a script that bundles the whole node_modules if I want to create a JavaScript action. I think this is not good practice, so I try to find another way.

Categories: guide
Tags: javascript, github

Oct 10, 2024

Some understanding of Grassmann numbers out of intuition

I was briefly introduced to Grassmann numbers when I studied quantum field theory. I then had the natural question of how we can formally define them. In this article, I went with my intuition and tried to answer this question.

Categories: math
Tags: quantum field theory, linear algebra, abstract algebra

Sep 10, 2024

The notational convenience of imaginary time in the derivation of the metric in Poincaré coordinates

In general relativity, people usually choose one of the two major metric signatures. However, in certain cases, the imaginary time convention can be more convenient. Here is one of such cases: the derivation of the metric in Poincaré coordinates for the anti-de Sitter space.

Categories: physics
Tags: general relativity, ads space

Jul 1, 2024

The smallest wave packet in the lowest Landau level

The smallest wave packet in the lowest Landau level exists, and is a Gaussian wave packet. This turns out to be related to the coherent state of the harmonic oscillator.

Categories: physics
Tags: quantum mechanics, condensed matter physics

Jun 30, 2024

Regularizing the partition function of a hydrogen atom

The partition function of a hydrogen atom diverges (only considering bound states). However, we can regularize it to get finite answers. Different regularizations give the same result. They largely agree with the physical arguments for the case of the hydrogen atom at room or cold temperature, but this should be considered a mere coincidence. The results from regularized partition functions cannot generally be trusted.