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CHAPTER 6
No Digging Required:
Quantum Tunneling
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We’re sitting in the garden, enjoying a beautiful sunny afternoon. I’m lying on a deckchair reading a book, and Emmy is sprawled out on the grass, basking in the sun and keeping an eye out for squirrel incursions.
“Can I ask a question?” she asks.
“Hmm? Okay, go ahead.”
“What do you know about tunneling?”
“Tunneling, eh?” I put my book down. “Well, it’s a process by which a particle can get to the other side of a barrier despite not having enough energy to pass over the barrier.”
“Barrier? Like a fence?”
“Well, metaphorically, at least.”
“Like the fence between this garden and the next?” She looks really hopeful.
“Oh. Is that what this is about?”
“There are bunnies over there!” She wags her tail for a minute, then looks crestfallen. “But I can’t get to them.”
“True, but I don’t think tunneling is the answer. It works for small particles, but wouldn’t work for a dog.”
“Why not?”
“Well, you can think of a barrier in terms of potential and kinetic energy. For example, right now, all your energy is potential energy, because you’re not moving. But you could start moving, say, if you took off after a squirrel, and turned that potential into kinetic energy.”
“I’m very fast. I have lots of energy.”
“Yes, I know. You’re a great trial to us. Anyway, whether you’re sitting still, or moving, you have the same total amount of energy. It’s just a question of what form it’s in.”
“Okay, but what does this have to do with the fence?”
“Well, you can think of the fence as being a place where you can only go if you have enough energy. For you to be at the spot where the fence is, you would have to jump very high or else occupy the same space as the fence, and either would take an awful lot of energy.”
“I can’t jump that high. That’s why I can’t get the bunnies.”
“Right. You don’t have enough energy to get over the fence. And because you don’t have enough energy, you can’t end up in the neighbors’ garden, and everybody is much happier that way, believe me.”
“Except me.” She pouts.
“Yes, well, except you.” I scratch behind her ears by way of apology. “Anyway, quantum mechanics predicts that even though you don’t have enough energy to go over the fence, there’s still a chance that you could end up on the other side. You could just sort of … pass through the fence, as if it weren’t there.”
“Like the bunnies do!”
“Pardon?”
“The bunnies. They go back and forth through the fence all the time.”
“Yes, well, that’s because they fit between the bars of the fence. It has nothing to do with quantum tunneling.” I stop for a moment. “Of course, it’s not a bad analogy. The bunnies don’t have enough energy to go over the fence, either, but they can go through it, and end up on the other side. Which is sort of like tunneling.”
“So how do I tunnel through the fence?”
“Well, you could eat fewer treats, and become thin enough to pass between the bars like the bunnies do.”
“I don’t like that plan. I’m a good dog. I deserve the treats I get.”
“And you get the treats you deserve. The other option would be quantum tunneling through the fence, but quantum tunneling isn’t something you do, it’s something that just happens. If you send a whole bunch of particles at the barrier, a small number of them will show up on the other side. But which ones go through is completely random. It’s all about probability.”
“So, I just need to run at the fence enough times, and I’ll end up on the other side?”
“I wouldn’t try it. The probability of a particle tunneling through a barrier depends on the thickness of the barrier and the quantum wavelength of the particle. The probability of a fifty-pound dog passing through a half-inch aluminium barrier would be something like one over e to the power of ten to the thirty-six. Do you know what that is?”
“What?”
“Zero. Or near enough to make no difference. So don’t go throwing yourself at the fence.”
She’s quiet for a minute.
“Anyway, I hope that answers your question.” I pick my book back up.
“Sort of.”
“Sort of?”
“Well, the quantum stuff was interesting, and all, but I was thinking of classical tunneling.”
“Classical tunneling?”
“I was going to dig a hole under the fence.”
“Oh.”
“It’s a good plan!” She wags her tail enthusiastically, and looks very pleased with herself.
