Category Archives: Quantum mechanics

Quantum mechanics

Quick Calculation: Trig Identities

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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, Seminar Announcements, Seminar Summaries, Statistical mechanics

Quantum Mechanics Homework #1

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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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BPSDB, Electromagnetism, Favorites, Neuroscience, Pseudoscience, Quantum mechanics, Video

Michael Egnor: 2400 Years Behind the Times

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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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Neuroscience, Pseudoscience, Quantum mechanics

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).

Quantum mechanics, Seminar Announcements, Seminar Summaries

Friday Quantum Mechanics

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“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, Quantum mechanics, String Theory, Topology

Category Theory on the Wobosphere

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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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About this Blag, Quantum mechanics

Comment Policy

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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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Neuroscience, Pseudoscience, Quantum mechanics

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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Favorites, Popularization, Quantum mechanics, String Theory

Polchinski on Smolin on Polchinski on Smolin

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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 →

Popularization, Pseudoscience, Quantum mechanics

The Unapologetic Mathematician on Popularization

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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.”

Astronomy, Cosmology, Popularization, Quantum mechanics, String Theory

A Quarter of Everything

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I have just two quibbles with this New Yorker article on the Large Hadron Collider to which Scott Aaronson directed my attention. First, throughout her informative story, Elizabeth Kolbert consistently abbreviates “Large Hadron Collider” as “L.H.C.” I’ve yet to see anybody in the physics community use the periods when they write “LHC.” Is this some official policy which we, the project’s website and the rest of the Internet just are too lazy to follow? Or are these strange little dots the product of a New Yorker in-house style guide demanding their presence based on some holier-than-Sinai rule about tiny ink specks? If the latter is the case, the foolish prescriptivist responsible needs an introduction to my friend the clue-by-four.

My second and marginally more serious complaint involves the following passage from page five:
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Bad Math, Neuroscience, Pseudoscience, Quantum mechanics

Recycled Rant: the Quantum Mind

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This is a topic which seems to come up with depressing regularity, so I figure I should put my stance on the record. The bulk of this post is recycled from a Memoirs of a Skepchick comment thread.

For various reasons, I am extremely skeptical of the notion that consciousness could be rooted in quantum phenomena. Of course, the entire world is quantum, in a sense: it’s the principles of quantum mechanics which determine the properties of materials out of which the world is made. Like Democritus of Abdera said twenty-five hundred years ago, “Nothing exists save atoms and the void”, and quantum physics constitutes the rules by which atoms play.

The challenge, then, is not to say “all is quantum” (a statement with no more content, by itself, than saying “all is love”). In what way do the strange and esoteric mathematical descriptions of the atomic and sub-atomic world build up the everyday stuff with which we are so familiar? This is a deep problem, one with many mysteries left to resolve, and physicists spend lots of time worrying about it. One thing which we do know is that when you put a lot of quantum particles together, at a certain point they stop acting in the quantum way and become better approximated by Newton’s laws of classical mechanics. This is odd, because if you put a pile of classical pieces together, you get a bigger classical object! Newton’s laws reproduce themselves at higher scales, but the quantum laws do not.

It’s a bit like discovering that all the ordinary houses on your ordinary street are made of bricks from Faerie.
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Evolution, Quantum mechanics

Favorite Paper Title of the Day

J. D. Bashford, I. Tsohantjis, P. D. Jarvis (3 February 1998). “A supersymmetric model for the evolution of the genetic code” PNAS 95 (3): 987–992.

The application of group theoretical methods to spectroscopy is today part of the accepted set of techniques for the analysis of many body systems in physics. In the present paper, we consider models for the symmetries of the genetic code by using the classical Lie superalgebras. We claim that, beyond the physical language of spectroscopy in which the method is couched, the group theoretical technique is indeed able to give a succinct account of many of the currently understood aspects of the evolution of the genetic code and the observed degeneracy structure of the codon-amino acid correspondences. We present one particular model, based on the Lie superalgebra A(5,0) ≅ sl(6/1), which has many natural structural features for this purpose that conform with observation. We argue that the model is susceptible to numerical verification by using the wealth of data that are available on biologically important molecules related to DNA.

See related and citing articles in Google Scholar.

Popularization, Quantum mechanics, String Theory

Press Release from AdS/CFT Territory

The good folks at Princeton have been beavering away at the esoteric, abstrusely mathematical yet infinitely tantalizing relationships between string theories and gauge theories. The latter are the rather well-respected mathematical descriptions of how the bits and pieces of atomic nuclei interact; the former is what you get when you look at the dawn of time, the centers of black holes and the other places where our understanding throws up its hands, and you jump in with both feet. Or, at least that’s what string theory was, back in 1997 or so.

Today, we’ve come to recognize that physics has a strange property: ideas you invent in one place pop their heads up where you never expected. Thus, supersymmetry — a mathematical concept invented in the 1970s to make string theory look a little more like the real world — branched off to become its own field of inquiry. In trying to figure out the implications of supersymmetry, Ed Witten and company invented supersymmetric quantum mechanics, which (among other things) gives you a wickedly delightful insight into all the problems professors use to torture their undergraduate physics students in Quantum Mechanics I.

The journey from string theory to SUSY QM leaves behind the essential “stringiness” of the original theory, but over the past several years, we’ve seen a whole slew of results which suggest that the math invented to quantize gravity, break the hearts of black holes and hold the Big Bang in our hands is also applicable to other, more accessible situations. Does this mean that string theory is the right path to quantizing gravity and all the rest? No, not necessarily. Does it mean that we can get our teeth into the equations and use experiments to see if at least some of our ideas work? Yes. Is knowing about this arena of activity a key part of understanding what physicists are doing today? Again, the answer is yes.

You can really hear the writer stretching for metaphors in the ScienceDaily story based on Princeton’s press release: “Between the two road sections lay a seemingly unbridgeable mathematical gulf”, etc. If you look at the paper which provoked this release, or even its abstract, you can appreciate why the press office’s language gets so tortured:

In two remarkable recent papers the planar perturbative expansion was proposed for the universal function of the coupling appearing in the dimensions of high-spin operators of the N=4 super Yang-Mills theory. We study numerically the integral equation derived by Beisert, Eden, and Staudacher, which resums the perturbative series. In a confirmation of the anti–de Sitter-space/conformal-field-theory (AdS/CFT) correspondence, we find a smooth function whose two leading terms at strong coupling match the results obtained for the semiclassical folded string spinning in AdS5. We also make a numerical prediction for the third term in the strong coupling series.

Clear as a kegger in a mud pit.

The press release tries to draw a layman-friendly picture, as I mentioned. At a slightly higher level of mathiness are Barton Zwiebach’s String Theory for Pedestrians lecture videos. (Yes, that’s the same Zwiebach who taught the class and wrote the book.) The AdS/CFT correspondence stuff appears in the second and third lectures of the three-lecture series.

(Tip o’ the string theorist’s beret to Peter Steinberg.)