Quantum Mechanics brought up to date

kw: book reviews, nonfiction, quantum mechanics, overviews

I've just finished reading Beyond Weird: Why Everything You Thought You Knew About Quantum Physics is Different, by Philip Ball. Prior to reading it, every book I've read that "explained" quantum mechanics has used the knowledge and language of the 1940's. Philip Ball brings things up to date, and it is about time someone did so!! The field of quantum (we can leave off the various modifying nouns for the nonce) has been the hottest of hot topics in physics for more than a century, so incredible numbers of experiments have been performed, and astronomical numbers of hypotheses have been proposed and defended and attacked and re-defended in tons and tons of published articles. So folks have learned a thing or to.

To cut to the chase. Werner Heisenberg got things rolling with his Uncertainty Principle: you can't specify both the position and the momentum of any object with accuracy more precise than a tiny "delta" that is related to Planck's Constant (ℏ); it yielded a Nobel Prize for him. Albert Einstein showed that light is quantized in his publication on the photoelectric effect, for which he received his Nobel Prize. In general terms, Heisenberg and Einstein agreed that a moving particle had a genuine position and momentum, but that no tool for determining these quantities could operate without changing them.

We can summarize the way this is explained in the Feynman Lectures on Physics thus:

A beam of electrons directed through two holes will produce an interference pattern on a screen. The electrons are behaving like waves. You want to determine which electron goes through which hole. Electrons can be 'probed' by shining short wavelength light on them, and recording where the scattered light came from. When you turn on the light beam, the interference pattern disappears, and is replaced by a smooth diffraction curve, as though there were one hole, not two. Weakening the beam doesn't help. So you try using a longer wavelength, then longer and longer wavelengths. Finally, you find a wavelength that will scatter off the electrons, but allows the interference pattern to form. However, the wavelength of the light you are now using is greater than the spacing between the holes, and you cannot tell which hole any electron goes through.

Niels Bohr thought differently. I have not determined whether he focused on the ubiquitous presence of diffraction patterns in focused beams (of light, electrons, or whatever; I'll come back to this). Whatever, he concluded that the observation of an electron's position or its direction of motion didn't just reveal that quantity, but created it. His published material shows that he dwelt on the matter of measurement; that no quantum theory could be admitted that said anything more than what measurements could say: "There is no quantum world. There is only an abstract quantum physical description. It is wrong to think that it is the task of physics to find out how nature is. Physics concerns what we can say about nature." (p.73) He went beyond this, stating that nothing really happens until it is observed.

This led Erwin Schrödinger to make fun of Bohr with his Cat story: Put a cat in a box, along with a device that has a 50% chance of killing the cat via poison gas in the next hour. Just before opening the box, can you say whether the cat is alive or dead? He asked if Bohr would say, "Both," until the box is opened. Most likely, Bohr would say that the wave function, which Schrödinger had proposed for calculating probabilities of quantum events, would "collapse" and you would only then observe either a dead cat or a living cat (which you would whisk out of the box and close it in case the mechanism were to go off just then and kill you both).

Bohr was a bully. He didn't just promote what became the first of many versions of his Copenhagen Interpretation, he evangelized it, in a totally obtrusive way. He used no violence (of the fisticuffs variety, anyway). He'd just talk you to death. It was like water torture, a torture to which many of that generation of physicists eventually submitted. Still, fewer than half the physicists of any generation, then or since, really believes it. Murray Gell-Mann, for one, thought he simply brainwashed people.

The thing is, there is a bit of support for this view, though it calls into question the definition of "observer". And I confess, when I first heard of the Cat story, I asked, "Isn't the cat an observer?" Anyway, anyone who has done technical photography, and especially astronomical photography, knows of Diffraction. The smaller the hole (the camera diaphragm, for instance) through which a beam of light passes, the more that beam spreads out. It isn't much, in most cases, because visible light has such short wavelengths, between 400 and 700 billionths of a meter (nanometers or nm). However, the pixels in a modern digital camera are really small and closely spaced. In my Nikon D3200 camera, for example, there are about 24 million cells in a sensor that measures 12mm x 18mm, spaced 0.003mm apart. That is 3,000 nm. For technical photography I'd prefer for the light to spread out no more than the distance from cell-to-cell, so that "every pixel counts". For a lens of focal length 100mm, we want a half-angle α to be less than arcsin(.003/200) = 0.00086°. For such calculations it is common to use a wavelength of 500 nm. We find that an aperture of 33.3mm, or larger, is needed to get full utility from the camera's sensor. That implies an f/stop of f/3. Make the aperture any smaller, and the picture gets successively fuzzier. If you have an SLR and can control the aperture, try taking a picture outdoors at f/22 or f/32 (many will allow that). It will be pretty fuzzy.

This is why astronomers like telescopes with a wide aperture. Not just because they are efficient "light buckets", but because the bigger the hole the light goes through, the less it spreads out, and the sharper an image you can obtain. Of course, a telescope with a focal ratio of f/3 or less is hard to build and expensive. But for a large instrument with a long focal length, those little pixels are very small "on the sky", allowing you to see finer detail in distant galaxies.

Now, if there is no sensor at the focus of the telescope, does the diffraction still occur? Here I attempt a détente between Bohr and Heisenberg. Heisenberg would say that the aperture, no matter its size, is "disturbing" the light beam from some distant object, spreading it out. The effect is never zero, no matter how big the aperture. This implies that the whole universe affects what happens to every bit of light, every photon, as it makes its way from source to wherever it is "going". But, whether we are watching what happens or not, it must still be happening. Bohr would have to admit that the "observer" is effectively the aperture, and by extension, that the universe itself is an observer, or is constituted of observers. Effectively, everything is an "observer" of everything!

On page 84, the author writes, "…the idea that the quantum measurement problem is a matter of 'disturbing' what is measured is exactly what the Copenhagen Interpretation denies." (Author's emphasis) I think this example shows that such a position is untenable. For my part, I think the Copenhagen Interpretation, as stated by Bohr and most of his followers, is simply silly. Photons go where they go and do what they do. So does anything else in motion. It is amazing that the environment, all of it, has an effect on where they go. But diffraction experiments show that it is so: Everything disturbs everything. However, for most of the universe out there, the disturbance doesn't seem to amount to much.

One quibble I have with the book is that it lacks a table of contents. The 19 chapters just have a 2-page heading with a snappy title. In the chapter titled "The everyday world is what quantum becomes at human scales" (the 11th), the environment is brought in, and the matter of "decoherence". Prior chapters have discussed all the things we find "weird", such as "entanglement" (for example, two particles that carry equal but opposite values of some characteristic such as polarization, even to the ends of the universe if they don't bump into anything). They get us ready for this chapter, the key chapter of the book.