“No, it’s not. Only bad dogs dig holes.”
“Oh.” Her tail stops, and her head droops. “But I’m a good dog, right?”
“Yes, you’re a very good dog. You’re the best.”
“Rub my belly?” She flips over on her back, and looks hopeful.
“Oh, okay …” I put my book back down, and lean over to rub her belly.
“Tunneling” is one of the most unexpected quantum phenomena, where a particle headed at some sort of obstacle—say, a dog running toward a fence—will pass right through it as if it weren’t there. This odd behavior is a direct consequence of the underlying wave nature of quantum particles seen in chapter 2.
In this chapter, we’ll talk about the essential physics concept of energy, and how energy determines where particles can be found. We’ll see that the wave nature of matter allows quantum particles to turn up in places that classical physics says they can’t reach, passing into or even through solid objects. We’ll also see how tunneling lets scientists build microscopes that can study the structure of matter, making possible revolutionary developments in biochemistry and nanotechnology.
THE ABILITY TO GET THINGS DONE: ENERGY
In order to explain quantum tunneling, we need to first talk about the classical physics of energy. While the term “energy” has passed from physics into more general use, its physics meaning is slightly different from its everyday, conversational use.
A one-sentence definition of the term “energy” in physics might be: “The energy content of an object is a measure of its ability to change its own motion or the motion of another object.” An object can have energy because it is moving, or because it is held stationary in a place where it might start moving. Every object has some energy simply because it has mass (Einstein’s E = mc2) and because its temperature is above absolute zero.* All of these forms of energy can be used to set a stationary object into motion, or to stop or deflect an object that is moving.
The most obvious form of energy is kinetic energy, the energy associated with a moving object. The kinetic energy of an object moving at an everyday sort of speed is equal to half its mass times the velocity squared, or as it’s usually written:
KE = ½ mv2
Kinetic energy is always a positive number, and increases as you increase either the mass or the speed. A Great Dane has more kinetic energy than a little Chihuahua moving at the same speed, while a hyperactive Siberian husky has more kinetic energy than a sleepy old bloodhound of the same mass. Kinetic energy is similar to momentum, but it increases faster as you increase the velocity, and unlike momentum, it doesn’t depend on the direction of motion.
Objects that are not already moving have the potential to start moving due to interactions with other objects. We describe this as potential energy. A heavy object on a table has potential energy: it’s not moving, but it can acquire kinetic energy if a hyperactive dog bumps into the table and it falls on her. Two magnets held close to each other have potential energy: when released, they’ll either rush together or fly apart. A dog always has potential energy, even when sleeping: at the slightest sound, she can leap up and start barking at nothing.
Energy is essential to physics because it’s a “conserved quantity”: the law of conservation of energy says that while energy may be converted from one form to another, the total amount of energy in a given system does not change. This turns some difficult problems into bookkeeping exercises: the total energy (kinetic plus potential) has to be the same at the end of the problem as at the beginning, so whatever energy is left over when you subtract the final potential from the total has to be kinetic energy.*
To get a better feel for how energy works, let’s think about a concrete example: a ball thrown up in the air. As any dog knows, what goes up must come down, and a ball that’s thrown up with some initial velocity will slow down, stop, and then fall back down. You can see this in the figure on the next page, which shows the height of a ball at a series of regularly spaced instants. At low heights, the ball is moving fast, and covers a lot of ground from one picture to the next. Near the top of its flight, the ball moves very little, and at the very peak, it’s perfectly still for a split second.
We can describe this flight in terms of energy. An instant after the toss, the ball is moving, so it has lots of kinetic energy, but it’s near the ground, and has no potential energy. The total energy is thus equal to the kinetic energy. We can think of this as being a kind of energy supply, like a jar full of treats, shown by the black bar in the figure. As the ball moves upwards, its kinetic energy decreases (because it’s not moving as fast), and its potential energy increases (because it’s higher off the ground). The kinetic energy level drops, replaced by potential energy (shown in gray), but the total energy remains the same.
