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Uncertain Principles

Physics, Politics, Pop Culture

Wednesday, April 06, 2005

Thick as Two Short...

In the previous post on this topic, I talked about how a simple model of black-body radiation leads to what's known as the "Ultraviolet Catastrophe": the prediction that the amount of radiation emitted by an object at a fixed temperature should increase as you go to shorter wavelengths. This prediction more or less works for long wavelengths, but is a miserable failure at shorter ones.

this was a huge and troubling problem for physics in the late nineteenth century. There's nothing obviously wrong with the derivation, but the result is almost completely wrong. Something was missing from the description of the physics, but nobody was sure what.

The solution was eventually provided by Max Planck, using a somewhat unorthodox method. Planck took what was known about black-body radiation (the Rayleigh-Jeans result, which works well at long wavelengths, the Wien displacement relation, which gives the peak of the black-body spectrum at a given temperature, some other observations), and fiddled around with them until he found a single expression that combined all the available observations (the Planck radiation formula). Then he tried to find a way to get that specific expression from first principles.

He eventually found it, but he had to use a mathematical trick to do it. The problem really comes in when you start assuming that the energy in the radiation field is smoothly and evenly distributed among the available modes. So what he did was to imagine that in comes in discrete chunks. These "quanta" of energy depend on the frequency of the radiation in a very simple way-- some constant h times the frequency (or hc/ λ where λ is the wavelength, and c the speed of light).

If you think about it a bit, you can see how this would fix the problem. The amount of light at a given wavelength depends on the number of quanta associated with radiation of that wavelengths, not the total energy allocated to modes in that region. As you move to shorter wavelengths (higher frequencies), the amount of energy allocated to a given range of wavelengths increases (because the number of possible modes in that range increases), but the number of quanta needed to account for that energy decreases (because the energy per quantum increases).

At long wavelengths (low frequencies), you have very few modes, so there's not a lot of energy, and very few quanta, so there's not much light. At short wavelengths (high frequencies), there are lots of possible modes, which means a lot of energy, but there are very few quanta, because each quantum contains a large amount of energy. Somewhere in the middle, you expect a peak-- there'll be a point in the middle where the total energy is fairly large, and the energy per quantum isn't too big, so you end up with a maximum number of quanta of radiation.

(I should note that this is a huge hand-wave-- what's actually going on is a little more complex (obviously, the energy of the field isn't divided perfectly uniformly, because eventually you get to a point where the energy per quantum is greater than the energy per mode), but this gets the basic spirit of the solution without being too ridiculously wrong. I think-- if you want to call me an idiot, you know where the comments are.)

It's a neat trick (Planck borrowed it from statistical physics, which was his main area of research), and it works beautifully. The problem is, there's no real reason to expect it to be correct. In fact, Planck himself didn't really believe that it had any physical reality-- he thought it was just a slick bit of mathematics that he could use to make the solution work, and that eventually somebody else would work out the right way to get there. It turns out, he had the right way all along, but it took a while for that to be appreciated. Einstein played a big part in that, which is how this all got brought up in that Boskone panel, but I'll save his part of the story for the next post.

I'll end this by noting that Planck's name has since been attached to h, the constant that he introduced in order to make his mathematical trick work. Planck's constant is a tiny, tiny number-- 6.626 10-34 joule-seconds (that's 0.0000000000000000000000000000000006626 J-s), but it shows up all over the place in quantum mechanics (including the uncertainty relations over in the left-hand column). In fact, the presence of an "h" in a formula is a pretty good indication that you're dealing with something quantum (unless the formula is something like mgh, which is gravitational potential energy in freshman mechanics), and if you'd like to make up your own quantum formula, it'd be a good idea to include an "h" somewhere in it.

The constant has units of energy multiplied by time, or kilograms times meters squared, divided by seconds. This means that you can put it together with the other major constants of the universe (the gravitational constant G and the speed of light c) to generate a set of "fundamental units" for length, mass, and time. These are sort of ridiculous for most normal purposes (the "Planck time" is 0.00000000000000000000000000000000000000000005 seconds), but they may be the natural units for describing certain extremely high-energy processes. If you hear string theorists talking about the "Planck scale," that's what they're talking about-- lengths comparable to the "fundamental" length you get from the proper arrangement of h, G and c (0.00000000000000000000000000000000002 meters, give or take).

