How to Teach Physics to Your Dog (17 page)

“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.

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 wave-function 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 wave-function 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.”

“Yeah, 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?”

“Yep. 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. Oooh! We should go 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 . . .”

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 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 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 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 . . .”

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.
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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.