A ball thrown up in the air starts out moving rapidly upwards, slows due to gravity, and turns around and falls back. The pictures show the position of the ball at regular intervals. The bars show the energy of the ball, with black indicating kinetic energy and gray indicating potential. Near ground level, all of the energy is kinetic energy, while at the peak of its flight, all of the energy is potential energy.
At the peak of its flight, the ball has potential energy, but no kinetic energy, because for a split second, it’s not moving at all. On the way back down, it goes through the same process in reverse: it starts with potential energy but no kinetic energy, and ends up with kinetic energy (the same amount it started with) but no potential energy.
“You’ve got this backwards, you know.”
“I do?”
“Yes, the jar full of treats should be the potential energy, because treats have the potential to give me energy, when I eat them. The empty jar should be kinetic energy, because I run all over the place after I eat treats.”
“You may have a point there. Of course, no analogy comparing energy to dog treats is ever going to be perfect.” “Why is that?”
“Because while you can convert potential energy to kinetic energy, you can also convert kinetic energy back to potential energy. Which would be like putting treats back in the jar.”
“Oh. I never do that.”
“Believe me, we’ve noticed.”
We also see from looking at the energy of a ball in flight that energy limits the motion of the ball. The ball starts with some total energy, all kinetic, and as it goes up, it converts that to potential energy. Once the initial kinetic energy has been turned into potential, the ball stops moving. The ball can’t go beyond a certain maximum height, because that would require its total energy to increase, and that can’t happen.* The maximum height the ball can reach with a given amount of energy is called the “turning point,” because the ball reverses direction at that point. Heights above the turning point are “forbidden,” because the ball doesn’t have enough energy to reach them.
FOLLOW THE BOUNCING WAVEFUNCTION:
A QUANTUM BALL
The thrown ball is a simple example of energy in action, and thinking about it in energy terms may not seem that helpful. Energy analysis can be applied to many more complicated systems, though, including situations that can only be described mathematically using energy. As a result, energy is one of the most important tools that physicists have for understanding the world.
Energy is especially important in talking about quantum mechanics. As we saw in chapter 2, quantum particles do not have a well-defined position or velocity, so there’s no way to keep track of those properties as we do with a classical system. Conservation of energy still applies, though, so we can understand quantum systems by looking at their energy. In fact, the Schrödinger equation uses the potential energy of a quantum object to predict what will happen to the wavefunction of that object, so every calculation done in quantum mechanics is fundamentally about energy.
We can see how energy relates to wavefunctions by imagining a quantum ball thrown in the air. We can predict some features of its wavefunction just by using what we know about its energy. Kinetic energy is similar to momentum, and we know from chapter 1 (page 10) that the momentum determines the wavelength. Near the ground, where the kinetic energy is high, the ball should have high momentum and thus the wavefunction should have a short wavelength. Higher up, where the ball is moving slowly, the ball has low momentum, and the wavefunction should have a longer wavelength. We also expect the probability of finding the ball above the turning point to be zero, because the ball should never go higher than allowed by its initial energy.
We can calculate the wavefunction for this system, and we find a probability distribution that looks like this:
The solid curve shows the probability distribution for finding a quantum particle at a given height. The dashed curve is the probability distribution for a classical particle.
Looking at this graph, we see more or less what we expect. The probability distribution oscillates more rapidly at low elevation (on the left), than at higher elevation. On closer inspection, though, we notice something strange: the probability does not go to zero exactly at the classical turning point (at about 17 units of height, where the dashed curve goes to zero). It drops off to zero, but there’s a range of heights above the turning point where the probability is still significant. There’s some probability of finding the ball at heights it should never be able to reach!
