Quantum Tunneling

Classical Mechanics

Quantum tunneling allows us a fascinating glimpse at the workings of the subatomic world. We have learned in the last one hundred years or so that atoms aren't really the miniature solar systems that we all imagine them being—a relatively huge nucleus with tiny electrons swirling around it like planets. In fact, we can't really say much of anything about where electrons are or where they're going. It turns out that all we can really say is where they might be at a given time. Let's back up a little and try to illustrate this principle with an example.


Imagine a roller coaster on a track shaped like the black line pictured to the right. It's pretty easy to imagine that if you started on the left side where the red line intersects the slope that when they drop, the cars would easily roll over the middle hump and up the other side to the red-black intersection point on the right side (in fact, it would go to that point exactly if we neglect air resistance and friction). Even though the roller coaster dips pretty deeply just to the left of the center hump, it's not difficult to imagine that the cars would be going fast enough to get over the hump with speed to spare.


Now imagine that we start on the left side again, but this time at the point where the blue line intersects the track. Dropping from this height, we can see that the cars don't have enough energy to get up over the hump. Starting on the left side means never getting to the right side and vise-versa. We'd need some kind of chain (like the ones that they use in roller coasters to pull you up the first hill) to do enough work on the cars to get them over the hill. In other words, unless the roller coaster cars got energy from somewhere else (like being pushed at the beginning or dragged up the slope with a chain) they will never see the right side of the track.

Everything we just discussed comes from "classical" mechanics, which imagines that everything is solid and exists where it's supposed to exist in the way that it's supposed to exist. Unfortunately classical mechanics breaks down when the system is really small (such as in an atom). We'll talk about one way that it stops working after a little lesson in terminology.

Terminology

The term "potential well" refers—in effect—to the dips in the track. In the case of a roller coaster, the "well" is formed by gravity in that the deeper you go into it, the more energy you need to get out. We can see a more solid example of this in the system of the earth. Imagine that our planet is resting on a large rubber sheet, causing a depression. To get into outer space, we must climb out of the hole first which means that we need to have enough energy to get out without falling back in. Once out, we are free to shut off the engines and simply coast (which is how we got to the moon) because there is no danger of falling back into the earth's potential well.

Potential wells can be made from all sorts of things. For instance, a large chunk of positive charge (such as an atomic nucleus) creates a potential well into which negative charges, like electrons, fall (the reason they don't ever fall into the absolute center is a complex question that we won't get into here). In the same way as with gravity, for an electron to escape a nucleus' potential well, it must have at least enough energy to climb out of the hole.

The other term that we need to understand is the "wave function." We refer more to the wave functions of tiny particles than to where they are or how fast they're moving because it allows us a little more accuracy in measurements. In effect, we say that particles have a tendency to behave like waves under the right conditions. For example, electrons (which have definite mass) have a wavelength and a frequency and can even diffract (bend around obstacles) like water waves do. As a result, when an electron is in a potential well, instead of thinking of it as a ball rolling back and forth between to peaks of maximum energy (like a roller coaster), we think of it as being a wave bouncing back and forth between two walls (think of dropping a pebble in a bowl of water and watching the waves bounce around). The size of the wave function at a certain point represents the probability of the electron being found at that spot. That is, the bigger the wave function, the more likely it is to find an electron there if we choose to measure it.

Quantum Tunneling

Remembering the blue track example, imagine an atomic potential well of that shape. Perhaps there are two nuclei of differing charge (so that one is deeper, or has a greater propensity to pull an electron than does the other) close enough together to affect a single electron. We can imagine that the larger of the two nuclei could, under the right circumstances, trap the electron in the deeper well, making it impossible to overcome the hump and orbit the other, smaller nucleus. Classically, this is exactly how we'd explain it.

However, we have observed that sometimes the electron, after having been trapped in a potential well and without enough energy to get out of it, sometimes escapes. This is evidence that the electron is behaving like a wave. Imagine clapping (thus making a sound wave) in a room with closed doors. Can a person outside the room hear you? If you clap loud enough, the sound wave will hit a wall and make it vibrate a little, causing the air on the other side of the wall to vary in pressure a little. That pressure variance becomes a pressure (sound) wave that can travel to another person's ear. It's not as loud, but it's still the same sound.

So also with an electron, when it's wave function hits against a wall, it sometimes tunnels into it. If there is another potential well on the other side close enough and deep enough, the electron will be transmitted to the other side (though with a much weaker wave function). Unlike with sound, the fact that the wave function has a smaller amplitude does not indicate that the electron is any less of one. In fact, it is the same electron that was on the other side of the well. The fact that it has a smaller wave function only means that it is less likely to be there if we chose to measure it. However, that probability only applies to the exact time of measurement. In all other times, the two nuclei behave as if the electron were always with it (even though measurements may seem to indicate that it is with one particular nucleus 80% of the time).

