Two Physics Questions

Posted by jns on January 18, 2006

Angry Professor at “A Gentleman’s C” asked two questions that caught my attention:

• Is there a difference between an electromagnetic field and an electromagnetic wave?
• Why does the addition of particle detectors in the two-slit experiment cause the collapse of the wavefunction?

Good, physics type questions. Naturally I started thinking about answers. I was finally provoked to write the answers down, though, through irritation. My irritation was at the willingness of the people in her comments to provide “answers” who didn’t really understand at all what they were talking about. It reminded me of my early days with Usenet newsgroups when (and this is still prevalent, of course) person A would ask a question and persons B-ZY would say that, although the didn’t know anything about the subject and had no idea about the answer, they’d be happy to offer their ill-informed opinions anyway. Person ZZ would invariably point out to person A that person A should really be interested in an entirely different question and was wasting time asking this one.

So, on the off chance that anyone might be intereted, here are my answers as I wrote them in Angry’s comments:

EM fields and EM waves are really two distinct, but intimately related concepts. A field in general is just a mathematical concept that assigns numbers to every point in space. They may be scalar fields or M-dimensional vector fields, continuous or not, static or time varying. Hydrodynamic fields specify a fluid’s velocity throughout space. EM fields specify the EM force as a function of spatial variables and time.

Stationary electric charges are the sources of static electric fields. Electric currents (steadily flowing electric charge) are the sources of magnetic fields. This connection between E and M is why they’re called EM fields. EM fields are completely described by Maxwell’s equations: four, first-order partial differential equations.

EM waves are time-dependent, propagating disturbances in the EM field, if you want to think of them that way. Propagating means that they travel in some direction and carry energy in the direction of travel. They are created by accelerating electric charges and are solutions of the second-order, partial differential equation, the “wave equation”, which is impicit in Maxwell’s equations. In modern thought, EM waves are identified with photons (just as “gravitons” are identified with the waves that presumably propagate through gravitational fields when masses accelerate).

The results of the double-slit experiment have nothing to do with interactions with photons from the detectors — this is some sort of mistaken impression [that was mentioned in the original question, but I cut it out]. It is an entirely quantum-identified effect. In short, the surprising result was that with one slit there was no interference effect; with two slits there was. This was thought to demonstrate the unequivocal wave-nature of light because how could a particle “know” which slit to go through? Waves, having spatial extent, can sense the presence of the second slit, provided the spacing of the slits is near the wavelength of the waves. The experiment was first done with light, but it can also be done with particles with suitable “slits”.

The answer to the conundrum from quatum mechanics is that the “particle” is really a “probability wave” that can sense both slits when they are present. According to the “Copenhagen Interpretation” of QM, the probability wavefunction has physical reality while the “particle” it represents has none until the act of observation, which instantaneously “collapses” the wavefunction and localizes the particle (in accordance with Heisenberg’s Uncertainty Principle) due to the act of detection itself. However, this result has nothing to do with the nature of the detector, and certainly nothing to do with interacting photons.

The Copenhagen Interpretation is not part of QM; the mathematical formalism works fine without it. It is, indeed, an interpretation of physical processes, and not everyone who uses QM believe the CI. It is claimed as the source of all those new-age ideas about how our minds alter the universe by “observing it”, ideas that demonstrate more clearly a lack of understanding of QM than an understanding of the universe.

Not All Things Freeze

Posted by jns on July 20, 2005

Some time ago I started reading¹ Robert Wolke’s What Einstein Told His Cook 2. It is a collection of very short pieces about food and cooking from a chemist’s point of view, assembled from his Washington Post columns.

Rather early on though, he made a small error of fact. I point this out not to chastise the author, but as an excuse to talk about helium, one of my favorite science topics.²

He wrote:

[In answer to a reader's question about why frozen cola exhibits separated ice crystals:]

All liquids turn into solids — that is, they freeze — when they get cold enough.

[Robert L. Wolke, What Einstein Told His Cook 2: The Sequel -- Further Adventures in Kitchen Science (W. W. Norton & Company, New York, 2005) p. 5.]

Before I make my point, there are a couple of preliminaries to discuss about freezing.

“Freeze” itself is easy enough: it just means turning from a liquid or a gas — better to say “from a fluid”, because “fluid” encompasses both³ — into a solid. “Solid” generally implies some sort of crystaline structure, but we can be generous and include amorphous solids like glasses.⁴ In other words, we can take “freeze”, as a synonym for “solidify”, to mean much the same as it means in casual English.