Why doesn’t the probability go to zero right at the turning point? Well, if it did, there would be a sudden change in the wavefunction at that point. We know from chapter 2 (page 47), though, that making a sudden change in the wavefunction requires adding together a huge number of wavefunctions with different wavelengths. Many wavelengths means a large uncertainty in the momentum, and therefore a large uncertainty in kinetic energy. But we don’t have a large uncertainty in the energy—we know how hard we threw the ball. A small energy uncertainty leaves us with a large uncertainty in the position of the turning point, which means no sharp changes in the wavefunction and a wavefunction that extends into the forbidden region.
The ball can’t both have a well-defined energy and turn around exactly where classical physics says it should. If we want a small uncertainty in the energy, we have to accept more uncertainty in the position, and that means that there will be some chance of finding the ball at higher elevation than classical physics allows.
“Why do the wiggles get bigger near the top?”
“I already said that. The ball is moving slower, so the wavelength gets longer.”
“Yes, but they get taller, too.”
“Oh, that. That’s also because the ball is slowing down. The ball spends more time near the top of its flight where it’s moving slowly than it does near the bottom where it’s moving fast, so there’s a higher probability of finding it near the top. You see the same thing with a classical ball, if you work out the probability distribution—that’s the dashed curve.”
“So, wait, the most likely position for the ball is way up in the air?”
“Yes. You can see that by looking at the figure on page 125 showing the ball in flight. There are a lot more pictures of the ball at high elevations than at low ones. That’s why when we play fetch, you usually catch the ball when it’s at the top of its flight—it’s not moving very fast, so it’s easier to get.”
“I’m very good at catching things. We should go and play fetch! That’s fun!”
“After we finish this chapter, okay? I haven’t talked about tunneling yet.”
“Oh, right. Let’s talk about tunneling. Passing through solid objects is even more fun than fetch.”
“Don’t get your hopes up too much …”
LIKE IT’S NOT EVEN THERE:
BARRIER PENETRATION AND TUNNELING
How does uncertainty in the turning point lead to particles passing through solid objects? Well, the inside of a solid object is a forbidden region—interactions between the atoms making up the two objects make the potential energy enormous for one object inside another. It’s a little like trying to stick a second dog into a kennel that already contains one unfriendly dog—you’ll have a hard time getting the second dog in there, and if you do, you’ll see a lot of extra energy, in the form of growling and barking and snapping.
In quantum mechanics, though, wavefunctions can extend into forbidden regions, and that works even for solid objects—there’s a tiny probability of finding one object inside another. Better yet, if the forbidden region is very narrow, quantum mechanics predicts a small probability of one object passing through the other, even though it doesn’t have enough energy to make it into the forbidden region, let alone to the other side.
The simplest example of this is an electron hitting a thin piece of metal, where the potential energy is much higher. Classical physics tells us that the kinetic energy of an electron outside the metal determines what happens when it reaches the edge of the metal. If the electron’s kinetic energy is large, it can convert most of its energy to potential, and still have kinetic energy left to move through the metal. If the kinetic energy outside is less than the potential inside the metal, though, there’s no way the electron can enter without increasing its total energy. The edge of the metal becomes a turning point, and the metal is a forbidden region: an electron coming in from the left bounces off the surface and goes back where it came from. An electron coming in from the right bounces off the other surface in the same way.
According to quantum mechanics, though, we can’t have a sharp turning point at the edge of the metal. As with the thrown ball, the electron’s wavefunction extends into the forbidden region where the potential energy is greater than the energy of the incoming particles. There’s some probability of finding the electrons inside the metal, even though classical physics says it’s forbidden. The probability of finding a particle in the forbidden region is highest near the edge, and decreases rapidly as you move farther in. If the forbidden region extends over a long enough distance, the probability drops to zero,* and that’s the end of it.
