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I’m continuing my series of articles for the Princeton Companion to Mathematics with my article on the Schrödinger equation – the fundamental equation of motion of quantum particles, possibly in the presence of an external field. My focus here is on the relationship between the Schrödinger equation of motion for wave functions (and the closely related Heisenberg equation of motion for quantum observables), and Hamilton’s equations of motion for classical particles (and the closely related Poisson equation of motion for classical observables). There is also some brief material on semiclassical analysis, scattering theory, and spectral theory, though with only a little more than 5 pages to work with in all, I could not devote much detail to these topics. (In particular, nonlinear Schrödinger equations, a favourite topic of mine, are not covered at all.)
As I said before, I will try to link to at least one other PCM article in every post in this series. Today I would like to highlight Madhu Sudan‘s delightful article on information and coding theory, “Reliable transmission of information“.
[Update, Oct 3: typos corrected.]
[Update, Oct 9: more typos corrected.]
I’d like to begin today by welcoming Timothy Gowers to the mathematics blogging community; Tim’s blog will also double as the “official” blog for the Princeton Companion to Mathematics, as indicated by his first post
which also contains links to further material (such as sample articles) on the Companion. Tim is already thinking beyond the blog medium, though, as you can see in his second post…
Anyway, this gives me an excuse to continue my own series of PCM articles. Some years back, Tim asked me to write a longer article on harmonic analysis – the quantitative study of oscillation, transforms, and other features of functions and sets on domains. At the time I did not fully understand the theme of the Companion, and wrote a rather detailed and technical survey of the subject, which turned out to be totally unsuitable for the Companion. I then went back and rewrote the article from scratch, leading to this article, which (modulo some further editing) is close to what will actually appear. (These two articles were already available on my web site, but not in a particularly prominent manner.) So, as you can see, the articles in the Companion are not exactly at the same level as the expository survey articles one sees published in journals.
I should also mention that some other authors for the Companion have put their articles on-line. For instance, Alain Connes‘ PCM article “Advice for the beginner“, aimed at graduate students just starting out in research mathematics, was in fact already linked to on one of the pages of this blog. I’ll try to point out links to other PCM articles in future posts in this series.
For the past several years, my good friend and fellow Medalist Timothy Gowers has been devoting an enormous amount of effort towards editing (with the help of June Barrow-Green and Imre Leader) a forthcoming book, the Princeton Companion to Mathematics. This immense project is somewhat difficult to explain succinctly; a zeroth order approximation would be that it is an “Encyclopedia of mathematics”, but the aim is not to be a comprehensive technical reference, nor is it a repository of key mathematical definitions and theorems; it is neither Scholarpedia nor Wikipedia. Instead, the idea is to give a flavour of a subject or mathematical concept by means of motivating examples, questions, and so forth, somewhat analogous to the “What is a …?” series in the Notices of the AMS. Ideally, any interested reader with a basic mathematics undergraduate education could use this book to get a rough handle on what (say) Ricci flow is and why it is useful, or what questions mathematicians are trying to answer about (say) harmonic analysis, without getting into the technical details (which are abundantly available elsewhere). There are contributions from many, many mathematicians on a wide range of topics, from symplectic geometry to modular forms to the history and influence of mathematics; I myself have contributed or been otherwise involved in about a dozen articles.
If all goes well, the Companion should be finalised later this year and available around March 2008. As a sort of “advertising campaign” for this project (and with Tim’s approval), I plan to gradually release to this blog (at the rate of one or two a month) the various articles I contributed for the project over the last few years. [As part of this advertising, I might add that the Companion can already be pre-ordered on Amazon.]
I’ll inaugurate this series with my article on “wave maps”. This describes, in general terms, what a wave map is (it is a mathematical model for a vibrating membrane in a manifold), and its relationship with other concepts such as harmonic maps and general relativity. As it turns out, for editing reasons various articles solicited for the Companion had to be removed from the print edition (but possibly may survive in the on-line version of the Companion, though details are unclear at this point); Tim and I agreed that as wave maps were not the most crucial geometric PDE that needed to be covered for the Companion (there are other articles on the Einstein equations and Ricci flow, for instance), that this particular article would end up being one of the “deleted scenes”. As such, it seems like a logical choice for the first article to release here.