Entanglement, just mentioned above, is one kind of Coherence between quantum entities. A laser beam and a Bose-Einstein condensate are, or express, coherent states among numerous entities. Coherence is thought to be fragile. It is actually quite robust, and even infectious. Particles that interact with their environment spread their quantum states around. The problem is, any instrument we might use to measure such quantum states is part of the environment, and so partakes of that state, becoming unable to detect it. That is what is meant by Decoherence. It expresses our inability to keep a pair of quanta, for example, in a given entangled state because they "want" to spread it around. The longer we want them to stay in coherence, the more it will cost. However, it is this phenomenon of decoherence that leads directly to the human-scale, everyday behavior of objects. The author concludes that the entire universe "observes" everything that goes on unless we take great pains to isolate something from the environment, so we can measure it. It is the error of Bohr and others in not recognizing the influence of the multitude of non-conscious "observers" known as the universe, that led to the silliness of the Copenhagen Interpretation.

Or, perhaps I ought to be more charitable to Niels Bohr. Maybe he was right, that things only happen when they are observed by a conscious entity. But diffraction shows that every photon, of whatever wavelength, that passes through the universe; every electron, neutrino, etc., etc., etc., is an observer, and produces the universe that we see, in which most quantum phenomena require costly apparatus to observe and maintain (except diffraction!). If the universe required conscious observers only, or things could not happen, that would imply that God made the universe only after there were observers to keep it functioning! And that's funny. It may even be true! The Bible, in the book of Job (38:4-7), mentions multitudes of angels that observed the "foundation of the Earth". The Copenhagen Interpretation agrees with Job.

A late chapter discusses quantum computing, that it is being over-hyped (so what else is new?). Near the end of the discussion, I read that all the ways of making a quantum computer, so far discovered, are special-purpose. One can search, one can encode or decode, and so forth. It appears that, at the moment at least, no general-purpose quantum computer can be produced, that would be analogous in its breadth of function to our general-purpose digital computers. So don't sell your stock in Intel or Apple just yet!

I was much refreshed by this book. The author's point of view is still somewhat "Copenhagenish", but that's OK with me. If decoherence is what he says it is, then it really is true that what we see at our scale of a meter or two is just the consequence of quanta doing what they do, in all their multitudes, and spreading their characteristics about quite promiscuously, so that the universe just keeps keeping on, as it has since the Big Bang, at the very least.

Wu Li: Circular reasoning to the max

kw: book reviews, nonfiction, physics, cosmology, buddhism, copenhagen interpretation, quantum mechanics

From time to time I have heard about The Dancing Wu Li Masters: An Overview of the New Physics, by Gary Zukav, since it was published in 1979. I had never read it until now. As a student of all the sciences, particularly the "hard" sciences (those amenable to experimental verification), since before 1960, I have at least a reading familiarity with physics, which is a hard science, and cosmology, which is not. Now having read the book, I find it contains no surprises, at least, none of a scientific nature. Of course, a lot has happened in physics and cosmology in the past nearly forty years.

The author, an admitted outsider to the field of physics, conceived of the book while on a retreat at Esalen along with a real mixed bag of folks including numerous scientists and science hangers-on (some would consider me more of a hanger-on, though I am a working scientist, even in "retirement" from a career in the sciences). Al Huang, who was teaching T'ai Chi at Esalen when Zukav was there, introduced him to the concepts of Wu Li. That is concepts, plural.

I have a great many Chinese friends. The Chinese languages, primarily Mandarin, the principal written Chinese language, abounds in homophones, words that sound the same, at least to a Westerner. Most basic Chinese words consist of one syllable, and very few require more than two syllables. Spoken Chinese sounds to us like a long string of only a few syllables repeated various ways, with a "sing-song" quality that means nothing. What Westerners miss is that the "sing-song" variations in tone are meaningful and are part of the proper pronunciation of Chinese words. Thus, the syllable "MA", depending on the tone, and its context in a sentence, has at least these meanings:

• Mother.
• When doubled, an affectionate term for Mother, just as in English, at least when pronounced with two flat tones.
• Horse, using a different tone.
• The verb "ride", when the context demands a verb rather than a noun, and using still another tone.
• The pronounced question mark that ends (nearly) all Chinese questions, spoken with a rising tone.

The familiar greeting "Ni Hao Ma" is a lot like the New Jersey, "How are ya?" The Chinese sentence, "Ma-ma ma ma ma", with the proper string of tones, means, "Is mother riding the horse?" (Chinese has no articles, so "the" is implied).

Depending on tone and context, "WU", pronounced "woo", has about 80 meanings, and "LI", pronounced "lee", has a great many, primarily focused on pattern. Different written Chinese characters (ideographs) are used for the various meanings of wu and li. In combination, the word wu li is the primary Chinese term for "physics". But when other combinations of ideographs with the same pronunciation (except for tones) are used, there are other meanings. In the context of this book, Al Huang gathered five. The literal meaning of the ideographs used for wu li meaning "physics" is "patterns of organic energy". The other four are "my way", "nonsense", "I clutch my ideas", and "enlightenment".

The book is structured around these five concepts, with each section containing two or three chapters. As I might have expected from a book inspired at Esalen, each chapter is numbered 1.

The "new physics" on which the book is centered is quantum mechanics and its relationship to Einstein's theories of relativity (special and general). The core message is the ambiguity of quantum phenomena—when any single "particle" is studied—coupled with the exactitude of the predictions the mathematical theories of quantum mechanics make regarding the statistics of interactions when many particles are subjected to the same set of conditions. The "scripture" of quantum mechanics is the Copenhagen Interpretation, that of Niels Bohr and his followers (I almost wrote "disciples").

Thus, for example, when light is shined through a pinhole, which spreads the beam by diffraction, and this beam is passed through a pair of narrow slits, an interference pattern emerges. This works best when monochromatic light is used, such as from a laser, but "near-mono" filtered light works well enough for visual purposes. The intensity in each part of the interference pattern can be exactly calculated by the Schrödinger wave equation, although the calculations are formidable; various simplifications of the wave equation yield very precise results with less arithmetical grinding.

I mentioned diffraction. This matter is first mentioned on pages 64-65 of the book. In the upper half of an illustration, a series of waves in a harbor are shown exiting a rather broad opening, and those that get through are shown going straight onward, with a sharp edge to their pattern. In the lower half, the opening of the harbor is smaller, and the waves exiting are shown as semicircular wave fronts spreading beyond the opening. There are two major errors here. Firstly, the upper pattern should show a little spreading at the edges of the "beam" of waves exiting the harbor (you can verify this using a wave tank, as I was shown decades ago in a Freshman physics class). In other words, diffraction occurs when waves pass through any opening of any width, not just very narrow ones. Secondly, for the lower wave pattern, the wavelength of the exiting waves is drawn as much shorter than the waves in the harbor.

In actuality, diffraction produces a nonzero probability of the waves at every angle. They seem to "go straight" through a larger opening only because the off-axis waves lose energy with angle very rapidly in such a case. When a wave front passes through an opening of a size similar to the wavelength, or smaller, there are significant amounts that are found at nearly every angle, making a much more divergent beam. Zukav seems to have been ignorant of this.

