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After realizing that the 2024 polyfill.io supply chain attack affected this website, I upgraded from Mathjax 3 to Mathjax 4. Also, Mathjax is now included without the previously recommended reference to polyfill.io.

For a brief time, starting after 01 June 2026 (when there was no apparent problem) and 18 June 2026 (when the problem first came to my attention), visitors to this site were faced with a spurious “Sign-In” pop-up, identifying itself as “polyfill.io”.

This was a (late) consequence of the 2024 polyfill.io supply chain attack. If, by chance, you entered any credentials, you may want to change them where necessary.

The content and hosting of this site was (to my knowledge) never compromised.

The Schrödinger equation in a wedge potential is one of the “standard” problems in Quantum Mechanics — standard, but not routine, as its solution involves the Airy equation, which is likely to be unfamiliar, at least until this precise moment.

Because of this, it seemed interesting to look into this problem, and to find both its exact solution (involving the Airy function), but also to solve it using the WKB approximation: another topic that is “standard” in Quantum Mechanics classes, but less often seen in the real world. Here it provides an additional benefit, as a way to avoid the Airy function and to express the solution in terms of “elementary” functions.

Gmail has its own ideas how to do things. Here are two small tricks that make it easier to use email with a local email client set-up.

I occasionally fantasize about an “ideal math curriculum” for Physics graduate students, based on my experience, in school and out. Which topics make sense, which don’t, what should count as reasonably expected knowledge, what is actually useful?

There are also some textbooks, several of which having been published after I left school, that I would like to use (actually: like to have used) in the appropriate classes.

Besides the Normal Distribution Function, I occasionally need the Airy Function Ai(x): it arises in perturbation theory and some other contexts. This function is most definitely not part of most standard numerics libraries! While high-quality implementations of it are part of most “serious” numerics libraries (such as SciPy or GSL), these libraries are not always available or convenient.

Here is a really simple numerical approximation: it is numerically “good enough” for casual work, and simple enough to be implemented on the fly as needed.

I recently got interested in singular perturbation theory , and to get help with the algebra, I turned to SymPy. I had tried to use Mathematica in graduate school in the early 90s, but the experience had been sufficiently frustrating that I had steered clear of computer algebra systems since.

SymPy presents itself as a “friendly”, less intimidating alternative, with a more familiar and conventional language and operating model.

It turns out, Caltech has made the Feynman Lectures available online on their website.

Occasionally, I like to drink hot chocolate — that is, real hot chocolate, from pure cocoa powder and whole milk; not some (sugared, flavored) drink mix.

So, I was somewhat devastated when I learned that the venerable, time-honored Droste (those of the Droste Effect) seems to have gone tragically out of business.

Some replacement was required: at least as good as the original, and satisfying my various purist predilections.

I always thought that the “types” that are created using Go’s type keyword were, in some vague sense, new, unique, original; but in any case separate and distinct types. It turns out, this is a misunderstanding.