# SUSY QM 1: Superalgebra

A couple months ago, I stumbled across an amusing bit of academic woo: “Quantum Mind and Social Science.” The misrepresentations, false dichotomies and nons sequitur of that piece prompted me to wonder what a good litmus test for knowing quantum mechanics might look like. Joshua offered a simple criterion: be able to pick the Schrödinger Equation out of a line-up. At a slightly higher level, I suggested being able to describe in the Heisenberg picture the time evolution of a harmonic oscillator coherent state, and explaining why states of the hydrogen atom with the same n but different angular momentum number l are degenerate. You can’t discuss the relationship between classical and quantum physics without bringing up coherent states eventually, and a good grounding in the basics should include the Schrödinger and Heisenberg pictures. (That’s why I wrote problem 5 in this homework assignment.)

The excited states of the hydrogen atom are our prototype for understanding how the periodic table works, and it’s often the first place one runs into the mathematics of angular momentum. Unfortunately, too many standard treatments of introductory QM say that hydrogen has “accidental degeneracies”: these states have the same energies as those states for no spectacularly interesting reason. But we are trained to associate degeneracies with symmetries — when two sets of eigenstates have the same eigenvalues, we expect some symmetry to be at work. So, is there a symmetry in the hydrogen atom above and beyond the familiar rotational kind, a symmetry which They haven’t been telling us about?

I’d like to explore this topic over a few posts. First, I’ll build up some very general machinery for solving problems, and then I’ll apply those techniques to the hydrogen atom; by that point, we should have a fair amount of knowledge with which we can move in any one of several interesting directions. To begin, let’s familiarize ourselves with the behavior of a superalgebra.
Continue reading SUSY QM 1: Superalgebra

# A Not-SchrÃ¶dinger Joke?

Today’s xkcd puzzles me.

It’s a cat in a box, but it’s not a SchrÃ¶dinger joke? Dood! UR IN MY HEAD, MESSIN WIT MAH MEEMZ. And, judging by the comments, I’m not the only one to be suffering in perplexity this morning.

Let’s just say that if I were expecting an office chair and opened the box to discover a live SchrÃ¶dinger bobcat, my hopes would collapse, and my ensuing comments would be rather decoherent.

# Scott Aaronson’s Outreach Program

What’s that sound? Could it be the information content of the popular discourse about quantum physics rising by a detectable degree?

Oh, dear. An advertising agency took a passage from Scott Aaronson’s quantum computing lecture notes and used it in a commercial.

Model 1: But if quantum mechanics isnâ€™t physics in the usual sense â€” if itâ€™s not about matter, or energy, or waves â€” then what is it about?

Model 2: Well, from my perspective, itâ€™s about information, probabilities, and observables, and how they relate to each other.

Model 1: Thatâ€™s interesting!

Naturally, the commercial doesn’t mention where they got the text from, but thanks to the quintessential Internet appeal of this story, it looks like Prof. Aaronson is already getting some server-crashing free publicity out of this. Should he be demanding more? Like, say, private tutoring sessions with the fashion models who show such an interest in quantum theory? Each of them have already shown a far greater command of the material than, for example, Deepak Chopra, and it would be a shame to waste such an opportunity.

You can read the ad agency’s craven response in this Sydney Morning Herald article:
Continue reading Scott Aaronson’s Outreach Program

# More on New Scientist

I felt sort of bad saying all that stuff about Wired when the guy who wrote the piece I did like showed up to say “Thanks for the link.” But hey, I’m not going to stop criticizing bad science reporting, nor can I imagine shutting myself up about the practices which I think cause bad science journalism. (Nor do I have the vanity to think that by myself, I’ll make any difference.) I’d feel considerably more uncomfortable if Greg Egan didn’t go and provide a whole new plateful of reasons to be upset with pop science.