Interestingly, if a double-slit setup using extra-sensitive photographic film is set up, you can get a surprising result. The best photo film can record the capture of each photon, as long as the light is blue enough, meaning the photons are energetic enough. One silver halide grain is exposed by the capture of a single photon. If the light is dimmed enough that only a few photons per second pass through the apparatus, and you let it run for less than a minute before extracting the film and developing it, the developed film will have one or two hundred tiny exposed grains that are seemingly scattered at random over the film. If instead, you leave the film in place for an entire day, there will of course be many more exposed grains, tens of thousands of them. They will show a very clear interference pattern, identical in form to the one you could see when the light was shining brightly and tens of trillions of photons per second were passing through the apparatus.

Interference is a wave phenomenon. Photons are particles; each carries a specific amount of energy and has a specific momentum (these are all the same for monochromatic light). It took me and all my fellow students a long time to become comfortable with the fact that light has both wave and particle characteristics. Eventually we thought of a photon as a "wavicle", a small wave bundle, that could somehow "sense" that both slits were open and "interfere with itself", when passing through a two-slit apparatus. It seems that light behaves as a wave when wave "behavior" is demanded of it (the two slits), and as a particle when particle "behavior" is required (exposing a silver grain in the film).

Where does Gary Zukav take this, and several other experimental results of quantum mechanics, special relativity, and general relativity? Straight to the door of a Buddhist sanctuary. The language he uses is usually as ambiguous as the language physicists typically use to describe concepts like the "collapse" of a wave function when an "observation" is made. He compares some conclusions and statements of physicists to similar statements of Buddhist doctrine, though I could seldom recognize the resemblance. The core of the Copenhagen Interpretation, at least as it is explained in this book, is that the Observer is central. But, to date, nobody has adequately defined "Observer". That doesn't stop Zukav from equating the one-is-all-all-is-one that he believes the new physics is trending toward to Buddhist teachings of the pre-Christian era. I have a question or two about observers, or Observers.

Must an Observer have a self-aware mind? Can the photographic film described above be an observer, or has no observation been made until the film has been developed and a human (or other self-aware entity) has looked at it to see the pattern? If I understand the Gary Zukav presentation of the Copenhagen Interpretation, there is no "collapse" of the wave function into an actual "event" without an observer. It is as though, outside your peripheral vision, nothing exists until you pay attention to it. Taken to an extreme, it means there was no Universe until humans evolved to be the Observers to bring it into existence. This is the reason for the title of this post. If this is actually what Niels Bohr believed, I have to say to him and his disciples, as Governer Festus long ago said to the Apostle Paul, "Much learning has driven you insane!" Paul was not insane, but I think Zukav might be. More on this anon…

At the time The Dancing Wu Li Masters was being written, some "newer" new physics concepts were arising, such as the Quark/Gluon resolution of the Particle Zoo, and the theory of the Multiverse. To take up the former: It appears that the quark is truly fundamental. All the hadrons seem to be made up of various combinations of quarks and anti-quarks. However, it takes such enormous energies to generate interactions that give evidence of the existence of quarks—and they apparently cannot be brought into independent existence—that we may need to await a particle accelerate wrapped around the equator of the Earth to achieve energies sufficient to determine whether quarks do or do not have any substructure. Apparently, electrons have no substructure, so maybe they and quarks are as fundamental as it gets. But our experiments have reached "only" into the range of 10 to 100 TeV. What might be achieved with an energy a thousand times as great, or a million? Fears have been expressed already that the current experiments at CERN could trigger destruction of the Universe. Maybe the Multiverse is real, and we inhabit a surviving Universe that didn't get destroyed.

The notion of the Multiverse is simple. Rather than the wave function for a particle "collapsing" into some actual event, an entirely random outcome within the statistical framework described by the wave function, perhaps every possible outcome actually occurs, and a new Universe is spawned to contain each of those outcomes. This is simple enough if the "outcome" is that a particular photon passes through either the left slit or the right slit of a two-slit apparatus. Two universes result. I one of them, the photon passes to the left, and in the other, it passes to the right. But there is detail in the interference pattern, and when I have done the experiment with a laser pointer and a home-made pair of slits cut in aluminum foil, I could see more than twenty interference fringes. Now what? Did each photon create twenty or more universes to accompany each outcome? When the light is bright enough to see, trillions of photons per second are "in use"; the beam of my laser pointer emits 200 trillion photons or deep red light per second. Did I inadvertently create a few quadrillion new universes, just by shining my laser pointer through a pair of slits? Were new universes being created at the same rate even when I wasn't looking?

So what are the chances that the search for the Higgs boson at CERN caused the creation of truly enormous numbers of universes, nearly all of which were immediately destroyed, and we inhabit one of those that survived. I think you can see where such thinking can lead.

And some folks say that I am crazy to believe in God, a God who knows a level of physics (if it is called that) that can resolve this stuff, without the insanity of Multiverse speculations. I think it is fair to say that "modern physics" has reached a point of adding more and more epicycles to a group of theories that seem to produce very precise results, but that they are really analogous to pre-Copernican cosmology. Actually, Copernicus used epicycles also, because he thought orbits were based on circles. It took Kepler and others to work that part out.

Another item or two that have arisen in physics since 1979:

