Last time, we found that the problem of the hydrogen atom could be split into a radial part and an angular part. Thanks to spherical symmetry, the angular part could be studied using angular momentum operators and spherical harmonics. We found that the 3D behavior of the electron could be reinterpreted as a 1D wavefunction of a particle in an effective potential which was the two-body interaction potential plus a “barrier” term which depended upon the angular momentum quantum number. Today, we’re going to solve the radial part of the problem and thereby find the eigenstates and eigenenergies of the hydrogen atom.
The technique we’ll employ has a certain charm, because we solved the first part, the angular dependence, using commutator relations, while as we shall see, the radial dependence can be solved with anticommutator relations. Continue reading SUSY QM 5: The Hydrogen Atom→
A couple weekends ago, some friends and I watched Jurassic Park (1993) for the first time in mumble-mumble years, and the experience brought a few things to mind.
First of all, I remembered reading Michael Crichton’s novel, and while the helicopter was flying to the island, I pondered how the movie took us back to a more innocent age, when the world and we were young, while in retrospect the book seems to look forward to Crichton the global-warming denier, Crichton the anti-scientist, Crichton the general jerk. (Yes, looking forward in retrospect is an odd feeling.) What do I mean by “innocent age”? Well, consider: back then, dinosaurs were just cool because they were dinosaurs, but if Jurassic Park came out today, in Year Two After Dover, the Young-Earth Creationists would bitch and moan on national TV that the movie accepts a 65-million-year gap between dinosaurs and people, and of course, the national media would give them equal time with the real scientists. Meanwhile, over at the Discoverup Institute, we’d have Dembski, Egnor and Behe saying, “Cloning dinosaurs from DNA is really Intelligent Design,” to which O’Leary and Cordova would add, “Yes, and Charles Darwin was a flatulent puppy-killer who didn’t know algebra and who was directly responsible for the Holocaust. Uh-huh. Jesus came to me and said that if I said potty-mouth things about Darwin, I’d get a pony in Heaven.” Continue reading You Bet Jurassic’s A Pun→
Mrs. Bad Astronomer has a guest post on her husband’s weblog, pondering why the press typically refers to Hillary Clinton by her first name. This called to my mind a related question in astronomy, appropriately enough, which I’d like to toss out as thought-food:
Two of the greatest figures in the Scientific Revolution are Johannes Kepler and Tycho Brahe. The former was among the greatest mathematicians of his day, and the latter had comparable prowess with astronomical observations. And, for some reason, everybody calls the first one “Kepler,” while the second is known as “Tycho.” Look on a map of the moon: there’s a crater called Tycho, where a monolith was found nine years ago, and one called Kepler (the names for these and other craters were doled out by an Italian astronomer, Giambattista Riccioli, in 1651, who was a big fan of Tycho and attached his name to the most prominent crater). Both men invented models for how the solar system worked; nowadays, historians of science speak of the “Tychonic universe,” which had the Sun going round the Earth but other stuff going around the Sun, and everybody still talks about Kepler’s Three Laws.
Is it just that “Johannes” is too generic, while “Brahe” sits roughly on the tongue?
Given their tempestuous working relationship — Kepler and Tycho were not the sort of personalities which could get along — I’m sure the celebrity magazines of their day would have blessed them with a joined name, something like “Jo-cho” (sounds like Yoko), if those magazines had only existed.
I was busy with something or other, so I didn’t get to see the event Dennis Overbye describes in the New York Times, where the director and the star of the new film Jumper chatted with MIT professors Ed Farhi and Max Tegmark before a live audience in lecture hall 26-100. Not having been there, I don’t have very much to say, but I do feel the need to quote one paragraph of Overbye’s article and add just a tiny bit of emphasis:
The real lure, [Farhi] said, is not transportation, but secure communication. If anybody eavesdrops on the teleportation signal, the whole thing doesnâ€™t work, Dr. Farhi said. Another use is in quantum computing, which would exploit the ability of quantum bits of information to have different values, both one and zero, at the same time to perform certain calculations, like factoring large prime numbers, much faster than ordinary computers.
In any other circumstance, I’d probably pontificate on how exponential parallelism is not the source of quantum computing’s calculation-fu, but I think we have a few other points to address first. . . .
Attentive readers will recall that Dr. Farhi was the guy who signed my paperwork when I was an undergraduate. For an amusing story from those days, see my post of last November, “Pay No Attention to the Man.”
Today, on a very special episode of Science After Sunclipse, Mary Sue discovers that for a quantum-mechanical system with a central potential, the eigenfunctions of the Hamiltonian can be separated into radial and angular factors, and the angular dependence can be understood using angular momentum operators. Continue reading SUSY QM 4: Separation of Variables→
I’m sure I’ll find something broken amongst my plugins and customizations, but to a first approximation, Science After Sunclipse appears successfully migrated to its new home. To celebrate, let’s have ourselves some Wilkins Coffee!
When the bus is stuck in traffic, or when I’m curled restlessly in bed during the darkest hours of morning when sleep will not come and all the old wounds on my heart ache like they were newly made, I sometimes think back over my education and add up all the time my teachers wasted. The worst, perhaps, was the mathematics, for which entire years of schooling went for naught. “Pre-Calculus”? Pfft. AP Computer Science? Pfft++. All in all, I’d say that upwards of a third of my mathematics schooling before university was a waste of time, and another third was so incompetently done that any student who hadn’t already been hooked on science and learning would have been completely sunk.
So, I find it easy to sympathize with people who say that math education needs a severe overhaul. I’m willing to contemplate big curriculum changes, but of course, you have to convince me that the specific changes you have in mind will actually do any good. When a proposal comes down the wire to eliminate fractions, I reserve the right to chortle and guffaw. Continue reading Fractions→
"no matter how gifted, you alone cannot change the world"