How to Teach Physics to Your Dog (25 page)

*
Strictly speaking, then, before its state is measured, Schrödinger’s cat can be in one of two states: “alive plus dead,” or “alive minus dead.”

*
The same Anton Zeilinger was seen in
chapter 1
heading the group that demonstrated diffraction of molecules, and
chapter 5
doing quantum interrogation. He has made a long and distinguished career doing experiments to demonstrate the weird and wonderful features of quantum mechanics.

*
In the seven years between the two experiments, Professor Zeilinger moved from Innsbruck to Vienna.

†
There’s no inherent problem with sending photons over very long distances—light manages to reach us from distant galaxies, after all—but interactions with the environment can destroy entangled states through the process of decoherence discussed in
chapter 4
. The Vienna experiment shows that decoherence can be avoided long enough to send entangled photons over useful distances.

*
A classical computer uses millions of tiny transistors on silicon chips; quantum computers could use anything with at least two states—atoms, molecules, electrons.

†
The “qu” is for “quantum.” Physicists are not widely admired for their ability to think up clever names.

*
In fact, the National Security Agency is one of the largest funders of quantum computing research in the United States.

Emmy is standing at the window, wagging her tail excitedly. I look outside, and the backyard is empty. “What are you looking at?” I ask.

“Bunnies made of cheese!” she says. I look again, and the yard is still empty.

“There are no bunnies out there,” I say, “and there are certainly not any bunnies made of cheese. The backyard is empty.”

“But particles are created out of empty space all the time, right?”

“You’re still reading my quantum physics books?”

“It’s boring here when you’re not home. Anyway, answer the question.”

“Well, yes, in a sense. They’re called virtual particles, and under the right conditions, the zero-point energy of the vacuum can occasionally manifest as pairs of particles, one normal matter and one antimatter.”

“See?” she says, wagging her tail harder. “Bunnies made of cheese!”

“I’m not sure how that helps you,” I say. “Virtual particles have to annihilate one another in a very short time, in order to satisfy the energy-time uncertainty principle. A virtual electron-positron pair lasts something like 10
^-21
seconds before
it disappears. They’re not around long enough to be real particles.”

“But they can become real, right?” She looks a little concerned. “I mean, what about Hawking radiation?”

“Well, sure, in a sense. The idea is that a virtual pair created near a black hole can have one of its members sucked into the black hole, at which point the other particle zips off and becomes real.”

The tail-wagging picks back up. “Bunnies made of cheese!”

“What?”

She gives an exasperated sigh. “Look, virtual particles are created all the time, right? Including in our backyard?”

“Yes, that’s right.”

“Including bunnies, yes?”

“Well, technically, it would have to be a bunny-antibunny pair . . .”

“And these bunnies, they could be made of cheese.”

“It’s not very likely, but I suppose in a Max Tegmark
*
sort of ‘everything possible
must
exist’ kind of universe, then yes, there’s a possibility that a bunny-antibunny pair made of cheese (and anticheese) might be created in the backyard, but—”

“And if I eat one, the other becomes real.” She’s wagging her tail so hard that her whole rear end is shaking.

“Yeah, but they wouldn’t last very long before they annihilated each other.”

“I’m very fast.”

“Given the mass of a bunny, they’d only last 10
^-52
seconds. If that.”

“In that case, you’d better let me outside. So I can catch the bunnies made of cheese.”

I sigh. “If you wanted to go outside, why didn’t you just say that?”

“What fun would that be? Anyway, bunnies made of cheese!”

I look out the window again. “I still don’t see any bunnies, but there is a squirrel by the bird feeder.”

“Ooooo! Squirrels!” I open the door, and she goes charging outside after the squirrel, who makes it up a tree just in time.

Back in
chapter 2
, we saw that the wave nature of matter gives rise to zero-point energy, meaning that no quantum particle can ever be completely at rest, but will always have at least some energy. Incredibly, this idea applies even to empty space. In quantum physics, even a perfect vacuum is a constant storm of activity, with “virtual particles” popping into existence for a fleeting moment, thanks to zero-point energy, then disappearing again.

The idea of “virtual particles” popping in and out of existence in the middle of empty space is one of the most compelling and bizarre ideas in modern physics. In this chapter, we’ll talk about quantum electrodynamics (“QED” for short), the underlying theory that gives rise to the idea of virtual particles. We’ll also talk about the experiments that make QED arguably the most precisely tested theory in the history of science. Ironically, though, our discussion of this ultraprecise theory needs to start with the Heisenberg uncertainty principle.

