How to Teach Physics to Your Dog (16 page)

Whether you think of it in terms of collapsing wavefunctions, or a single expanding wavefunction undergoing decoherence, the quantum Zeno effect is a dramatic demonstration of the strange nature of quantum measurement. Unlike classical measurement, the act of measuring a quantum system changes the state of that system, leaving it in only one of the allowed states, which is very different than what we expect classically. With a clever arrangement of the experimental situation, this can be exploited to prevent a system from changing states, or even to extract information from a system without interacting with it directly.

“That’s really interesting. Weird, but interesting.”

“Thanks.”

“Now, if you’ll excuse me, I need to go look in my bowl.”

“Why is that?”

“Well, I’m going to use the Zeno effect to get more food. I figure, if I keep measuring my bowl to be full of kibble, I’ll always have kibble, no matter how much I eat. That will be fun.”

“Of course, if you keep measuring your bowl to be empty, it’ll always be empty, and you’ll never have kibble.”

“Oh. That would be bad. I didn’t think of that.”

“Anyway, you’d need to have some natural quantum process that caused kibble to appear in the bowl for that to work. Things aren’t going to appear for no reason, just because you want to measure them.”

“Well, you sometimes put kibble in my bowl, right? And you’re a natural process.”

“In a manner of speaking.”

“So, how about putting some kibble in my bowl?”

“Oh, all right. It’s almost dinnertime. Come on.”

“Oooh! Kibble!”

*
While the summing of infinite series is accepted as the resolution of Zeno’s paradox by physicists and engineers and most mathematicians, some philosophers do not accept this as a sufficient resolution of Zeno’s paradox (
Stanford Encyclopedia of Philosophy
). This just proves that philosophers are crazier than mathematicians, or even cats.

*
A better Greek literary allusion might be the myth of Sisyphus, who was condemned to spend eternity pushing a boulder up a hill, only to have it slip free and roll back to the bottom again. The name “Sisyphus effect” was used for something else, though, so this is called the quantum Zeno effect.

†
A quarter of a second seems pretty fast to humans or dogs, but it’s really slow for an atom. Atoms usually change states in a few billionths of a second.

We’re sitting in the backyard, enjoying a beautiful sunny afternoon. I’m lying on a lounge chair 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? Sure, 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 yard 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’ yard, 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 get skinny 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 aluminum 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.

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 = mc
²
)
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 mv
²

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 upward, 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 upward, 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 backward, you know.”

“I do?”

“Yeah, 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.”