• On page 119 we read, "No one, not one person has ever seen an atom." If you are talking about direct visual sight without the use of a microscope, you could say the same thing about bacteria or viruses. But we have microscopes of several kinds that can show us what they look like in rather amazing detail. Since about 1981, highly refined transmission electron microscopes have been able to show atoms directly, and since the invention in 1982 of the scanning tunneling microscope and the atomic force microscope, we now have three methods for seeing where the atoms lie in a surface. Whatever point the author wished to make based on the above statement is now moot.
• Beginning on page 292 we find an illustration using polarized light. Simply put, when light is passed through a polarizer (such as the special plastic in some sunglasses), the light that emerges is now all vibrating in the same plane (for convenience, we use the electric vector as the "direction" of polarization, though the magnetic vector could be used equally well, and is at 90° to the electric vector. Zukav does not mention this). When you place a second polarizer with its polarizing axis at 90° to the first, it blocks all the light. If you rotate it to various angles, some of the light gets through, in accordance with an elliptical formula. Now, if you set the two polarizers so their polarization axes are at precisely 90° so that no light is getting through, then put a third polarizer between them, with its axis oriented at 45° to the other two, quite a lot of light gets through! This goes on for several pages and is presented as quite a mystery. Strangely, elsewhere in the book we find the tools to solve this mystery (I didn't look up page numbers):
• In a discussion of Feynman Diagrams and the S-Matrix (Scattering Matrix) we read that physicists consider every interaction to entail the destruction of all the impinging particles and the creation of new ones that exit the interaction locus at the appropriate angles with appropriate velocities. Thus, when a photon reflects off a mirror or any shiny surface, it is actually absorbed and a new photon is released at the appropriate angle. So they say. Refraction works similarly. Thus, the polarizer absorbs the incoming photons and releases a somewhat smaller number of photons, all with the appropriate polarization.
• As I recall, a polarizer made of stretched plastic film passes 38% of the original light. A Nicol prism can actually split light into two beams with nearly no loss, so that 50% exits with horizontal polarization at one angle, and 50% with vertical polarization at a different angle. This would make no sense according to the "picket fence" analogy, because very, very little of the original light could get through any polarizer: only that which is already polarized the "right" way. Thus, a Nicol prism, in particular, "tests" each photon, and either twists its polarization to match the nearest direction (and shifting its exit angle according to the one or the other), or annihilates the photon and emits one of appropriate polarization and exit angle.
• Polarizing plastic is less efficient, passing only light of one polarization, but obviously changing whatever the polarization was of most photons to match its orientation. Thus, what is happening with the 45° polarizer is this: it absorbs some photons entirely, and twists the polarization of the rest of them by 45°. Then when they reach the last polarizer, they are now subject to a further absorption or twisting, so that the "twisted ones" get through, with perhaps 5% of the original beam intensity. That is a lot more than the fraction of a percent that "sneaks through" the original set of crossed polarizers because plastic film polarizers are not perfect.
• So polarizing devices do not just passively allow certain photons to pass and block all others, but they change the polarization of the photons that they allow to pass.
• I cannot pass by the chance to mention circular polarization. A thin piece of calcite or quartz (or, indeed, any colorless crystalline material that does not have cubic molecular symmetry) rotates the polarization of the incoming light. What is more, if it is just the right thickness, it will produce circularly polarized light. This is sometimes thought of as two streams of photons that are related to one another. Think of a vertically polarized photon coupled with a horizontally polarized photon, and their "waves" are out of phase by a quarter of a wavelength. Then, in effect, their polarization will rotate as the go.

As interpreted by Gary Zukav, physics was becoming one with Buddhism. I wonder what he would make of today's situation, with the great popularity among physicists of cosmological string theories (at the moment, they can't decide which of the potential 10⁵⁰⁰ possible string theories to favor!), the supposed detection of increasing cosmological expansion that may lead to a "big rip" in which all things will be literally shredded to their composite quarks, and the theory of cosmological inflation (developed in the early 1980's) that supposes that the initial expansion of the big bang took off at several trillion trillion trillion times the speed of light for just a tiny fraction of a second, during which the Universe grew to a size somewhere between that of a grapefruit and a galaxy (nobody can pin that down too precisely).

In my view, coupling physics theorizing with Buddhism is tantamount to solipsism. Let us accept as a first premise that what exists, does indeed exist, and go from there. Then the extreme versions of "New Physics" simply vanish, like an unobserved photon.

Exploring the quantum boundary

kw: book reviews, nonfiction, quantum mechanics, quantum theory, popular treatments

When I was about ten, I was disappointed in a picture I'd taken. I had been too far from the person I was "shooting", so he looked like no more than a couple of dots. Having recently learned about enlargements, I suggested getting the middle of the picture enlarged. My father remarked that the photo shop charges a lot for enlargements. Then I suggested putting it under my microscope and taking another picture, then getting that printed—I'd already been setting up a clumsy rig with a tripod holding Dad's camera at the eyepiece and making photos of the cells in thin-sliced carrots and leaves. He said I could try, but it would be very blurry, then explained about the grain in the print and in the negative. I looked, and sure enough, even at 25X the film grain made the picture look like it was printed on sand.

The next year he and I made a small telescope (I still use it), and I learned about diffraction and the magnification limit of an optical system. I realized, even if the film and print grain were a hundred times smaller, and even if the optics of the camera were flawless, diffraction would limit how much I could enlarge the final image.

This is an illustration of the Rayleigh criterion for resolving star images in a telescope. I downloaded it from the Angular Resolution article in Wikipedia. The upper section shows that the Airy Disks of the two stars are fully separated. The Airy Disk is everything inside the first dark ring (first null). The lowest section shows serious overlap, and the middle section shows the Rayleigh criterion, at which point the first null of one Airy Disk passes through the center of the other. This is the accepted resolution limit of a telescope system, or indeed, any optical system, including the eye.

What causes this pattern? It results from the interaction of light from a distant point source (or multiple sources) passing through a circular aperture. Just by the way, if you should get the notion to make a telescope with a rectangular aperture, under high magnification you'll get a diffraction pattern more like this:

Such diffraction patterns, I realized one day, are a visible manifestation of quantum-mechanical effects. If you could solve the Schrödinger Wave Equation for this system, the square of its solution would look like this image. In the SWE, the solution is in complex space, and represents probabilities, while the square of the complex probability at any point is the intensity of, for example, a beam of light or electrons, as it is spread through space by diffraction. One characteristic of the SWE is that, while there will frequently be numerous nulls, or zeroes, in the solution, there is no greatest angle or maximum distance beyond which its solution is always zero. This is why even huge telescopes such as the 10m diameter Keck telescopes in Hawaii still have a diffraction pattern once all other aberrations are accounted for (the atmosphere is a much bigger light scatterer "down here", though).

So, think of it. The yellow-green light that our eyes are most sensitive to has a wavelength of 0.55µ, or 550 nm. That's pretty small, about 1/1800 mm. And, even if we are comfortable with photons, the minimal packets of light, we think of them as having a similar "size". But diffraction patterns show us that a photon can somehow "sense" the entire aperture as it "chooses" by how much to change its direction of travel. A certain experiment that has been done with both photons and electrons proves it:

• Set up a very, very light-tight box with a dimmable light source at one end, a sheet with a hole in it about midway, and either a sheet of film or an array of sensitive detectors (e.g. a digital camera sensor) at the opposite end.
• Let's assume the light source is accompanied by a lens system that makes a uniform beam larger in diameter than the hole in the sheet.
• Set the "brightness" of the light source such that there will very seldom be more than one photon inside the box at any one time. That's pretty dim!
• A 550 nm photon has an energy of 2.254 eV.
• A 1 mw yellow-green laser set to that wavelength (you can do that with dye lasers) emits 2.77 quadrillion photons per second.
• Light traverses a 1-meter box in about 3 ns.
• The 1 mw laser thus emits 8.3 million photons in those 3 ns.
• Thus you must dim the beam by a factor of more than 8 million. That is 23 f/stops, or an ND of 6.9. Two pieces of #9 welding glass is about right.
• Close the box, turn on the light, and wait about 3 hours.
• Develop or download the resulting image. It will have the same diffraction pattern as if you'd left off the filters and shot a picture in 1/1000 sec.

The experiment has been done many times, usually using a two-slit setup. Either way, it shows that both a photon and an electron somehow "self-interfere" as they are influenced by everything along the way from emitter to "final resting place."

All the above serves to get my mind in gear to write about The Quantum Moment: How Planck, Bohr, Einstein, and Heisenberg Taught Us to Love Uncertainty By Robert P. Crease and Alfred Scharff Goldhaber. The authors, professors at Stony Brook University, aim to demonstrate that "quantum stuff" keeps things from either collapsing or flying apart. That we owe our lives to it. Dr. Goldhaber, in particular, draws upon classroom experience, for he teaches a course that uses optics to introduce quantum mechanics.

