Category Archives: Quantum mechanics

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:
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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

[tex]\{A,B\} = AB – BA.[/tex]

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.
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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.
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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:
Panda reading Feynman, vol. 3
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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.
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Quantum Mechanics Homework #1

Well, in the past two days I’ve linked to an Internet quiz and some anime videos, so in order to retain my street cred in the Faculty Lounge, it’s time to post a homework assignment. Don’t worry: if you haven’t met me in person, there’s no way I can grade you on it (unless our quantum states are somehow entangled). This problem set covers everything in our first two seminar sessions on QM, except for the kaon physics which we did as a lead-up to our next topic, Bell’s Inequality. I’ve chosen six problems, arranged in roughly increasing order of difficulty. The first two are on commutator relations, the third involves position- and momentum-space wavefunctions, the fourth brings on the harmonic oscillator (with some statistical mechanics), the fifth tests your knowledge about the Heisenberg picture, and the sixth gets into the time evolution of two-state systems.

Without extra ado, then, I give you Quantum Mechanics Homework #1.
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Michael Egnor: 2400 Years Behind the Times

BPSDBI nearly sprayed my breakfast across my friend’s new flatscreen monitor when I saw the latest from Michael Egnor:

Clearly the brain, as a material substance, causes movement of the body, which is also a material substance. The links are nerves and muscles. But there is no material link between our ideas and our brains, because ideas aren’t material.

Mr. Spock, are your sensors detecting any signs of intelligent life?
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PZ Myers on the Quantum Mind

This one is worth repeating:

Quantum effects in microtubules are going to be inconsequential relative to ion fluxes and chemical changes in membrane properties and channels, and there is no explained mechanism to regulate quantum effects. It’s like trying to explain the tides by speculating about the dabbling of gnats in estuaries.

The people who talk about this stuff usually seem to have absolutely no knowledge of neuroscience.

For sheer disdain, this might be second only to Patricia S. Churchland‘s remark, “The want of directly relevant data is frustrating enough, but the explanatory vacuum is catastrophic. Pixie dust in the synapses is about as explanatorily powerful as quantum coherence in the microtubules.”

My most extensive essay on this subject can be found here, and a good paper with many references into the literature is A. Litt et al.,Is the Brain a Quantum Computer?” (Cognitive Science, 2006).

Friday Quantum Mechanics

“So, Blake,” I sez to myself. “You’ve been selected for multiple editions of the Skeptic’s Circle. You’ve been linked, twice, from Pharyngula. Clearly, you’re rising to astonishing heights of science-blogebrity. What worlds are left to conquer?”

“Well,” I replied. “There’s going out for a milkshake with Rebecca Watson.”

I shook my head. “Not gonna happen — she’s just too picky counting tentacles. Anything else?”

“Well, you could do what Revere warned you not to do.”

“Ah, yes, write a sixteen-part series on mathematical modeling! But the modeling of antiviral resistance isn’t really my field.”

“True, but didn’t you spend your spring break in Amsterdam a few years ago, writing that paper which was the first article Prof. Rajagopal ever graded with an A-double-plus?”

“Hey, yeah, on supersymmetric quantum mechanics and the Dirac Equation!”

“So,” I suggested to me, “why don’t you break that paper down into several blag posts, interleave it with some Bill Hicks videos so not all your readers wander away, and have yourself a continuing physics series?”

“Could work, I suppose. But that paper was written for third-term quantum mechanics students, so I’d probably have to build up to it, even just a little.”

“Bah,” I said. “At least you’ll have a purpose in life. And you can start by expounding on the canonical commutation relation for position and momentum. That’ll be your warm-up, after which you can do angular momentum and central potentials —”

“Which I do have written up somewhere,” I interposed, “since I discovered I could type LaTeX as fast as my professors could lecture.”

“Weirdo,” I said.
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Category Theory on the Wobosphere

Our seminar series might or might not be getting into category theory in the coming months. (We’re already drawing diagrams and showing that they commute; not everybody knows it yet!) To facilitate this process should we ever go in that direction, and to provide a general public service, I’m compiling a list of useful category-theory resources extant on the Wobosphere. My selection will be pedagogically oriented, rather than emphasizing the latest research; I’d like to collect reading material which could plausibly be presented to advanced undergraduate or beginning graduate students in, say, their first semester of encountering the subject. I’ll be both happy and eager to update this list with any beneficial suggestions the Gentle Readers have to offer.
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Comment Policy

I walked away to give my lecture on quantum mechanics, and I came back to find a brief, affronted note from a creationist.