Egan has been masochistically plowing through New Scientist ever since the EmDrive incident, when he had found himself “gobsmacked by the level of scientific illiteracy” the magazine had put on display. Now, commenting at The n-Category Café, he gives two additional recent “absurdities.”
Continue reading More on New Scientist

# Non-materialist Neuroscience

Trent Toulouse writes to say that RationalWiki has finished switching its service providers, so that it is once again available at rationalwiki.com. I had previously pointed to their growing article on non-materialist neuroscience, the domain of Michael Egnor and Denyse O’Leary. The article is clearly a work in progress, but it’s already a good read.

I added some stuff on the abuse of quantum mechanics, too.

In the brief interlude between my morning of debugging PHP code — Semantic MediaWiki isn’t compatible with Cite.php, the bastards! — and my afternoon of category theory, I’d like to call attention to a few items.

First, an observation: for some reason I can’t quite fathom, I was able to adapt myself to using HTML entities for punctuation marks, writing &mdash; for — and the like, but my brain didn’t process the fact that HTML entities also exist for accented letters. Instead of typing, say, &agrave; to get à, I would hit Ctrl+T to open a new Firefox tab, hit the Tab key to move to the Search bar, type a French phrase which I knew had the accented characters in question, copy the characters I needed from the search-result summaries, and paste them where I needed them.

Searching was easier than typing. Now, that’s either a sign of advanced Internet-induced brain rot, or an indication that our interconnected world has definitively left TwenCen far behind.

OK, it could be both.

Next, interesting items recently spotted on the Weboblagospherenet:

# Dangerous Ideas

I must admit that when I hear somebody talking about “dangerous ideas,” one of my eyebrows will — without voluntary intervention on my part — lift upwards, Spock-style. Such talk invariably reminds me of my old film-studies professor, David Thorburn, who said, paraphrasing the acerbic Gerald Graff, “if the self-preening metaphors of peril, subversion and ideological danger in the literary theorists’ account of their work were taken seriously, their insurance costs would match those for firefighters, Grand Prix drivers and war correspondents.”

Still, when Bee at Backreaction says something is interesting, I take a look. Today’s topic is the Edge annual question for 2006, “What is your Dangerous Idea?” Up goes the eyebrow. I don’t want to go near the Susskind/Greene spat about “anthropic” reasoning; frankly, without technical details far beyond the level of an Edge essay, “anthropic” talk rapidly devolves into inanities which resemble the assertion, “Hitler had to lose the war, because otherwise we wouldn’t be sitting around talking about why Hitler lost the war.” Suffice to say that neither Susskind nor Greene mentions NP-complete problems or proton decay.

So, moving on, let’s get to what Bee calls “the more bizarre pieces.” I was particularly drawn to and repelled from (yeah, it was a weird feeling) the essays of Rupert Sheldrake and Rudy Rucker. The latter goes off about “panpsychism,” which sounds like a fantastic opportunity to ramble about quantum mechanics, the inner lives of seashells and the dictionary of Humpty Dumpty, in which words mean exactly what the speaker wants them to mean, reason and usage notwithstanding.

Hey, “consciousness” is just one tiny part of what living things do, and life is a teensy fraction of what the Universe does. Why not give the rest of the biosphere a little attention and support “panphotosynthesism” instead?

# Behe on The Colbert Report

Last night, Michael Behe was Stephen Colbert’s guest on The Colbert Report. It was, shall we say, educational.

BEHE: Nobody was searching for the limits of Newton’s theory when Newton first proposed it. He thought that he had solved all of physics. But then when —

COLBERT: You mean about how — how apples fall?

BEHE: Apples fall, cannonballs go. But then —

COLBERT: Mm-hmmm.

BEHE: But then when —

COLBERT: He invented the cannonball? He invented the dive — the cannonball?

[audience laughs]

BEHE: Cannonballs fly.

Oh, yes. It’s nice to know that nobody checked to see if Newton was right, or if “universal gravitation” was really universal.