The book is filled with mini-histories and mini-biographies of the "physics greats" of a century ago who wrestled with the findings of phenomena that revealed that Newtonian mechanics are not up to the task of explaining all the little stuff that underlies our everyday experience. Optical diffraction is just one such phenomenon. If there were no diffraction, you could put a really powerful eyepiece on an ordinary pair of binoculars and see to the end of the universe...if your eyes were sensitive to really, really dim light (telescopes are big mainly to collect more light; high resolution is also good, but is secondary in many cases).

Einstein imagined riding a beam of light from emitter to absorber. Nowhere have I read an explanation that, from the photon's point of view, nothing happens at all. The special theory of relativity, with length compression by Lorentz contraction, and time dilation, only applies to non-photons, and in particular, particles with mass. If you take Lorentz contraction and time dilation to their limits at v=c, the photon travels no distance at all, and does so in zero time. So there is nothing to experience! From a photon's point of view, the entire universe has zero size and time has no meaning; the big bang may as well never have happened!

What if we step back a tiny bit, and imagine the neutrinos that arrived in 1987, heralding the core collapse of an immense star in the Large Magellanic Cloud, Supernova 1987a (SN1987a). I haven't read any analysis of their apparent velocity, but it must have been only the tiniest whisker slower than c. Neutrinos do have some mass, perhaps a few billionths of the mass of an electron, so they tend to have near-c velocities. It is likely that the "clock" of those neutrinos registered only a few minutes during their journey of 187,000 light years, and the distance seemed at most a few hundreds or thousands of kilometers. Now, that is relativistic.

What did Einstein and Planck and Heisenberg do that got everyone (among physicists) all in a dither for the first half of the Twentieth Century? First, Planck applied a minimum limit to the "packets" of energy radiating from a heated object, in order to combine two competing, and incompatible, mathematical models of "black body radiation" into a single formula. Einstein later showed a simpler derivation of that formula. But at first, physicists just thought of it all as a mathematical trick. In between, Einstein had described a good theory of the photoelectric effect, which seemed to require that light be in finite packets, that we now call photons.

Photons are usually small in terms of the energy they convey. As mentioned above, the yellow-green color seen at 550 nm wavelength is carried by photons with an energy of 2.254 eV (electron-Volts). An eV is a 6 billionth-billionths of a joule, and a 1-watt current is defined as one joule per second. But molecules are also small, and the energies that underlie their structure are similarly small. UVb radiation from the sun, just half the wavelength, and thus twice the energy, of "550 nm yellow-green", breaks chemical bonds in your skin, causing damage that can lead to cancer. So use sunscreen! (The middle of the UVb band is close to 275 nm, with a photon energy near 4.5 eV; more than enough to knock a carbon-carbon bond for a loop.)

Book after book is filled with the stories of the founders and discoverers of quantum physics. This book puts it all into a context that the authors call the Quantum Moment. They use the word "moment" the way a historian uses "era". From 1687 until 1927, the Newtonian Moment dominated about 240 years of physics discovery. Once a critical mass of physicists had to accept that quantum phenomena were real, not just mathematical tricks, the Quantum Moment arrived. The stories of the epic battle between Bohr, who formulated the Copenhagen Interpretation, and Einstein, whose work stimulated Bohr and others, but from which Einstein then recoiled, is told here with more feeling and clarity than any other I've read.

Scientists have an emotional bond with their science. For many of them, it is their church, which they defend as keenly as any ardent fundamental Christian defends his church's theology. In the Newtonian Moment, phenomena whose initial state could be perfectly described were thought to be perfectly predictable. The math might be gnarly, but it could, in principle, be done. Quantum theory, and then quantum mechanics, blow by blow cracked open this notion and showed it to be a fantasy.

This is not just the problem of imperfect knowledge, rounding errors, or the need to simplify your equations to make them solvable. Heisenberg's Uncertainty Principle is not just a description of the way a measurement apparatus "kicks" a particle when you are measuring its location or velocity. What is Uncertain is not your measurement, but the actual location and velocity of the particle itself, at least according to Bohr. One implication of this with more recent application is the "no-quantum-cloning" principle, which makes certain applications of quantum computing impossible. However, they also make it very possible to create unbreakable cryptographic codes, which has the governments of the world (or their equivalents of our NSA and CIA) all-aquiver.

Then there's the cat. The authors give us the luscious details of Schrödinger's Cat satire, which he proposed as a slap against the notion of an "observer". Bohr and others needed some instruction from optics: every quantum particle is sensitive to, very literally, everything in the universe. All at once, and with no apparent limitation set by c. Heck, half the time, the cat is the only observer that matters. The other half, the cat is dead, and it ceases to matter to him. But, the authors point out, the air in the box is an "observer": the exchange of oxygen, water and carbon dioxide around a breathing cat are quite different from those near a dead one. So all we can say from outside the box with the cat in it, is that we can't decide the status of the cat without looking inside. We just need to remember that the term "observer" is very squishy.

I recall reading that even a pitched baseball has a "wavelength", according to the deBroglie formula. It is really tiny, only a few thousand times larger than the Planck limit of 10-35 cm, in fact. That means the deBroglie wavelength of a jet aircraft is much, much smaller than the Planck limit, which is why "real world" phenomena are easily treated as continuous for practical matters.

But the Cat, and the Uncertainly limit, show that the boundary between quantum and "classical" worlds is hard to pin down. Since that is the core of the Copenhagen Interpretation, it is seen to be weak at best, and in the eyes of some physicists, simply wrong. But there is no well-attested competing theory.

We must remember that the theories and mathematics of quantum "stuff" describe lots of "what" and a little bit of "how". They tell us nothing about "why". We don't know why there is a Pauli Exclusion Principle, that two electrons, and two only, can coexist in an atomic "s" shell, but only if they have opposite spins (and that "spin" is oddly different from the way a top spins). But we do know, that if it were not so, atoms would collapse in a blast of brightness, almost immediately, and the universe would collapse back into a reverse of the big bang, all at once and everywhere.

One scientist's work is not mentioned in this book, probably because he wasn't directly involved in the quantum revolution. But his work is pertinent in another way. Kurt Gödel formulated his Incompleteness Theorems in 1931, early in the Quantum Moment. Together, they show that no mathematical system can "solve" every problem that can be stated using its postulates, and that no mathematical system can be used to describe its own limitations. For example, there are rather simple polynomials that can be formulated using Algebra, but can only be solved using Complex Analysis. Even weirder if you know only Algebra, the simple formula X²=1 has two answers (1 and -1), but we tend to think that Xⁿ=-1 has only the answer -1 when n is odd, and is "imaginary" when n is even. But in Complex analysis, when n=3, for example, there are three answers, two of them involving an "imaginary" part.