For a very narrow barrier, though, there is some probability of finding the electrons at the opposite edge of the forbidden region from where they entered. Beyond that point, they’re no longer forbidden to be there—they’re back out in empty space, and move off with the same energy they had at the start. Somebody watching the experiment would see a tiny fraction of the incoming particles—one in a million, say—simply pass through the barrier as if it weren’t even there. This is called tunneling, because the electrons have passed the forbidden region even though it’s impossible for them to be inside it. In a sense, they’ve ducked under the barrier, like a bad dog who tunnels under a fence.
The wavefunction for this situation is shown above. On the left, we have an incoming electron with some momentum and energy, represented by a wave with a well-defined wavelength.† When the electron reaches the edge of the metal, it enters the forbidden region, and the probability decreases rapidly. It doesn’t get to zero before it reaches the right edge of the forbidden region, though, so it emerges there as another wave with the same wavelength as on the left.
The probability distribution for an electron coming in from the left, hitting a barrier where the potential energy of an electron would increase above the total energy of the electron. The probability drops off rapidly inside the forbidden region, but does not reach zero, so there is some probability of finding the electron to the right of the barrier.
The smaller height of the wave to the right of the barrier indicates that the probability of finding the electron on the right is much lower than the probability of finding it on the left. The probability of tunneling decreases exponentially as the barrier thickness increases—if you double the thickness, the probability is much less than half of the original probability. On the other hand, as the energy of the incoming electrons increases, they penetrate farther into the forbidden region, and the probability of one making it all the way through increases.
“So the electrons just drill holes through the barrier?”
“No, they pass through it as if it weren’t there at all. They don’t have enough energy to punch through.”
“But how do you know that?”
“Well, the electrons show up on the far side of the barrier with exactly the same energy as before they hit it. If they were boring little holes through the barrier, they would lose some energy in the process, and we’d be able to detect that.”
“Maybe they’re just really tiny holes?”
“No, we can look at that with a scanning probe microscope, and there aren’t holes.”
“What’s a scanning probe microscope?”
“That’s an excellent and very convenient question …”
FEELING SINGLE ATOMS:
SCANNING TUNNELING MICROSCOPY
Tunneling has a more direct technological application than most of the other weird quantum phenomena we’ve discussed. Tunneling is the basis for a device called a scanning tunneling microscope (STM), which uses electron tunneling to make images of objects as small as a single atom. The STM was invented in 1981 at IBM Zurich, and has become an essential tool for people studying the atomic structure of solid materials. Its inventors, Gerd Binnig and Heinrich Rohrer, won the Nobel Prize in Physics in 1986.
An STM consists of a sample of electrically conducting material and a very sharp metal tip brought within a few nanometers of the surface of the sample. The tip is held at a slightly different voltage than the sample, so electrons in the tip want to move from the tip into the sample. The electrons can’t flow directly from the tip into the sample, though, because the small gap between the tip and the sample acts as a barrier preventing the movement of electrons.*
If the gap between the tip and the sample is small enough, though—a nanometer or so—there’s some chance that electrons will tunnel from the tip to the sample. That produces a small current, which can be measured. The tunneling probability (and thus the current) increases dramatically as the tip gets closer to the surface, so changes in the current can be used to detect tiny changes in the distance between the two—changes smaller than the diameter of a single atom.
Making an image with an STM is like running your finger across a surface, and feeling the bumps and scratches. You scan the tip back and forth over the surface of the sample, keeping the height of the tip constant, and as you move the tip, you monitor the current flowing between the tip and the sample. The current increases whenever there’s a small bump sticking up from the surface making it easier for electrons to tunnel across, and decreases whenever there’s a small dip in the surface. If you take a large number of height measurements at points on a grid, you can put them together to create an image of the individual atoms making up the surface of your sample.
A schematic of a scanning tunneling microscope. A sharp tip is positioned close to the surface of a material, and moved back and forth in a regular pattern. Electrons from the tip will tunnel across the gap between the tip and the surface, producing a small electric current that is amplified and measured. The amount of current depends very sensitively on the distance between the tip and the surface, allowing a reconstruction of the surface sensitive enough to detect single atoms.