Wait. You say that it was Edmund Halley who used Newton’s laws to predict that comets travel in elliptical orbits, and that the comet seen in 1456, 1531, 1607 and 1682 would return in 1758? How could Halley say such a thing, after Newton had made his view clear that all comets travel in parabolic paths? It’s in the Principia, for Heaven’s sake! And you say that Halley was the one who realized that the stars are not fixed to a “celestial firmament” but instead move through space? How dare you imply that the views of one person are not the entirety of science! Sir, how dare you have the temerity to insist that people did not take Newton at his word but instead used his theories to make predictions about the world which they could then compare to observations to — I can hardly even articulate such a heretical notion — see if Newton was wrong.

What! Are you telling me it was the French, those wine-swilling, toad-munching surrender monkeys, who had the audacity to test Newton’s prediction that the Earth is an oblate spheroid? Sir, you could tell me all you want about the 1735 expeditions to Peru and Lapland under Charles-Marie de La Condamine and Pierre-Louis Moreau de Maupertuis respectively — the former of which incidentally brought back the first rubber and curare Europe had ever seen — but the mere suggestion that Newton’s word was not good enough is so repugnant I refuse to consider the matter further.

It gets better:
Continue reading Behe on The Colbert Report

# Late Enough at Night

Late enough at night, category diagrams start looking like adinkras and network motifs.

# Overbye on Hunting the Higgs

Dennis Overbye has an article in today’s New York Times on the search for the Higgs boson, and naturally, I’ve got complaints about it. It’s a pretty good piece: Overbye can do solid work (he went a little overboard looking for journalistic “balance” in the Bogdanov Affair, but that was a while ago). Still, I wouldn’t be myself if I couldn’t gripe and grouse.

First, I’m definitely not alone in asking people to please stop saying “God particle.” Leon Lederman has a great deal to answer for after coining this term; I’ve never heard or seen physicists use it seriously, and it keeps inviting unwarranted metaphors. (Incidentally, there was once detected an “Oh-My-God Particle,” a cosmic-ray proton of astonishingly high energy; for recent developments in this ultra-high-energy regime, see here. Physicists joke about the term, but they don’t use it.)

Second, this part rubs me the wrong way:
Continue reading Overbye on Hunting the Higgs

# Einstein Summation and Levi-Civita Symbols

PUBLIC SERVICE ANNOUNCEMENT: if any of you saw me wearing black corduroy pants and a purple T-shirt emblazoned with a picture of my friend Mike wearing a squid on his head, yes, it was laundry day. Rest assured, the reality disruption was only temporary, and normal service should be resumed shortly.

Now, to the business of the day. Earlier, we took a look at rotations and found a way to summarize their behavior using commutator relations. Recall that the commutator of A and B is defined to be

$\{A,B\} = AB – BA.$

For real or complex numbers, the commutator vanishes, but as we saw, the commutators of matrices can be non-zero and quite interesting. We recognized that this would have to be the case, since we used matrices to describe rotations in three-dimensional space, and rotations about different axes in 3D do not commute. Looking at very small rotations, we also found that the commutators of rotation generators were tied up together in a way which involved cyclic permutations. Today, we’ll express this discovery more neatly, using the Einstein summation convention and a mathematical object called the Levi-Civita tensor.
Continue reading Einstein Summation and Levi-Civita Symbols

# Rotation Matrices

My group theory teacher, Prof. Daniel Freedman, had some interesting professorial habits. When invoking some bit of background knowledge with which we were all supposed to have been familiar, he would say, “As you learned in high school. . . .” Typically, this would make a lecture sound a bit like the following:

“To finish the proof, note that we’re taking the trace of a product of matrices. As you learned in high school, the trace is invariant under cyclic permutations. . . .”

Prof. Freedman also said “seventeen” for “zero” from time to time. After working out a long series of mathematical expressions on the blackboard, showing that this and that cancel so that the overall result should be nothing, with the students alternating their glances between the board and their notes, he would complete the equation and proclaim, “Equals seventeen!” At which point, all the students look up and wonder, momentarily, what they just missed.

