Our goal in this series is the solution of the hydrogen atom using the methods of supersymmetric quantum mechanics. Last time, we constructed the following picture of the procedure:
If the potential we wish to study satisfies a certain criterion, which we called “shape invariance,” we can construct a hierarchy of Hamiltonians, each missing the lowest-energy eigenstate of the last, and find the complete spectrum of the original Hamiltonian by “working leftward” in the state diagram. We shall see that with the hydrogen atom, each state in the diagram corresponds to a physical eigenstate of the system, but in order to get there, we have to turn the three-dimensional Coulomb potential of the hydrogen atom into the kind of problem we can study with the SUSY QM machinery we’ve built up so far. Two steps will be necessary to do this: first, moving to the center-of-mass reference frame, and second, separating the radial and angular dependencies. In this post, we’ll tackle the first of those two tasks.
While the SUSY part isn’t widely taught, these preliminary steps are more familiar. This brief note is based on Chapter VII of Cohen-Tannoudji, Diu and Laloë.
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Jorge Luis Borges once wrote, in a 1960 essay called “Mutations,”
In a hallway I saw a sign with an arrow pointing the way, and I was struck by the thought that that inoffensive symbol had once been a thing of iron, an inexorable, mortal projectile that had penetrated the flesh of men and lions and clouded the sun of Thermopylae and bequeathed to Harald Sigurdson, for all time, six feet of English earth.
This line came to mind when, a few years ago, I had to write a poem for a poetry workshop class. Looking back, it was the easiest course credit I ever got.
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Salvo Assenza, Jesus Gomez-Gardenes and Vito Latora say in their e-print, “Enhancement of cooperation in highly clustered scale-free networks” (arXiv:0801.2416),
We study the effect of clustering on the organization of cooperation, by analyzing the evolutionary dynamics of the Prisoner’s Dilemma on scale-free networks with a tunable value of clustering. We find that a high value of the clustering coefficient produces an overall enhancement of cooperation in the network, even for a very high temptation to defect. On the other hand, high clustering homogenizes the process of invasion of degree classes by defectors, decreasing the chances of survival of low densities of cooperator strategists in the network.
Suddenly, with regard to cooperation issues, I have gone from [tex]N[/tex] papers behind to [tex]N + 1[/tex] papers behind. Question: If we construct a time-dependent network model of a population, can we represent both kin recognition and the effect of spatial distribution by node clustering?
There are only 1.27 times as many straight people as gay people in the world, and most people aren’t either.
How do I know? Google says so. Just count the hits.
lesbian porn: 9,930,000
straight porn: 610,000
heterosexual porn: 360,000
homosexual porn: 284,000
Now that I’ve gotten Science After Sunclipse banned from every school room in America (oh, like it wasn’t already, me being an ill-tempered, illiterate evilutionist and all), you can just go about your business. Or you can hang around while Windy and I resolve the group selection controversy.
As luck would have it, my day job will require me to teach a mini-class on writing computer simulations for scientific purposes next week, and one of the examples I thought I’d include turned out to be just the thing I needed to provide an illustration for the review of John Allen Paulos’ Irreligion (2007) I’ve got in the works. Since the details are a little beyond the scope of that review, I might as well make another post out of them.
First, the punchline:
This is a fifty-frame animation of Conway’s Game of Life which begins with a 32×32 grid of random values and successively advances by applying the transition rules of that famous automaton. The purpose of the exercise is twofold: first, to demonstrate some basic programming techniques and show how much you can do in Python with half an hour of free time; and second, to see how persistent and cyclic features arise from random configurations.
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I wonder how many instances of the phrase “quantum leap” Scott Bakula is personally responsible for? There is, as many others have observed before me, a hefty dose of irony in calling a major transition a “quantum leap” or “quantum jump,” as the original leaps to gain the name were the transitions of electrons between energy levels. We’re talking about an electron’s “orbit” changing its diameter from one zillionth of a centimeter to four zillionths of a centimeter. But science never stands in the way of evocative, quasi-scientific jargon!
It’s old irony, but it’s still worth a chuckle, as when India’s prime minister declares, “We need a quantum jump in science education and research.” Start at the top, my friend, start at the top. OK, points for effort:
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It’s Monday. . . and is it ever a Monday.
Instead of a real post, here is another installment of “found poetry” — the search queries by which people have found their way to Science After Sunclipse:
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