Not only can you see single atoms, but if you bring the tip into direct contact with the surface, you can push individual atoms around. Scientists have used this ability to make a number of incredible structures, such as the oval-shaped “corral” shown in the picture on the next page, made at IBM’s Almaden research laboratory. The bumps making up the “corral” are individual iron atoms on a copper surface, which have been dragged into place by the STM. Such structures can be used to study the quantum behavior of electrons inside the “corral,” which accounts for the wavy features seen on the copper surface.
Scanning tunneling microscopes have revolutionized the study of solids and surfaces, and the technology may lead to new manufacturing techniques for tiny devices. Other scientists have used STMs to study and manipulate individual strands of DNA, providing a more detailed understanding of the behavior of genetic material and possibly new drugs or treatments for genetic damage. All of this is made possible by the underlying wave nature of matter.
Iron atoms on a copper surface, arranged in a “corral” pattern using an STM. The wave pattern inside the corral is due to the wave nature of electrons on the copper surface. Image courtesy of IBM.
“That’s nice and everything, but I’m not interested in microscopic bunnies. What has quantum tunneling ever done for me?”
“Well, for one thing, you wouldn’t be able to enjoy a nice sunny day if not for tunneling.”
“What do you mean?”
“Well, the Sun shines because of fusion reactions in the core, right?”
“Everybody knows that. Even the beagle down the road knows that, and that dog is really stupid.”
“Yes. Well. Anyway, fusion works by sticking protons together to make helium from hydrogen. Because protons are positively charged, they repel one another, and that repulsion sets up a barrier. And as hot as the Sun is, the protons in the Sun still don’t have enough energy to get over that barrier directly.”
“So they tunnel through?”
“Exactly. The probability of any given proton tunneling through the barrier is pretty low, but there are lots and lots of protons in the Sun, and enough of them do tunnel through to keep the reaction going. So it’s really tunneling that lets the Sun shine.”
“Hmm. I suppose that is pretty impressive.”
“I’m so glad you approve …”
“Can we go and play fetch, now?”
* Temperature measures the energy due to the motion of the individual atoms making up an object, and “absolute zero” is the imaginary temperature at which that motion would cease. No real object can be cooled all the way to absolute zero, though, and even if one could it would still have zero-point energy, as discussed in chapter 2 (page 49).
* Potential energy is generally much easier to calculate than kinetic energy. Potential energy usually depends only on the positions of the interacting objects, while the kinetic energy depends on the velocity, which depends on what has happened in the recent past. The easiest way to tackle an energy problem is usually to calculate the potential energy using the position, and find the kinetic energy by process of elimination. For example, when a roller coaster pauses at the top of a big hill, we know that all of its energy is potential energy. Later on, we can easily calculate the potential energy from the height of the track, and that lets us find the kinetic energy (and thus the speed) without needing to know what happened in between.
* The total energy of an object can be increased, by adding energy from some other source, in the same way that a dog’s treat jar can be refilled by a friendly human. The extra energy does not come for free, though—the energy of the outside object has to decrease, in the same way that a human’s bank balance will decrease in order to supply the treats. The total energy of the entire universe—balls, dogs, treats, and humans—is a constant, and has not increased or decreased in the fourteen billion years since the Big Bang.
* Strictly speaking, the probability is never exactly zero—the mathematical function describing the probability is an exponential, and while it gets closer to zero as the electron moves into the barrier, it never gets all the way there. Quantum physics predicts a tiny probability that a ball thrown in the air will tunnel through the forbidden region that starts at its classical maximum height, and end up on the Moon. That’s not a good bet, though—the probability is so small that it’s indistinguishable from zero, for all practical purposes.
† Notice that the uncertainty in the position is very large—the electron could be just about anywhere to the left of the forbidden region.
* A potential energy barrier doesn’t have to be a solid physical object. An air gap will do just fine, which is why you can’t make a lightbulb light up by just holding it close to the socket.
