Jason Underdown’s Blog

PASADENA, CA—Sequestered in a private booth at a Pasadena-area Cheesecake Factory for nearly 25 minutes, a party of eight California Institute of Technology physicists emerged exhausted but visibly excited Friday evening after successfully splitting the bill…

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Larry Cusick, a professor of mathematics at Cal-State Fresno has written a good introduction on how to write proofs. There are lots of great examples. I wish I had found this before I took my introductory analysis classes.

If you have ever wanted to learn some higher mathematics, but don’t want to slog through the long tedious proofs associated with calculus and analysis, then you might like this gem of a book by Edwin H. Connell. He is an emeritus mathematics professor at the University of Miami, and has graciously made his book available for download on the web.

I am currently taking a course in abstract algebra and thus not well qualified to review the book yet, but I have read the first two chapters and find that it presents the material succinctly and quite readably. It is definitely at the undergraduate level and probably accessible to ambitious high-school students as well. I also agree wholeheartedly with the following comment which comes from his web site.

The present situation with college textbooks is a national disgrace. Textbooks are too big and too expensive.

If you need a cheap origami CD case then try these folding instructions from Thomas Hull. But if you want to create a really nice mix CD for that special someone then use this web app which allows you to enter artist, title and track information.

The Hertz Foundation is accepting applications for Graduate Fellowships
with preference given to seniors and first year graduate students. The
amount awarded ranges from $28,000 to $33,000 per year. The application
deadline is October 27, 2006. For more information and application
forms, go to www.hertzfoundation.org.

According to Steve Yegge’s latest blog post, JavaScript is the most important language in the world today. After some arguments to convince you that his outrageous claim has real merit, he then goes on a rant about how much JavaScript sucks, and how he wishes it was more like Ruby on rails. Here is a great quote from his post:

Well, if you happen to be doing web programming, Ruby on Rails defies classical language mechanics by actually being a lower energy state. That’s right; it’s more lazy to learn Rails than it is to try to get your web framework to be that productive, so people are just tunneling over to it like so many electrons.

Excellent stuff!

This blog post has a short collection of very creative mathematics done by students. The last one is my favorite.

My younger brother Jeff asked me what a wave function is the other day. The following is my response.

What is a wave function? The short answer is that it is a probability amplitude, that also happens to solve Schrodinger’s equation, but I’m sure that won’t help you much.

When physicists started looking at really small things like atoms, they discovered that the rules of classical mechanics no longer apply. That is that dynamical variables like position, velocity, momentum, acceleration, etc. can no longer be computed via Newton’s laws. They found that these dynamical variables come in pairs. For example position and momentum form a pair, so do time and energy.

The really big idea in Quantum Mechanics is Heisenberg’s uncertainty principle which states that the more carefully you measure one variable the more uncertain you will be about its corresponding partner variable. Imagine you were to use very precise lasers to measure the position of an atom, well the more precisely you know the position, the less knowledge you will have of its momentum. This is exactly analogous to ordinary waves. If you create a standing wave on a rope for example, you know exactly its frequency and wavelength, but it doesn’t make sense to even ask where the wave is because it is everywhere. Now imagine sending a single wave pulse along a taut rope, in this case you will have a fairly good idea of its position, but measuring its frequency is now problematic. Waves inherently predict the uncertainty principle.

After Einstein proposed that light behaves like particles with definite or quantized energy (in his famous photoelectric effect paper of 1905), Louis de Broglie posited that perhaps matter can behave like a wave as well. This was the pivotal intuitive leap that paved the way for Schrodinger. He decided that if de Broglie was right then there must be some wave equation that governs matter waves just like there is a wave equation that governs light waves discovered by Maxwell). So he set about trying to figure it out by applying de Broglie’s famous relations (E=hbar*omega and p=hbar*k which relate dynamical quantities usually associated with matter to dynamical quantities related to waves) to Maxwell’s wave equation. He found that if he slightly modified it and allowed the matter waves to be complex-valued, then the equation could be solved. Complex-valued simply means that the function returns two values, or in other words a complex number of the form a+bi where i is the square root of -1.

After physicists had a new equation to play with, they found all sorts of solutions, but for a long time, no one knew how to interpret what the solutions of the equation actually meant physically. Max Born proposed that solutions of Schrodinger’s equation are probability amplitudes.

If you square a probability amplitude you get a probability density which can be used to predict the likelihood of getting a certain value when you measure that dynamical quantity. That’s very abstract so let me make it more concrete. Once you have psi (a function which solves Schrodinger’s equation), you can predict the likelihood that your quantum system will have energy E, or the likelihood that the particle will be found at position x.

William Chace, a former president of Emory and Wesleyan Universities, and the author of “100 Semesters: My Adventures as Student, Professor and University President, and What I Learned Along the Way” wrote an excellent op-ed piece for the New York Times.

Every Friday there is a show on NPR (radio) about various science topics. Last week Ira Flatow, the host of the show, talked to Sean Carroll, a cosmologist at CalTech about the recent discovery I mentioned in my last post. You can download and listen to the discussion from the Science Friday website.