At present, then, science has three boundaries to infinite exploration:

• Heisenberg Uncertainty. You can't know everything to infinite precision.
• Schrödinger Undecidability: You can't predict quantum phenomena on a particle-by-particle basis. Even if you could escape the Uncertainty Principle, you couldn't do anything of great use with the results (which would fill all the computers in the known universe, just describing a helium atom to sufficient precision).
• Gödel Incompleteness: You can't solve most of the questions being asked in the framework of quantum mechanics, not now, not ever, using the methods of quantum mechanics. QM appears to be the most Gödelian of mathematical systems, in that it asks so few questions that can be answered!

For scientists who grew up in the Newtonian Moment, it is like finding out that your church has no roof, and the rain and raccoons are getting in and taking over the place. No wonder Einstein was upset! We are in the Quantum Moment, nearly 90 years into it, and it may be another century or two before a new Moment supersedes it. Get used to it.

Everything affects everything

kw: book reviews, nonfiction, quantum mechanics, quantum theory, entanglement

A couple of quotes are in order:

"Entanglement is not one but rather the characteristic trait of quantum mechanics." —Erwin Schrödinger

"The Universe is not only queerer than we suppose, but queerer than we can suppose." —J.B.S. Haldane

Mathematics professor Amir D. Aczel, whose books illuminate the mysteries of science and mathematics for "the public", has tackled the fundamental scientific mystery in Entanglement: The Greatest Mystery in Physics. He has taken on the unenviable job to explain something nobody can understand to everybody else. This is truly to unscrew the inscrutable!

"Entanglement" has a specific meaning in physics. Certain processes, such as the mutual annihilation of an electron and a positron, or the two-photon cascade in the decay of "excited" states of many kinds of atoms, produce a pair of photons that have exactly opposite values of certain quantum properties such as spin or polarization. Then, if you measure the property of one of them, you know that the other one has the opposite property. This is most of the aforesaid meaning; to make it complete, two provisos are needed: firstly, that entanglement is possible between more than photons, but also electrons and any other kinds of particles that can be produced in such linked pairs (for example, the Cooper Pairs of electrons that make superconductivity possible); and secondly, that the actual values of the property to be measured exist in superposition on both particles, until a measurement is taken on one of them, and then in some way the other particle takes on the opposite property, instantly, with zero delay. This last proviso, the core of the Copenhagen Interpretation, is what so bothered Einstein, and he went to his grave believing it could not be so. He called it "spooky action at a distance".

Dr. Aczel uses twenty chapters filled with stories, mini-biographies, explanations, and an occasional formula, to tease out the development of the ideas and experiments that have led to the inescapable conclusion that entanglement really occurs, and that there are not some "hidden variables" that determine the values we will measure at the time of our choosing. This is a subtle point, and one I find hard to imagine, let alone describe.

However, superposition of states is not confined to entanglement. It is everywhere. It is the reason we cannot see with infinite clarity. We call it diffraction. Most people never encounter diffraction to any bothersome extent. But anyone who owns a microscope or telescope knows about it. However, you don't even need one of those. A pinhole will do.

Try this. Take three pieces of aluminum foil a few cm in size. For ease of handling, make suitable holes in cardboard and tape the foils over the holes. Pierce one with a 3-penny nail or a sharpened piece of 14-gauge wire. If carefully done, you get a 2mm hole (what works best for me is holding the foil against a piece of Styrofoam to pierce it). Pierce the second with the thinnest pin or needle you can find. With luck, you can make a hole in the range 1/2-3/4 mm in diameter. With the third, hold it against a piece of glass, and just barely poke it with the tip of your sharpest pin. You may need to twist the pin to get the point to just go through. With luck, you will get a hole 1/10 mm in diameter. In a darkened room, shine a flashlight through the largest hole, holding it about half a meter from the wall or a light-colored surface (such as a piece of paper taped to the wall). The light spot will be about the same size as the hole. Then shine the light through the middle-sized hole. The spot will be dimmer, but nearly the same size; definitely larger than the hole itself. Now shine the light through the third, tiny pinhole. You may not see much at first. Move the hole closer to the wall until you can see the spot. Even with it held rather close to the wall, the spot will be much larger than the pinhole, and may be as much as 5mm across.

It is a matter of ratio. The width of the spot divided by the distance between the hole and the wall is the same as the diameter of the hole divided by the wavelength of the light. A 1/10 mm hole is 100 microns. Yellow light has a wavelength of 0.6 microns, so the ratio is about 160:1. If the hole and light are held 500mm from the wall, the spot's size will be about 500/160 or 3mm. Now, why are tiny photons, with a wavelength of 0.6 microns, disturbed as they pass through a hole so much larger than they are? Amazingly, even the Hubble Space Telescope, orbiting above the blurring atmosphere, with a mirror whose diameter is 2.4m, disturbs the photons entering its aperture such that it cannot see with infinite clarity, but has a "figure of merit" of about 1/25 arc second at visible wavelengths. It cannot record an image with details smaller than that. Thus, when it looks at a galaxy a billion light years distant, the smallest features seen in the images it records are nearly 200 light years across.

The quantum mechanical explanation for diffraction is that the photon (or any other moving particle) has a rather diffuse "edge". Though it's wavelength is less than a micron, it has an extension and can "feel" the size of a hole it is passing through. The full consequence of diffraction is that there is no limit to the size of the "hole" that a moving particle can "feel". This has also been confirmed with electrons. An electron microscope makes much sharper pictures, and thus can be used at much greater magnification, than an optical microscope. However, even electrons with a wavelength (called the de Broglie wavelength; it depends on mass and velocity) of 1/10,000 micron are diffracted as they pass through the aperture in the magnetic lens of an electron microscope, so it takes rather clever (and large) design to make an electron microscope that can directly see atoms. But this has been done.

Suppose there were no diffraction at all? Then, even a small telescope could see to the ends of the Universe. The Hubble, being above the atmosphere, would be able to see aliens walking on the surface of planets anywhere in the visible Universe, depending only on the cost of making lenses that could increase its angular magnification by a factor of a few million or billion. In fact, your 1/10 mm pinhole could be a telescopic camera. Just put film a meter or so away from the hole (in a dark box), put it on a stable mount (with a clock drive if you are looking at stars), and expose for a long, long time, because you are gathering so little light. No matter how far away you put the film from the pinhole, the spot would be 1/10 mm across, so for higher resolution, just go longer. But even a "pinhole telescope" one meter long would have an effective f/ratio of 10,000. It would take a very long exposure even to make an image of the sun! That's the main reason professional telescopes are wide, to gather more light.

Diffraction implies that every moving particle is affected by everything in the Universe! On page 127 of Entanglement, an illustration shows an electron passing by a closed cylinder. There is a magnetic field inside the cylinder, but not outside. Still, the electron's motion is affected by the magnetic field. Some part of the electron's wave nature still enters the cylinder, even though it may pass by some distance away (the distance used in the experiment is not stated, but is likely to be a few mm).

