Physics

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 rarely 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.

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.

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

Imagine a rod that is initially at temperature $T_1$ and then brought into an environment with a lower temperature $T_0 < T_1$. How quickly does the body cool down? When will it have reached the environment’s temperature? What is the temperature profile throughout the rod, as a function of time?

This is essentially a worked homework set: a complete, step-by-step solution of the diffusion (or heat) equation in one dimension.