Maybe this all sounds obscure as if it couldn't possibly matter to a normal person, but what I've just described is a covalent bond which is the kind of bond that keeps two hydrogen atoms attached to an oxygen atom in water molecules (and, consequently, the bond that makes most or all of the molecules in air stay together).

Conclusion

The take-home message of quantum tunneling is that small particles act very differently from large ones. I guess, technically that grapefruits have wavelengths too, but for reasons associated with the Uncertainty Principle, the effect that quantum mechanics has on large objects is negligible. But on very small scales, classical mechanics breaks down. Objects that we imagine to be solid become waves, Newtonian mechanics breaks down, and nothing seems to work the way we expect it. The usefulness of knowing exactly how they act on that scale is important, though, as we can see by examining its effects. Knowledge of quantum tunneling leads us to innovations such as flash memory (which the thumb drive in your pocket uses), semiconductors (a fundamental component in computers) and chemistry.

Universe Synthesis: Part III

In the last two installments of the Universe Synthesis series, I explained in relatively simple terms what happened during the first 380,000 years of the life of the universe. In the first trillionth of a second or so, the four major forces that we know today split from the single force that they started out as, and then the first inklings of matter formed. We left off with the creation of hydrogen and helium.

The problem with an expanding universe is that as is expands, it loses kinetic energy. Imagine two pots of boiling water (each with an equal amount of water in them). If you set one on the stove and dump the other one on the floor, which will cool down faster? Clearly, the water, as it expands, radiates (and conducts) more heat away from it at a faster rate. In the same sort of way—as the universe expands—it cools down. Unfortunately, it takes energy to force atoms together to make heavier atoms. At 380,000 years after the Big Bang, the universe is overwhelmingly (if not completely) devoid of any element heavier than helium. With the matter in the universe cooling and spreading out at an alarming rate, how did the rest of the periodic table form? Where do we get carbon, oxygen, nitrogen, iron, and gold?

The answer is—in the simplest form—gravity. The fast expanding space is filled, intermittently with enormous clouds of hydrogen (mixed with a very little helium). But the clouds aren't homogeneous; they're clumpy. In some places there are a few more atoms per cubic centimeter, making the region very slightly more massive than the areas around it. Believe it or not, this is the beginning of a star.

The slightly-more-massive clump has just a little stronger gravity than the other slightly less dense regions. As a result, other hydrogen atoms are statistically more likely to fall into the clump and join it. Over a very long time, the clump gets larger and larger, becoming more dense and more compact. As it gains matter (again, only hydrogen), the matter tends to fall towards the center of gravity, causing a particularly large mass of hydrogen gas to start forming there. The gas pushes on itself, or rather, it pulls it self together by its own gravity until the pressure is so great that it ignites.

Ignition, here, does not have the same meaning as it does on earth. The hydrogen is not burning, per se, it's fusing. The pressure is so great that the atoms are fused together. Two protons (which is just a hydrogen atom without its electrons) are fused into deuterium (still hydrogen, but with an extra neutron), deuterium and another proton make helium. Helium fuses into lithium, which fuses into beryllium. This is called the proton-proton chain. In heavier stars, there's enough thermal energy to initiate the CNO cycle, which creates primarily carbon, nitrogen, and oxygen. With each fusion reaction, a little bit of energy is released as light. The light you see when you look at the sun (note: don't look at the sun) is the byproduct of the proton-proton chain.

Atoms continue to fall toward the center of gravity. As they do, and because they do not fall uniformly in every direction, the whole mass begins to spin. As it does, a disc that is perpendicular to the axis of rotation begins to form around the newly formed star. Matter begins to collect into the disc. Soon a star is happily burning. Around it, other pockets of dense hydrogen have started to burn. The whole collection of them is now a galaxy.

In course of time, the star runs out of material to burn. Heavy stars can get big and hot enough to force helium, carbon, and other heavier elements to burn, but eventually the matter in the star ceases to fuse. Either gently, bit by bit, or in a violent explosion, the layers of new elements are ejected into the interstellar medium (left-over hydrogen), enriching the surrounding area with new elements. In the particularly large explosions, the atoms gain enough energy to fuse into extremely heavy elements such as gold, copper, tungsten, or mercury. Since the interstellar medium is still, even after all that, predominantly hydrogen, the whole process stars again. Only this time, the conglomerating gas is enriched.