“When they get cold enough” is also without serious hidden landmines, although usually a physicist will want to know under what conditions “they get cold enough”. One common condition is that “they get cold” just sitting around in the air, under normal atmospheric pressure. This is frequent and familiar to us: we are very familiar with water freezing under normal atmospheric conditions.

However, thermodynamically speaking, freezing under normal atmospheric conditions is not terribly interesting scientifically, even if it can be well defined. A particular condition of thermodynamic interest is when a substance freezes “under its own vapor pressure”.

Imagine a closed vessel — make it clear glass so we can see what’s going on inside — filled only with pure substance of interest and then sealed off. We put enough stuff into the vessel and lowered the temperature enough, that there is liquid stuff and gaseous stuff in the vessel, both visible at the same time. On the earth, all the liquid will be at the bottom of the vessel, and all the gas will be at the top, with the two phases separated by a “menicus”, or interface.⁵ Let’s also say, to avoid misunderstanding, that the meniscus is exactly in the middle (by volume) of the vessel. That way, as we vary the temperature there will always be gas and liquid in the vessel, and the meniscus will always be right in the middle. If we keep the temperature of the vessel constant for awhile, so that all the stuff is at the same temperature, then the two phases are said to be “coexisting in thermodynamic equilibrium”.

Now, lower the temperature of the stuff in the vessel and, sooner or later, solid stuff will appear and the material is said to have frozen “under its own vapor pressure”. In fact, there will be a unique temperature, called the “triple-point temperature”, at which gas, liquid, and solid phases can all coexist in thermodynamic equilibrium.⁶

To be precise then, our author probably meant by his statement that all liquids freeze under their own vapor pressure if they get cold enough.

However, this isn’t strictly true. The element helium has many interesting and surprising properties. Among them, helium is the only element that will not freeze under its own vapor pressure. It will indeed freeze, but it must be under at least 25 atmospheres of pressure to do so. This also implies that helium has no triple point, unlike any other elemental substance you can think of.

For many years around the turn of the 19th to 20th century, there was thought to be a class of substances called “permanent gasses”. These were gases that could not be caused to condense into droplets of liquid, no matter how much pressure was applied to them. Then it was discovered that they would condense, provided they were cooled to low enough temperatures first. The temperature below which each must be cooled before condensation is even possible is its “critical temperature”.⁷

The critical temperature is usually not the same as the “boiling point” temperature. “Boiling” usually implies that the substance is at atmospheric pressue. For example, nitrogen boils at 77 K⁸ but its critical temperature is about 126 K; oxygen has a critical temperature 155 K, but boils at 90 K.

The critical temperature of helium is 4.2 K, which is really, really, really cold. Until helium is cooled at least to that temperature, condensation is impossible and liquid cannot be produced under any pressure. As it turns out, the critical pressure is low, so helium will liquefy rather easily at atmospheric pressure, if one can get it cold enough.

Helium, the last of the “permanent gases”, was finally liquefied by the Dutch physicist Heike Kamerlingh Onnes in 1908 at his laboratory in Leiden.⁹ Some of us, who have been low-temperature physicists in previous lives, think of Kamerlingh Onnes as a sort of scientific grandfather, since we date the beginning of low-temperature physics to his liquefaction of helium. We also esteem the memory of Sir James Dewar(1842–1923), inventor of the Dewar flask (commercially known as a Thermos bottle), without a couple of which my own thesis experiment would not have been possible. Like atomic physics, low-temperature physics is a distinctly 20th-century discipline.

Helium does not freeze under its own vapor pressure, but it does do very odd things when it is cooled further below its critical temperature. At 2.17 K (at vapor pressure, i.e., at liquid-vapor coexistance), pure helium-4 (by far the most abundant isotope of helium¹⁰) undergoes what is know as a “superfluid transition”, also called the “lambda line” (the reason why to be explained in another essay sometime). The superfluid phase of helium exhibits many wondrous properties, like the lack of viscosity — the ability to flow through microscopic channels unimpeded¹¹ — and various other unusual behaviors.

However, we must save those topics for another time.¹²

———-
¹I’ve long since finished, too, and moved on to the prior volume in the series. I find them a little on the light-weight side, but considering the audience and the venue, that’s not altogether surprising. Regardless, they have been fun and informative reading.

²I have no doubt that this is because in my formative years, i.e., when I was a graduate student, I did low-temperature work (cryogenics) at liquid-helium temperatures (about 2–5 K, or 2–5 degrees above “absolute zero”) measuring properties of helium itself.

³Operationally, “fluid” is anything that “flows”, i.e., any substance which is subject to the equations of motion from fluid dynamics.