I think you can see from the above discussion that I view the essence of quantum mechanics to be non-locality. Every photon, every electron, every atom or molecule in an "atomic beam" experiment, even every Buckyball (C60 molecule) in an experiment Aczel describes on p24, is "spookily" connected to the entire Universe!! Entanglement is simply one rather puzzling embodiment of such connections.

OK, why doesn't a jogger's direction get "disturbed" while running between two buildings? The jogger's de Broglie wavelength is about 10-³⁶m. The ratio is so huge, that the runner, aiming for the middle of the sidewalk half a block ahead, will only "miss" by a trillion-trillionth of a mm. Not enough to notice. And the jogger will take a few dozen steps in that same half block. The disturbance of each step, and ensuing corrections by the jogger, make the only effective difference.

There is another large-scale effect that shows why Star Trek teleportation is unlikely. Quantum entanglement makes it possible to "teleport" certain quantum properties, such as spin or polarization, from one particle to another, effectively making particle #2 identical to particle #1 (while destroying that property for #1), but in a different location. In effect, particle #1 jumps from the first location to the other, instantaneously. What about multi-particle systems, such as a human body? The number of protons and neutrons and electrons in a human body of, say 50kg mass (my wife's size), is about 6x10²³ times 50,000, times 1.7 (for the electrons), or about 5x10²⁸. That is, 50,000 trillion trillion particles. You have to measure not just spin or polarization, but identity (proton, neutron, or electron), location (to the nearest nanometer, or maybe to the nearest femtometer, I am not too sure), and velocity for each and every one of them, and take no more than about a millionth of a second to do so, then perform the quantum transportation to that number of particles at your target location. The measurement operation would effectively focus many quadrillions of quadrillions of watts of energy on that 50kg body, and vaporize it in much less than the millionth of a second it takes to make the measurement. It would be greater than a multi-megaton nuclear explosion. Neither the Enterprise nor the planet you were sending Captain Kirk to visit would survive intact.

The explanations in the book are clear, or as clear as possible for our limited mind to take in. To be sure, the experiments that confirm that entanglement really takes place do not give us any indication how or why it occurs, they just confirm that it does. Practically speaking, "why" is a theological question anyway. Science describes, and to some extent it can predict (that is what theories are for). And, to a lesser extent, it can enable technological achievements. Will a "quantum computer" or "quantum encryption" become practical, using equipment smaller than a battleship, or perhaps a kitchen stove? Possibly. Unlikely in my view.

Controlled by the Unseen

kw: book reviews, nonfiction, quantum mechanics, analysis

In one of Bill Cosby's earliest comedy routines, he asked, "Why is there air?" and answered, "So we'll have something to fill up basketballs." When physicists get reflective, they might ask, "Why is there anything?" That metaphysical question may be unanswerable, but the related, seemingly simpler question, "How does the floor hold me up?" has at least a partial answer: "Because the floor, and you, are composed of fermions."

In the middle of my seven-year slog to get a Bachelor's degree, I spent two years as a physics major. Nearly all the senior-level physics courses were about particle physics and things like statistical thermodynamics, which is about averaging the behavior of large numbers of particles. I didn't realize it at the time, but Stat-Thermo is really about exploring the boundary between particle physics, AKA quantum mechanics, and classical physics. But at this point, at least some of you reading this are saying, "Hold on, back up a bit. What is a fermion?"

You may have heard of the rule that no two objects can occupy the same space at the same time. To a physicist, this is only half true, because all particles are objects, and all "human scale" things are made up of particles, and only half of the particles obey the rule. Maybe you've heard of photons. They are the particles that we call "light". We see because of light; our eyes are light detectors. Some photons are of infrared "light", which our eyes cannot see, but our skin can detect because we feel the warmth. Some are of ultraviolet "light", which our eyes also cannot see, but our skin is like a slow photographic film, and tans or burns when we are exposed to it. There are several other varieties of "light", such as x-rays or radio, all of them being photons of various energies.

Light violates the rule. You can shine two light beams across one another, and the photons don't bounce off one another. If you had a kind of high-speed super-microscope and could watch photons saunter by, you might see a pair of photons slip right through one another without interacting at all. Ask a physicist why, and the answer will be, "Photons are not fermions." Then what are they? Bosons. OK, now, so what???

There are two kinds of particles. Fermions obey Fermi-Dirac statistics, named for the famous physicists who worked out the rules they follow. They cannot pass through one another the way photons do. Instead, they collide, or scatter. All the things we call material objects are composed entirely of fermions. They are "why" solid matter is solid, and why you don't drop through the floor. Bosons obey Bose-Einstein statistics, named for the famous physicists who worked out the rules they follow. Bosons such as photons do not interact with each other, but they do interact with fermions. There are different kinds of fermions and different kinds of bosons. And a key point: when fermions interact, they do so by exchanging bosons. You are held up by the floor because gajillions of bosons are bouncing back and forth between "your" fermions and the ones that make up the floor.

This is just the beginning of the road into the quantum world, a world explored and explained in 101 Quantum Questions: What You Need to Know About the World You Can't See by Kenneth W. Ford, a retired physicist. The material I ran through in the prior few paragraphs actually takes up several of the Questions in the book. Dr. Ford is one of the world's great explainers. I already know a lot of physics, but I learned a great many interesting things from him.

One is, that in certain cases, under very carefully arranged conditions, particles of solid matter can be bosons, and it relies on a simple mathematical rule we all learned in grade school: The sum of an even number of odd numbers is an even number. But if you have any number of even numbers, and add them to any odd number, the sum is odd. In the quantum world, electrons and protons and neutrons are fermions. A pair of fermions such as electrons can, under the right circumstances, be a boson. Bosons have "evenness" as a characteristic. Pairing of electrons is responsible for superconductivity, and a superconducting magnet the size of a washing machine is the core of a MRI scanner. But groups of electrons and protons and neutrons can "add up" to an even number, and that means certain atoms can be bosons, and at very low temperatures, a gas of the right kind of atoms become a "Bose-Einstein Condensate" in which the atoms no longer bounce off one another, but behave more like very slow photons. Way cool.

Another important item is the Correspondence Principle (Question 3). It refers to the change in perception when you get lots and lots of quanta, and things average out until they can be quite accurately studied using classical physics. It's like this. Take a piece of chalk. Break it in half. While you could say you have two half pieces of chalk, you actually have two pieces of chalk, because the word "piece" is not divisible. Take one of them and break it again. Keep doing so. According to classical mechanics, you can do this forever. But we know it isn't so. The chalk is made of bits of tiny shells, about 0.01mm in size. If you start with a standard piece of chalk, and cut it with a cleaver carefully, switching direction of cutting from time to time, you can cut it about 32 times, at which point the two "pieces" each consist of two or three shell bits each. Under a microscope, pick one with two shell bits and cleave a 33d time. Now you need a different strategy.