Posted at 7:07 AM | link | follow-ups | 9 comments

Tuesday, April 05, 2005

Playing Songs from Their New Album, "Antithetical Squid"

It's been ages since I've done a physics-explaining post here, long enough that most of the people reading probably don't remember me doing them at all. I'm going to try to do a few in the coming months, and also start doing "Journal Club" posts again, now that I'm not teaching a double load any more.

(We'll see how long I can keep to either of those...)

Anyway, the "Einstein" panel I did at Boskone was pretty popular, and Kate pointed out afterwards that it might be interesting to write up some of the things we talked about there. I've been meaning to do that, but last term was so unbelievably hectic that I just haven't had a chance. Until now.

I've talked about Special Relativity before (here, here, and here), and I can't claim to understand General Relativity well enough to explain it to anyone, but we spent a fair bit of time in that panel talking about photons, and that's a story I can tell. So here's the first of a projected handful of posts explaining about photons, and the quantum theory of light.

Historically, the place to start is with black-body radiation and the "Ultraviolet Catastrophe" (which would be a great name for a band, or an album). The term "black body" is a little misleading the first time you hear it, because it has nothing at all to do with color. In fact, the most familiar examples of black-body radiation are not, in fact, black-- they're red, yellow, and white.

A "black body" in physics terms, is an idealized object that absorbs all frequencies of radiation equally well. Most real objects aren't perfect black bodies, having some frequencies of radiation that they absorb particularly well, and others that they hardly absorb at all, but it's not hard to come up with things that are reasonable approximations, and it's a place to start.

(You can make a reasonably good "black body" by getting a bunch of double-edged razor blades (fifteen or twenty) stacking them up, and clamping them together. If you look at the sides of the stack, they'll look basically black-- though the individual blades are reflective, light reflecting off the angles of the blades is reflected deeper into the stack, and never makes it back out to an observer.)

Of course, the fact that a black body absorbs all frequencies equally well does not mean that it only absorbs light-- if it did that, it would keep heating up until it melted, and that wouldn't be terribly interesting. Instead, a black body will emit radiation in a manner that allows it to reach equilibrium with its surroundings. The absorbed light heats the object, while the emitted light carries some heat away, and these heating and cooling rates balance out at some particular temperature.

A question that was very much of interest to physicists in the late 1800's was: What sort of spectrum do you expect from black-body emission? It ought to be possible to predict that spectrum, and how it should vary with temperature, and compare that to experimental observations. A number of people set out to predict the shape of the black-body spectrum, and they all got more or less the same result. The same disastrously wrong result.

To understand the problem, it's helpful to think of another example of a way to make a black body. Imagine making a box of some material-- it doesn't matter what-- and poking a small hole in it. Any light that gets into the hole is extremely unlikely to come back out without bouncing around lots of times, and being absorbed. But if the box is at a constant temperature (which we can safely assume), the radiation inside the box should be in equilibrium with it, so what leaks out through the hole should be a black-body radiation spectrum.

So the question becomes, what sort of distribution of radiation do you expect inside the box? As we've said nothing about what the box is made of, there's no reason to expect it to favor any particular frequency of radiation, so the energy content of the radiation inside the box should be divided more or less evenly over all the possible frequencies that can fit into the box. This doesn't sound like a big improvement, but it gives you a way to attack the problem.

At the time that people were working on this, everybody knew that light was made up of electromagnetic waves. And if you're dealing with waves hitting a surface (like one of the inside walls of the box), you know that the amplitude of the wave has to be zero right where it hits the wall. If the wave in question is between two walls (inside a box, say), it needs to be zero at both ends. The only frequencies of radiation that you can have in the box are those associated with a wavelength that fits neatly in the box (frequency and wavelength are inversely related-- if you double the wavelength, the frequency of the resulting wave will be half of the original frequency). All other frequencies are forbidden.

This is easier to understand if you think about waves on a string that's fixed at both ends-- a piano or a guitar string, for example. This is sort of like a one-dimensional "box"-- the string can't move at the ends, and waves traveling up and down the string have to satisfy the same condition as the electromagnetic waves trapped in the black-body box. If you've got one meter of string, a wave with a wavelength of one meter will work-- it starts at zero, goes up, comes back down, and ends up back at zero, over the course of one meter. A wave with a wavelength of half a meter works, too, as it does two full cycles before ending up back at zero. And it turns out that wavelengths of two meters and 2/3rds of a meter will also work, because they do a half, and one-and-a-half oscillations, respectively, but still end up at zero. There's no way to fit a wave with a wavelength of three-quarters of a meter in there, though-- it needs one end or the other to be pulled away from zero, and that's not allowed. (The allowed patterns are illustrated here.)