⁴There is controversy over whether “glassy solids” are a different form of matter from gas, liquid, and solid, but for the present purposes it’s not necessary to choose sides.

⁵Observing the meniscus, for example its curvature, can tell us many interesting things about the properties of the substance.

⁶Practically speaking, triple-points are very useful since they occur at a unique, well-defined and reproducible temperature. If one can contrive, say, to have a glass vessel filled with pure water in equilibrium with all three phases present, then one knows exactly what the temperature is of the entire system. This procedure is actually used in the definition of the “International Practical Scale of Temperatures”, a set of standard procedures for establishing nearly thermodynamic absolute temperature calibrations in the laboratory. Temperature, though, is a whole other story.

⁷Since “critical phenomena”, the study of elemental properties very near the critical point (i.e., near the critical temperature and critical pressure) was my area of research for some 15 years or more, there’s much more I could say about it, but this isn’t the place.

⁸“K” is the abbreviation for Kelvins, the units of the thermodynamic temperature scale. A Kelvin (NB, not “degrees Kelvin”!) is the same size as a Celsius degree; “0″ on the Kelvin scale is “absolute zero”, which is about -273 centigrade degrees, or -459 Farenheit degrees.

⁹Read a fascinating essay about this called “Heike Kamerlingh Onnes and the Liquefaction of Helium“, written by Jedtsada Laucharoen, a student at the Horace Mann School, The Bronx. His essay was the 1st place prize winner in the physics category of the “Laureates of Tomorrow Nobel Essay Contest”.

¹⁰The only other naturally occuring isotope is helium-3; I forget offhand what the relative abundances are. But, this does give me an excuse to quote (from memory, so my precision may only be close) my favorite first line from a book, J.D. Wilke’s Properties of Liquid and Solid Helium (my bible in graduate school): “Helium exists in three naturally occuring isotopes: helium-4, helium-3, and helium-6; as the latter has a half-life of only 0.67 seconds, it need concern us no further.” And, indeed, in the ensuing 700+ pages, helium-6 is never mentioned again.

¹¹Every low-temperature physicist’s nightmare is to develop a microscopic “superleak” in his apparatus. Below the superfluid transistion helium will pour out of the apparatus, but it is exceedingly difficult to do leak detection at such low temperatures, so the usual response is to throw the thing out and start over. Fortunately, I never faced encountered that problem.

¹²As a closing treat, I will thrill you with the title of my Ph.D. dissertation: Shear Viscosity and Thermal Conductivity in Liquid Helium-4 and Dilute Mixtures of Helium-3 in Helium-4 near the Lambda Transition

Polling: “Margin of Error”

Posted by jns on June 24, 2005

This is not a particularly recent poll, although the assertion is still true. But that’s not the point.

The New York Times > Washington > New Poll Finds Bush Priorities Are Out of Step With Americans

The poll was conducted by telephone with 1,111 adults from Thursday through Monday. It has a margin of sampling error of plus or minus three percentage points.

Have you ever wondered where that “margin of error of + or – 3 % points” comes from, or why the weird number of people polled (which most people react to by thinking it’s much too small)?
The simple answer is simple. Follow along:

1. The sample size is 1,111
2. The square root of 1,111 is 33.33
3. 33.33 / 1111 = 0.03, or 3%

The answer is that simple. In random sampling from a uniform population, the best estimate of how good the average result is will be

+/- [sqrt(N) / N] = [1/sqrt(N)],

where N is the number of [statistically independent] samples.
The inverse works too. If you are told that the error is E%, then

N = 1 / (E/100)²

is the original sample size.
There is no mystery about this relationship between error and sample size in polls, and it is not what determines careful or “scientific” polling. It is simply an unvarying, mathematical result giving the best guess you can make about the error in an average calculated from random (i.e., statistically independent) samples taken from the larger population that one is trying to characterize.
The trick, of course, is in that bit about taking “random samples”. That’s the part that polling organizations work very hard at: to convince their customers that they (and they alone among their competitors) know how to take very good, very nearly “random samples” from any given population — all Americans, all likely Republican voters, all women under 18 who watch MTV, all men over 50 who eat chocolate ice cream at least twice a week, whatever group the poll’s sponsor is interested in.
All the work, or artistry (some would like to say “science”) goes into selecting the samples so that they will be randomly drawn from the population of interest; none of it goes into calculating the margin of error.
So now, when you hear a margin of error quoted, you can amaze all your friends by revealing the exact number of people who were asked the question, and sound amazingly clever.