A little shell cube (for simplicity) 0.01mm on a side contains about 10²³ atoms of a mix of calcium, carbon and oxygen. That means, with the right equipment, including an Atomic Force Microscope (AFM), you can divide this bit of shell another 76 times, at which point you have either one atom or two atoms in the remaining "piece". Without an atom smasher, you're done dividing, and you've "only" reached division number 109.

By the way, if you do have an AFM handy, you can shortcut the whole process by tickling a single atom away from its location, but don't be surprised if getting a chosen atom of C, O or Ca to move is fiendishly difficult. You're in a realm where the electrons in the CaCO₃ molecules have been shared in such a way that the atoms strongly resist being relocated. You have entered the quantum realm. You are on the other side of the correspondence principle in which electron behavior is more important.

The physical characteristics of the original piece of chalk depend on the average behavior of all the little bits of shell, and they way that the shell pieces might break while you are drawing with the chalk on a blackboard depend on the average behavior of quintillions of molecules of CaCO₃. You can be sure, just by drawing with chalk, that you aren't breaking up any of the molecules. You need to cook the chalk at a few hundred degrees to do that!

OK, I used the term "average behavior" a couple of times there, rather loosely. The "strength" of the chalk, or of an iron bar, or of a piece of glass in the window, is a measurable quantity, and chances are that if several careful people make measurements of the strength of an object like an iron bar, they will produce results that are very, very similar. When that is the case with a quantity, we call it a fixed quantity. So the ultimate strength of iron is 68,000 psi (I don't know what that is in KPa). But if you "get into" an iron bar at the atomic level, you'll find several causes of variation. The strength of the force between two iron atoms depends on if their magnetic axes are in the same or opposite directions, on the possible presence of a grain boundary between them, and even whether one of them is a different isotope. But in a 0.1kg iron bar containing around 10²⁵ iron atoms, all those effects average out.

So where is the boundary between quantum randomness and classical fixedness? We can explore that with a thought experiment. Consider a baseball pitcher. They are famed (and paid millions) for being able to consistently hit the strike zone. The best pitchers not only hit it, but they can get most of their pitches near the lower inside corner where they are hardest to hit hard. Beware of a pitcher who throws too many in the "sweet spot" just a bit further out than the center of the zone. Home run hitters love a pitch that goes there; it'll soon be out of the park.

A pitcher's accuracy is based on the average behavior of many neurons in the brain, that control the motions of the pitching arm. A neuron is a very noisy signal processor, and it takes many neurons to produce consistency. Let's model one neuron as a on-off switch that "tries" to switch at a specific time. It may be off by a couple of milliseconds one way or another. Start with the world's worst neuron. It is so bad that, if it were the only control of the pitcher's arm, he could throw in the half-sphere that is "toward" home plate a little over half the time. Now if we combine three neurons (it takes three to calculate a standard deviation), the accuracy of the pitch may improve a little. Take larger and larger numbers, some firing earlier, some later, but the more you use, the closer to the right timing you get. We'll use random numbers to simulate the situation.

There are several ways to obtain random numbers. The modern RAND functions in computer software do an excellent job of simulating shot noise, the "gold standard" way to produce random numbers before there were computers. If you produce 10,000 RAND function results, and make a chart of the results, you'll find why it is called a Uniform distribution. Any number between 0 and 1 can occur. I did so, and gathered them into buckets from 0-0.01, 0.01-0.02, and so forth. The expected content of each bucket is 100; the actual range was 81 to 121, with no particular pattern (a graph below will show this).

Let's produce combinations. 10,000 sums of just three RAND functions have some interesting characteristics. This is a lot like our "worst neuron", if we assume that the entire range from 0 to 1 is a complete circle of 360 degrees. Half the "pitches" are within 43° of the strike zone, and most of them are in the general direction of home plate. The strongest "bucket" around the center of the strike zone now has 250 of the 10,000. Next step, multiply by four, using 12 numbers each.

Twelve RAND functions per sum gives us a lot better focus. A real strike zone is about two feet high and two feet wide. From 66 feet away, it is a square less than 2° on a side. In terms of our hundred buckets, just considering the right-to-left angle, the strike zone is smaller than one bucket; each bucket is 3.6° across. Of 10,000 12-RAND sums, 491 made it into the central bucket, and more than half the pitches are within 18° of center.

Another step of 4x: 10,000 sums of 48 RAND functions. Now the central bucket has received 950 pitches, and half of all pitches are within 7.2° of the center of the strike zone. 99% of the pitches are within 36°.

A final step of 4x: 10,000 sums of 192 RAND functions. The central bucket contains 1,830 pitches, or one out of 5.5. The true "strike zone", being 2/3.6 of a bucket, must have been hit about 1,000 times or a tenth of all pitches. What do we need for half the pitches to be in the strike zone? Continuing the trend we see in the numbers, it would take about 960 RAND functions per sample to produce this kind of accuracy, or 320 of the "world's worst neuron". The following chart shows the numbers behind these few paragraphs:

We expect the frequency distribution for a single RAND function per evaluation to be a uniform function, and we see that Freq 1 is quite uniform, with a little of the jiggle one would expect. Freq 3, the "world's worst neuron" has a visible central tendency, but only a little. By the time you get to Freq 192, there is a nice, sharp distribution. If we calculated and plotted a Freq 1 million, everything would be in the central bucket, a simple spike. At that point, re-running the experiment several times (lots of computer time), you'd find such tiny differences between the average values that you could say you have a very accurate pitcher! Probably, a real pitcher's throwing arm is controlled by a few thousand neurons or a few times ten thousand, which is why the good ones have control within the strike zone.

Physical properties of objects big enough to see, being based on the average of millions of millions of quanta, are similarly "fixed" to any level of accuracy we can measure. The little bit of shell we talked about, that is 0.01mm across, doesn't have millions of molecules, it has trillions of billions. It is hard to see except under a powerful microscope; it looks like a small chunk of glass at 400x. But it is already large enough that classical measurements of its properties are very accurate.

There is one quantum effect that you can see with a microscope: Brownian motion. It works best with pollen, but tiny, tiny bits of shell will also work. Get your microscope ready, and it is best to use dark field illumination, so the pollen grains or bits of shell look like bright points on a dark background. Stir an infinitesimal pinch of powder into a drop of water on a microscope slide, drop on a cover slip and look at it immediately. Focus on one of the dimmer bright spots (one of the smallest bits). You'll see that it jiggles a little. It is small enough that the difference in the number of water molecules hitting one side doesn't exactly balance the number hitting the other side, so it is moved back and forth, up and down, in a random fashion, just a tiny bit. Brownian motion provided the first clear evidence of the reality of atoms. It can't happen according to classical mechanics (explained with Question 5).

Well, these are just a couple of the ideas this book triggered in me. Read it and see what it triggers in you. I have hardly mentioned the many persons the author wrote about as he told the stories behind his 101 questions. There is a lot more to Fermi than just the way fermions behave, and there is a lot more to quanta than just that they do this or that thing when pushed into proximity. Read and enjoy!