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!
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