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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Friday Anime Remixes

WARNING: if you have a life, you might not “get” this post.

Sometimes, the creativity of the Internet astonishes me.

No, really. I know I’m a snarky and sarcastic person given to damnation-by-faint-praise, but on occasion, even I am honestly impressed. While much of what goes on in the Network’s twisty little forking passages doesn’t contribute to saving the world, we do have to assure ourselves our world is worth saving, don’t we?

And that’s why I’m embedding the theatrical trailer for The Matrix (1999), recreated using clips from Ghost in the Shell (1995).

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Where am I in the Inferno?

The Dante’s Inferno Test has banished you to the Sixth Level of Hell – The City of Dis!
Here is how you matched up against all the levels:

Level Score
Purgatory (Repenting Believers) Very Low
Level 1 – Limbo (Virtuous Non-Believers) High
Level 2 (Lustful) Moderate
Level 3 (Gluttonous) Moderate
Level 4 (Prodigal and Avaricious) Low
Level 5 (Wrathful and Gloomy) Low
Level 6 – The City of Dis (Heretics) Very High
Level 7 (Violent) Very High
Level 8- the Malebolge (Fraudulent, Malicious, Panderers) High
Level 9 – Cocytus (Treacherous) Low

Take the Dante Inferno Hell Test

Funny, I didn’t think I was that violent (and I took the non-violent option on the only questions I can think are relevant). I guess this means I won’t be joining Dave Bacon in Limbo. Lucky schmuck — he doesn’t even have to deal with the unbaptized infants anymore.

At least I won’t have to spend eternity in the tenth circle, Corpadverticus.

Quoting a Motl

Among other things, that when you talk to string theorists in person, they’re much more open-minded and reasonable than you’d expect! Of course, when your de facto spokesman is the self-parodying Luboš Motl — who often manages to excoriate feminists, climatologists, and loop quantum gravity theorists in the very same sentence — it’s hard not to seem reasonable by comparison.

— Scott Aaronson (21 December 2006)

If you inhabit the same corner of the Network that I do (and you’re reading this right now, so that seems likely) you’ve probably heard about NASA Administrator Mike Griffin’s intemperate remarks about global warming. (For a refresher, see the dependable Phil Plait.)

Via Tim Lambert, I hear that a denialist “press release” collects some approving comments from the usual suspects. Apparently, the “Science and Public Policy Institute” saved the best quote for last:
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Fr*ming in the CJR

The framing kerfluffle has reached the Columbia Journalism Review‘s website, with a piece entitled “Just the Facts, and Opinions Too” (5 June 2007). Curtis Brainard does a pretty good job of setting out the scenario, but his story leaves out an aspect which I think is significant.

I have come to hypothesize that there are the reasonable Mooney and Nisbet, who get written up in places like Brainard’s article, and then there are their insane twin brothers who keep trying to kick those uppity atheists back into the corner. (Similar statements about rational and bizarre twins have been made for Thomas Kuhn and Lee Smolin.) To a first approximation, if I heard some people saying what Curtis Brainard records Nisbet and Mooney as saying, my response would resemble the following:
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Behe’s Bad Arithmetic and Worse Science

Blogging on Peer-Reviewed ResearchI would like to celebrate the official release date of Michael Behe’s new book, The Edge of Evolution, by pointing out a grotesque factual error in its very heart.

A major theme, if not the major theme, of Behe’s diatribe is the disease known as malaria. He wants to prove that evolution by natural means of any but the most trivial features of living things is completely impossible, and he uses the parasites which cause malaria as his test case. Unfortunately for him, but happily for the decades of scientific discovery which he tries to condemn, the mistakes in his argument tear it asunder.

Since Behe’s argument is built upon malaria, and in particular the evolution of the parasite Plasmodium falciparum to be resistant to the drug chloroquine, we should look at what modern biology has learned about how chloroquine resistance evolved. As chloroquine was “the best and most affordable antimalarial drug” in the history of antimalarial drugs, the evolution of resistance to it — and the spread of that resistance throughout the world — poses a real problem, which scientists are naturally eager to investigate.
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CEO Kills 250 Million Americans

My day job, about which I am Not Yet At Liberty To Speak in detail, brings me into contact with people I didn’t expect to meet when I graduated MIT with a physics degree. I could understand rubbing elbows with software developers, but healthcare policy professionals were a whole ‘nother world. I mean, I managed to meet and mingle with a set technician for The Truman Show, not to mention Mary Prankster. (A friend of mine spent a year in Los Angeles, driving around with a Dresden Dolls bumper sticker. One day, a guy stopped her in a parking lot and exclaimed in wonder and joy, “You like the Dresden Dolls?” To which she replied, “I know the Dresden Dolls.” She assures me that I’ve met them too, at a party somewhere in this town, but it must have been an absolutely smashing party because I have no recollection of it whatsoever.) But corporate executives?

So, it is with considerable surprise that I find myself equipped to tell the tale of the CEO who killed 250 million Americans.
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Egnor, Farrell and Lemaître

I don’t want Science After Sunclipse to become, in Orac’s words, “All Egnor, all the time,” but after my recent intemperate remarks about Egnor’s misguided dualism, I figure I should follow with a link to John Farrell’s rebuttal to Egnor’s cosmology.

Following up on his insistence that science is always seeking to confirm the “design inference,” Egnor says this about Georges Lemaître, the first scientist to conceive of the Horrendous Space Kablooie:

Ironically, we owe much of our modern understanding of the universe to pro-intelligent design astronomers. Georges Lemaître was the astrophysicst who pioneered the Big Bang Theory. Fr. Lemaître (above, with Einstein) was a Belgian Roman Catholic priest, honorary prelate, and a professor of physics and astronomy. He famously described the moment of the Big Bang as “the day without yesterday”, referring to the first day of creation in Genesis, and he was explicit in his belief in the evidence for God’s design in the universe. His Big Bang theory met with considerable opposition because of its religious implications.

Farrell observes, “First, Lemaître was not referring to the day without yesterday as the first day of creation in Genesis. I’m sure it will surprise no one that Mr. Egnor offers no quotes to support his contention.” In fact, Lemaître stated the following at the 1958 Solvay conference:

As far as I can see, such a theory remains entirely outside any metaphysical or religious question. It leaves the materialist free to deny any transcendental Being. He may keep, for the bottom of space-time, the same attitude of mind he has been able to adopt for events occurring in non-singular places in space-time. For the believer, it removes any attempt at familiarity with God, as were Laplace’s chiquenaude or Jean’s finger. It is consonant with the wording of Isaias speaking of the “Hidden God”, hidden even in the beginning of creation. . . . Science has not to surrender in face of the Universe and when Pascal tries to infer the existence of God from the supposed infinitude of Nature, we may think that he is looking in the wrong direction.

Farrell goes on at slightly greater length, but I believe the point has been made.

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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