“Here, we’re summing over the indices of an antisymmetric tensor, so by exchanging i and j here and relabeling there, we can show that the quantity has to equal the negative of itself. The contraction of the tensor is therefore, as you learned in high school — seventeen!”

One day, I managed to best his line. I realized that the formula currently on the board had to work out to one, not zero, so when he wrote the equals sign, paused and turned to the class with an inquiring eye, I quickly raised my hand and said, “Eighteen!”

Incidentally, truly simple topics like Euler’s formula and trigonometric identities were supposed to have been learned in middle or elementary school.

Today, we’ll talk about one of the things Prof. Freedman said we should have covered in high school: the rotation matrices for two- and three-dimensional rotations. This will give us the quantitative, symbolic tools necessary to talk about commutativy and non-commutativity, the topic we explored in an earlier post.

# Rotation and Commutation

Today we will advance our coverage toward quantum mechanics by looking at an unusual feature of daily life. We’ll be looking at an aspect of the world which doesn’t quite behave as expected; though it won’t be as counterintuitive as, say, the Heisenberg uncertainty relations, it does tend to make people blink a few times and say, “That’s not — well, I guess it is right.” Furthermore, poking into this area will motivate the development of some mathematical tools which will remarkably simplify our study of symmetry in quantum physics.

Fortunately, then, I found an assistant to help me with the demonstrations. Please welcome my fellow physics enthusiast, here on an academic scholarship after a rough-and-tumble life in Bear City:

# Feynman on Quantum Mechanics

[DELETED; SEE BELOW]

[The video previously referenced here, one of Richard Feynman’s Messenger Lectures, is no longer available due to copyright concerns. I should make perfectly clear that I’ve never had a copy of this Feynman video or any other on my server; I found it one day during a bit of idle Google-searching, and the film to which I linked was stored on Google’s servers. Offhand, I don’t even know how to make a video stored on my own computer play in a nice little box.]

# Quick Calculation: Trig Identities

In my next quantum mechanics post, I’ll be talking about rotation matrices. My derivation of these mathematical objects will use some equations from trigonometry, the addition and subtraction formulas for sines and cosines. These are the sort of things one finds on the inside front cover of a trigonometry textbook, so if you’re not curious where anything comes from, that would satisfy you; however, if that’s what you find satisfactory, there’s precious little point waking up in the morning, so I’d like to give a little back story.

The addition and subtraction formulas give you the sine and cosine of the sum (or difference) of two angles, provided you know the sines and cosines of the angles themselves. Geometry tells us the sine and cosine of 45 degrees, by looking at an equilateral right triangle (whose internal angles are 45, 45 and 90 degrees). By looking at a 30-60-90 triangle, we can get the sines and cosines of 30 and 60 degrees. With all this information in hand, we’d like to get the sine and cosine of, say, 60 – 45 = 15 degrees, or 60 + 15 = 75 degrees.

One can extract these formulas out of a geometric argument, in the fashion of Euclid, but geometric arguments (while they lend themselves to spiffy pictures) tend to involve a certain amount of chicanery. One must find the proper “construction lines,” inscribe and circumscribe the correct circles and so forth. If one sees a geometric proof and, six months later, wishes to recover the result, remembering the necessary diagrams and manipulations can be quite the challenge.

I say “one must find” and “if one sees,” but really, this is me we’re talking about: I can see the proof, and I’ll remember that the final answer involves sine of this and cosine of that, but I’ve learned better than to trust my memory at getting all the plus and minus signs in the right places. (Talking to other people with college degrees in physics and math makes me suspect I’m not alone.) So, to contribute to the general welfare of the world, I’m going to go through the process I run through every time I need to use the addition and subtraction formulas. I’ve got it down to about fifteen seconds of pencil work, which I can do in the margin of my notebook, and I get all the damn minus signs in the right place.
Continue reading Quick Calculation: Trig Identities