You have to understand how upsetting I found such a transition. I love lecturing. I’ve got thespian blood — my grandmother performed with Orson Welles’ players — and every trip to the blackboard is a chance to shine. What’s more, I was speaking to people who had a strong math background, so I could employ matrices, commutators and other linear algebra trickery without fear. My lecture, part of our effort to get the math people up to relativistic speed with the physics we want to study, started with the canonical commutation relations between position and momentum, derived the form of the momentum operator in coordinate space, and solved for the position representation of momentum eigenstates. I then covered the particle-in-a-box and the simple harmonic oscillator, after which I did a little kaon physics to lead up to Bell’s Inequality, which we will discuss next time.

And after all that fun, I had to come back to my laptop and read indignant creationist snark. I considered my response during the walk home, and after due contemplation, I decided to embrace Scott Aaronson’s comment policy:
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Perakh on Barr: Rejoicing in Materialism

Via the Panda’s Thumb comes notice of Mark Perakh’s review of Stephen M. Barr’s Modern Physics and Ancient Faith (2003). I recommend reading the whole review; Perakh demonstrates that Barr’s book, like Ken Miller’s Finding Darwin’s God (1999), offers some lucid descriptions of modern science but devolves into poor reasoning and non sequiturs when it touches notions of faith.

None of Barr’s arguments or Perakh’s counter-arguments are particularly new (which is one sign of how decrepit a business this “natural theology” really is). Barr organizes his book by describing successive “plot twists,” discoveries which supposedly upset the tidy materialism of a century ago. You could guess that quantum mechanics figures prominently; a couple linear operators fail to commute, and people run around saying reality’s been undone. Kurt Gödel also makes an appearance:
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Polchinski on Smolin on Polchinski on Smolin

Over at Cosmic Variance, Sean Carroll has just posted a guest essay by Joe Polchinski replying to Lee Smolin’s response to Polchinski’s review of Smolin’s book. I managed to snag the first comment spot; I predict that Peter Woit will show up within ten. With any luck, the ensuing comments will contain much good talk about physics, though the signal-to-noise ratio is a perennial problem. (Even Sean admits that he doesn’t read every comment.)

I would like to skim past several details of the physics and pull out for special consideration a passage of Polchinski’s which concerns, ironically, what happens when you think in text instead of physics:

This process of translation of an idea from words to calculation will be familiar to any theoretical physicist. It is often the hardest part of a problem, and the point where the greatest creativity enters. Many word-ideas die quickly at this point, or are transmuted or sharpened. Had you applied it to your word-ideas, you would probably have quickly recognized their falsehood. Further, over-reliance on the imprecise language of words is surely correlated with the tendency to confuse scientific arguments with sociological ones.

Polchinski is speaking about the standards one must maintain while doing science, but similar concerns apply to the process of explaining science. Of course, the latter process is one ingredient in the former, but we often think of “popularizing” (or vulgarisation if we want to be Gallic) as a distinct enterprise from communicating with fellow researchers and educating the next round of students. John Armstrong’s recent post on this topic addresses the same question from the opposite direction: according to Polchinski, going from words to equations is the hard part of getting work done, while Armstrong points out that when “vulgarizing” the science, that’s the very step we omit!

Armstrong amplified his point in the comments here at Sunclipse:

Roger Penrose noted specifically in his introduction to The Road To Reality that modern physics is no longer accessible to anyone — specialists included — except through the mathematics. We understand quantum field theory as well as we do because we understand the mathematics. To avoid the mathematics in its entirely [sic] cuts the legs out from under any popularization of physics, and risks becoming The Tao of Physics or The Dancing Wu Li Masters.

Continue reading Polchinski on Smolin on Polchinski on Smolin

The Unapologetic Mathematician on Popularization

John Armstrong has a short and sweet post on popularizing mathematics and physics which is worth reading in its entirety. I don’t want to quote the whole thing, so just go read it.

Incidentally, he says he was inspired to write the post by the movie Mindwalk (1990). I had only ever heard of this flick because they’d stuck a preview for it on the Brief History of Time video I rented sometime in the late nineties. I then managed to forget about it until a few weeks ago, when I was poking through Wikipedia for articles containing pseudoscience. Somehow, in the tangled thicket of pages growing like weeds upon quantum mysticism and Choprawoo, I found Mindwalk. “Aha! I remember seeing a preview for that movie.”