So, there are only certain wavelengths of radiation allowed inside the box. Figuring out the spectrum of light emitted by the black-body box, then, just involves counting up the number of allowed wavelengths, and giving each one an equal part of the energy contained in the box. Well, ok, it's a little more complicated than that, since we're not dealing with a one-dimensional box, but that's the basic idea-- you just need to figure out how many ways there are to squeeze a wave of a given wavelength into the box so it's zero at both ends, and divide the energy among those. There'll be a little smearing-out of the spectrum, of course, but in general, the more ways there are to fit waves in a given wavelength rage into the box, the more energy you expect to find in that wavelength range.

And this is where disaster strikes. The shorter the wavelength, the more ways there are to fit those waves into the box. If you think about the string example, you can see how this works: if you've got a wave with a wavelength of one meter on a one-meter length of string, the next higher wavelength is all the way up at two meters, and the next shorter allowed wavelength is two-thirds of a meter. But if you've got a wave with a wavelength of one centimeter on that same one-meter string, well, two centimeters will work just as well, and half a centimeter is fine, too. And so will 1.0101 cm, as will 0.990099 cm, along with 1.0204 cm, 0.98039 cm, and lots of others.

So, by this logic, you would expect to find very little radiation at long wavelength (because there just aren't that many ways to fit long wavelengths into the box), and lots of radiation at short wavelengths (because there are lots of wavelengths that fit). In fact, the shorter you make the wavelength, the more radiation you would expect to find.

Of course, this is completely and disastrously wrong, and a good thing, too-- if it were true, we'd all be getting blasted by ultraviolet light emitted by everything around us, all the time. But that's not what real black bodies do-- there is an increase as you move from very long wavelengths to shorter ones, but the real spectrum reaches a peak, and then turns over. For an object at any reasonable temperature, there's less light emitted in the ultraviolet region of the spectrum than in the infra-red, even though the theory predicts that the ultraviolet emission should be many times greater. Hence, it picked up the name "Ultraviolet Catastrophe."

And even without comparing to experimental data, it's obvious that there's some sort of problem here. By the reasoning given above, we would expect an infinite amount of radiation at very short wavelengths, and that's just impossible. If nature abhors a vacuum, it absolutely detests infinities. Something is hugely wrong with this theory.

So, there has to be some reason why radiation at short wavelengths is less favorable than radiation at longer wavelengths. But there's no reason to expect that that should be the case, and physicists in the late 1800's couldn't find any hole in the reasoning that led to the "Ultraviolet Catastrophe" (also known as the "Rayleigh-Jeans Law," not because people were mad at Rayleigh and Jeans for coming up with this whole mess, but because it actually works reasonably well for very long wavelengths).

Fixing this problem required a really dramatic and completely unjustified leap, but the explanation will have to wait for the next post.

Posted at 10:10 AM | link | follow-ups | 6 comments

Fifth Time's the Charm

Congratulations to Roy Williams and North Carolina who won the national title last night, defeating Illinois 75-70. You might not have known this, but Roy Williams had never won a national championship before this. Also, the Pope is dead.

I'm happy to see Roy Williams win, because he's one of the class acts in college basketball-- he's unfailingly nice, he's never been accused of any impropriety that I know of, and his teams play the game the way it's meant to be played. I thought he was crazy to give up a good gig at Kansas for the insane expectations of North Carolina, but he's made it work, and good for him.

As for the game last night, the big story was the play of North Carolina center Sean May, who led the Tar Heels with 26 points and 10 rebounds. That performance was even more impressive if you watched the game, and saw the god-awful passes he was thrown by his teammates-- over his head, down by his feet, bouncing stupidly through traffic. If any of Carolina's guards had been able to make a halfway decent entry pass to the post, May could've had fifty-- Illinois just had no answer for him.

I'll probably go back to hating the Tar Heels next year, but for now, I'm happy to see them win one for their coach. Couldn't happen to a nicer guy.

