This fall I’m teaching on quantum mechanics for mathematicians, at the undergraduate level. There’s a web-page with more information here. I’ll be writing up lecture notes, which should appear on that web-page as the course goes on, starting Wednesday.
We’ll see how this works, but the plan is to teach many of the standard topics, although starting from a different point. Most quantum mechanics classes start out with classical mechanics, then somehow try and motivate quantum mechanics from there, following the historical logic of the subject. I’ll instead start with the simplest purely quantum systems, especially the two-state, spin-1/2 system, now famous as the “qubit” of quantum computation. This is also a central example for the theory of Lie groups, Lie algebras and representations, so something that every mathematician should become familiar with. Another advantage of starting here is that there’s no analysis, just linear algebra, and one can easily do everything rigorously.
Later on in the course I’ll get to the standard material about wave-functions and quantum particles in potentials. The emphasis will be though not on the analytical machinery needed as a rigorous foundation for this subject in general, but on specific problems and their symmetries, and the use of these symmetries to do real calculations, ending up with the spectrum of the hydrogen atom.
We’ll see how this goes, and what the students think of it. As lecture notes appear, corrections and suggestions of how to improve them would be appreciated.