When the disc forms around this new star, the heavier elements stay behind as the hydrogen and helium fall towards the center. Close to the star, almost all of the hydrogen falls into the giant fusion reactor leaving behind rocky clumps of carbon. These clumps eventually collide and conglomerate themselves, forming huge spinning rocks that eventually form terrestrial planets. Further out, lots of hydrogen and helium remains to collect into large, dense clouds not big enough to become stars; they become gas giants. The further away a gas giant is from the star, the more molecules are able to form in its atmosphere (being cool enough to form them without immediately breaking them apart again) such as methane (which is what give Neptune and Uranus that nice, blue color).

Lest you think that we'll one day run out of building materials, consider that after 14 billion years of element synthesis, the detectable matter in the universe is still 75% hydrogen and a little less than 25% helium. That is, everything else that is not those two elements comprises much less than 1% of the total matter in the universe. Even then, consider your car, your kitchen appliances, a gold deposit in a mountain, or the circuitry in your computer. A long time ago, in a galaxy far away (I couldn't resist, but seriously...) every single one of those atoms was being shoved together in the first few seconds after a violent supernova explosion. And every single breath of air you take is filled with atoms that were fused inside a star millions of years ago. We live and breathe stardust.
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So, that's how it happened. Or, at least, that's the best we can do at explaining it right now. This model is constantly being reformed and reworked, and new processed are constantly being discovered. One of the most amazing things to me is that almost all of this information was deduced by astronomers looking at the sun and other stars (note: do not look at the sun unless you are a trained professional). The only information from those sources that we can get is the light that they give off. In other words, astronomers found a way to deduce all of this just by looking at patterns of light given off by stars and combining it with what we already know about physics on earth. That, to me, is an amazing accomplishment.

Heisenberg Uncertainty Principle

Imagine that you take a picture of a moving car. Depending on the kind of camera you have, your pictures will develop in one of two ways. First, it is possible that your camera has a really slow shutter and that the car is blurry. If you knew the shutter speed of your camera, you could make a pretty good guess at how fast the car was going by studying the size of the blur. Even then, however you wouldn't be exact. The only problem would be to describe exactly where the car was at the moment you took the shot. In fact, you couldn't say that it was anywhere precisely at the time you took the picture. All you could do with any degree of certainty is decide that the car was between two definite points (the beginning and end of the blur) during the entire second that you took the picture.

The other kind of shot would have been taken with a camera that had a very, very fast shutter speed. The picture would turn out crisp, with almost no blurs at all. Finally, we know exactly where the car was at the instant the picture was taken. Unfortunately, by gaining this information, we've lost information that we could have known. Now, looking at the picture, we have no way of saying how fast the car was moving. For all we know, it could be standing still.

In either case, the picture cannot ever tell us everything we want to know about the car. We get one side or the other. And it has nothing to do at all with the quality of the camera. Even if we were using the best camera in the world, a slow shutter would tell us lots about velocity and a fast shutter would give us a good idea of position. This conundrum is the basic idea of the Heisenberg Uncertainty Principle.

Before the advent of quantum physics, it was believed that if we knew the exact position and velocity of a particle then we could determine exactly where it would be at any point in the future. I suppose we could still think of that as being true. The problem is trying to measure both of those quantities simultaneously. We encounter the same problems as we did with the camera. We can only simultaneously determine momentum and position to a certain degree of accuracy.

It's important to realize that the illustration that I gave above with the car is only a metaphor to help us describe the real Uncertainty Principle. In actuality, that "certain degree of accuracy" is an extremely small number (≈10^-34) and is thus only really an issue when we are talking about very small things like electrons.

The issue is not, however, as trivial as the particles are small. What are the implications of the Uncertainty Principle? First, we learn that it is impossible, regardless of the quality of the instrument, to learn everything about everything. The information provided to us on the sub-atomic scale is finitely limited. But, that's not necessarily a bad thing. Sometimes it is useful to know in precise terms that which we do not know. Such limits imposed on us by the universe have helped us to understand the shape, size, and configuration of an atom, and thus to describe more completely atomic interactions.

Further, the very idea that we cannot be exactly precise in our measurements caused a paradigm shift that defines the way we think about science today. Before this principle (and others such as de Broglie's wave mechanics and wave-particle duality), people were generally under the impression that the universe was deterministic—that every future event could theoretically be predicted. Now, we view the universe as being probabilistic instead—that we can only know the probability of a future event to happen. The probabilistic ideology, though seemingly less "correct" was something of a step away from perfect—though ultimately incorrect—answers and a step toward the best philosophy of understanding at which we can arrive.