Posted at 8:40 AM | link | follow-ups | no comments

Monday, April 04, 2005

Science Analogy of the Century

Michael Nielsen picks up on some comments by Dave Bacon regarding the frequent claim that "biology will be the physics of the 21st Century." Michael offers a list of open questions in physics in support of his contention that "physics will be the physics of the twenty-first century"

  • How can quantum mechanics and gravity be put into a single theory, preferably one integrating the usual standard model of particle physics?
  • What’s up with quantum mechanics and measurement? The fact that we don’t properly understand our most successful scientific theory always seems to me like something of an embarrassment.
  • How did our Universe start? How will it end? What is its structure?
  • There are many other puzzles – dark matter, the cosmological constant, the Pioneer anomaly, and others – which we don’t understand. It’s possible and maybe even probable that some of these are unimportant. Still, it seems pretty likely that one or more of these is the tip of a really big iceberg.

Those are all big questions, and I think it will be great for physics if some or all of them get answered in the coming decades. But I don't think they really rise to the level necessary for the analogy.

Of course, this all depends on what you mean by the phrase "physics of the twenty-first century," so it's important to be clear about the definition. I take it to mean an analogy to the state of science in the mid-to-late twentieth century, during which physics was the dominant science, both in research funding, and in the public imagination. And for the purposes of the analogy, the latter is more important than the former.

The crucial thing about physics in the middle of the last century was not the intellectual revolution that went on in the field, with Relativity and Quantum Mechanics supplanting the classical theories, but rather the material consequences of that revolution. Quantum Mechanics is important not because it forced scientists to re-think our relation to the universe, but because an understanding of quantum theory makes it possible to build devices like transistors and lasers. Relativity is important not because it transformed our understanding of space and time, but because understanding the theory makes it possible to build atom bombs and nuclear power plants. Everything that happened in the latter half of the twentieth century, from the Cold War to the Internet, is in some sense a result of the revolution in physics that took place in the first few decades of that century.

Seen in that way, none of the problems Michael mentions look like they stack up. Yes, a working theory of quantum gravity would be a major revolution in physics. But it doesn't seem likely to have material consequences for the average person, unless some quirk of the theory makes either free energy or levitation possible. Quantum measurement and cosmology are fascinating topics, and the people who nail them down will richly deserve their Nobel Prizes, but I don't think either is likely to have results that will re-make the world in the way that the transistor and the atomic bomb did.

The biological sciences, on the other hand, look (from the outside at least) like they might be on the brink of the sort of practical revolution that physics had in the mid-twentieth century. New tools have made it possible to pile up huge amounts of data on every kind of problem imaginable, and it seems like they're right on the cusp of figuring out how to do things that will really make a difference, and transform society: disease eradication, life extension, genetic therapy, all those buzz-words. It may be that some of those projects will prove to be more like nuclear fusion ("twenty years away, and expected to remain so for the next twenty years") than the transistor, but really, a breakthrough on any of them is likely to lead to a radical transformation of society.

Don't get me wrong-- I'm not saying that physics has no future. The questions that Michael cites are more than enough to keep my generation of physicists occupied, and probably the next two after us. They're fascinating questions, and they richly deserve to be studied.

But if somebody in 2105 is going to post on their weblog equivalent a sentence of the form "Everything that happened in the last fifty years is in some sense a result of the revolution in [science]," the science they mention is more likely to be biology than physics.

Posted at 7:44 AM | link | follow-ups | 12 comments

Something for Everyone

Lots of hot booklog action this weekend, for those who might care, but don't read the Library of Babel regularly. I covered most of the bases, too-- you've got mainstream literary fiction, really excellent genre fiction (edited by Teresa Nielsen Hayden, so thank her for a job well done...), and comic books. What more could you want?

OK, none of those reviews are really snarky pans. I'm working my way up to one of those. Stay tuned...

Posted at 7:36 AM | link | follow-ups | no comments

Sunday, April 03, 2005

Short Dadgum Memories

North Carolina won their semi-final game Saturday night, and have advanced to face Illinois in the championship game. This means that all the talking heads in sports get to tighten the screws on head coach Roy "Deputy Dawg" Williams, who as we all know has never won a title in four previous appearances in the Final Four.

On The Sports Reporters this morning, professional loudmouths Bob Ryan and Mitch Albom went on at great length about this, making frequent comparisons between Williams and Dean Smith. Now, this is a natural comparison to make, because Roy Williams is a Dean Smith product, and the sainted Dean took twenty-odd years to win his first title. But I've got another name to throw out, here: Mike Krzyzewski.

One of the things that drives me nuts about this conversation is that nobody ever brings up Krzyzewski. It's a classic example of the short historical memory of sports media types. I realize that, these days, Krzyzewski is deemed the second coming of John Wooden, and we're supposed to imagine heavenly choirs making "oooo-oooo-ooo" noises whenever somebody mentions his name, but that's not how he looked in 1991.

In 1991, Duke under Krzyzewski had been to four Final Fours (1986, 1988, 1989, 1990), and lost in every one of them. The previous year, they had been absolutely blasted by UNLV in the championship game. In 1991, Duke was the Denver Broncos of college basketball (in 1991, the Buffalo Bills weren't the Buffalo Bills yet, having only lost one Super Bowl, to the Giants), and Krzyzewski was John Elway (who, at that time, was a three-time Super Bowl loser, not a famous furniture designer).

In fact, I remember hearing the exact same conversations about "why can't he win the big one?" and "what will it mean for his legacy?" that we're getting now about Roy Williams. And, of course, Duke went on to beat UNLV in the semis, and then beat Kansas (under Roy Williams) for the title. And now, John Wooden, heavenly choirs, "ooo-ooo-ooo" noises.

So give ol' Roy a break (and really, how can you live with kicking a guy who says "dadgum" without irony?). Or, if you insist on comparing him to some other famous coach, pick one who's still in the business.

Posted at 11:15 AM | link | follow-ups | no comments

Scenes From a Semi-Final Saturday

Jim Nantz: Welcome back to CBS Sports' exclusive coverage of the Road to the Final Four ®. Michigan State and North Carolina are just about ready for the Pontiac Opening Tip, but first, let's check in with Bonnie Bernstein. Bonnie?

Bonnie Bernstein: Jim, I spoke to North Carolina coach Roy Williams just a minute ago, and asked him what his team needs to do to beat this Michigan State team. He said that he thinks they need to score more points than the Spartans in order to win. Back to you, Jim.

JN: Thank you for that special report, Bonnie. We're just about ready for the Pontiac Opening Tip ® here in St. Louis®-- back, after these messages.

{Commercials: Worried about taste loss? If the moment's right, will you be ready? Shouldn't you be buying insurance from State Farm?}

JN: Welcome back to St. Louis everybody. We're ready for the tip-off here in our second national semi-final game, between the State Farm Michigan State Spartans® of Coach Tom Izzo, and the University of North Carolina® Tar Heels®, under Coach Roy Williams, presented by Citibank®. Billy Packer, what should we expect during this game?

Billy Packer: Well, Jim, I'm looking at these two teams, and I really don't like what I'm seeing. I don't like any of it. I don't understand why Roy Williams and Tom Izzo are letting them do this. Their shorts are all way too baggy-- someone could get hurt.

JN: Thank you for that insight®. OK, the players are lined up, the referee is ready, and we'll be back for the Pontiac Opening Tip ®, right after these messages.

{Commercials: Shouldn't you be buying insurance from Allstate? If the moment is right for taste loss, will you be ready? Mike Krzyzewski drives ugly GM cars-- why don't you?}

JN: Welcome back, everybody. Goodyear Referee® Jim Burr® has the ball, and he throws it up, and we're underway. The ball is tipped, and controlled by Michigan State senior Kelvin Torbert, in what may be the last time he ever catches the ball off the opening tip in a Michigan State uniform. How must that feel?

BP: Jim, they're bringing the ball up on the offensive end, and I have to say, I don't like the set they're using. They'll never be able to score against North Carolina with this offense. I just don't like it.

JN: It looks like Tom® Izzo® doesn't like it, either-- he's called a Johnson & Johnson Time-Out ®. We'll be back, right after this.

{Commercials: When the moment is right, Allstate can help you with taste loss. The NCAA: taking advantage of athletes in lots of sports other than basketball. Mike Krzyzewski doesn't think of himself as a smarmy pitchman for AmEx, he thinks--}

Me: Hey, look, there's poker on ESPN...

{Fade to black.}

Posted at 8:42 AM | link | follow-ups | 2 comments

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