The best-known version of the uncertainty principle is the one that we talked about in
chapter 2
(page 48), which puts a limit on the uncertainties in the position and momentum of a particle.
At a very fundamental level, the more we know about the position of a particle, the less we can know about how fast it is moving, and vice versa.

Slightly less well known is the uncertainty relationship between energy and time, which says that the uncertainty in energy multiplied by the uncertainty in time has to be larger than Planck’s constant divided by four pi:

Δ
E
Δ
t ≥ h/4
π

As with position-momentum uncertainty, this means that the more we know about one of these two quantities, the less we can know about the other.

The idea that energy is somehow related to time may seem strange at first, but we can understand it by thinking about light. As we saw in
chapter 1
(page 21), the energy of a photon is determined by the frequency associated with that color of light. A low-uncertainty measurement of energy, then, requires a precise measurement of frequency.

So, how do you make a precise measurement of frequency? Imagine that you want to measure the rate at which an excited dog is wagging her tail, which is a fairly regular oscillation with small fluctuations in the frequency and amplitude: sometimes she wags a little faster, sometimes a little slower, sometimes farther to the right, sometimes farther to the left. What is the best way to measure the wagging frequency?

Frequency is measured in oscillations per second, so you need to count the number of wags that take place in some fixed time interval. If you wait five seconds, and count ten tail wags, that’s a frequency of two oscillations per second. Any such measurement will always have some uncertainty, though—when you counted ten oscillations, was it really ten full oscillations, or ten-and-a-little-bit? Had her tail gotten all the way to the right, or was she wagging it farther that time around?

To minimize that uncertainty, you need to look over a much longer time. The uncertainty in your count of wags will tend to be a constant—one tenth of a tail wag, say—so the more oscillations you count, the better you do in terms of the relative uncertainty.

If you watch a tail wagging for five seconds, and count ten oscillations, plus or minus one tenth, the frequency you measure is

f
= (10 ± 0.1 oscillations)/(5 seconds) = 2.00 ± 0.02 Hz
*

That is, the frequency is somewhere between 1.98 and 2.02 oscillations per second.

If you watch for fifty seconds (assuming the dog doesn’t explode from impatience), you’ll count a hundred wags, plus or minus one tenth, and the frequency is then

f
= (100 ± 0.1 oscillations)/(50 seconds) = 2.000 ± 0.002 Hz

Increasing the number of oscillations that you measure leads to a decrease in the uncertainty of the frequency, and a more precise determination of just how happy the dog is.

The cost of that decrease in frequency uncertainty is an increase in the time uncertainty. To get one tenth the energy uncertainty, you spent ten times as long making the measurement, which means you can’t say exactly
when
you measured the frequency. You know the
average
frequency over those fifty seconds, but you can’t point to a specific instant and say that the frequency right then was 2.000 Hz. All you can say is that over that fifty-second interval, the dog’s tail was wagging at
about 2 Hz, but at any given instant, it may have been faster or slower.

You could be more specific about the measurement time, by measuring for only half a second, say, and counting one tail wag, but then the frequency uncertainty becomes much larger:
f
= 2.0 ± 0.2 Hz. You can’t have a small uncertainty in both the frequency of an oscillation and the time when it was measured.

The same logic applies to measuring the frequency of light, though the oscillations are much too fast to count by hand. We directly see this uncertainty principle in action in the interaction between light and atoms. We know from chapters 2 and 3 that atoms will be found only in certain allowed energy states, and that they move between these states by absorbing or emitting photons.

When an atom moves from a high-energy state to a lower-energy state, the frequency of the emitted photon is determined by the energy difference between the two states. That energy difference has some uncertainty, though, that depends on how much time the atom spent in the higher-energy state. Two identical atoms placed in the same high-energy state can thus emit photons with very slightly different frequencies.

The difference is tiny—for typical atoms, it’s about one hundred- millionth of the frequency of the photons. This can be measured using lasers, though, and this tiny frequency uncertainty limits our ability to make certain measurements of atomic properties.

“So it takes you awhile to count things. Big deal. What’s that have to do with bunnies made of cheese?”

“The frequency-counting example is just one example of a more general principle.
Energy-time uncertainty
holds no matter what form the energy is in.”

“Why?”

“Well, all forms of energy have to be equivalent, because
you can convert one form of energy to another. So, if you have a photon of uncertain energy, and you use it to start an electron moving, the kinetic energy of that electron must be uncertain.”

“I still don’t see what this has to do with bunnies.”

“We know from Einstein’s theory of relativity that mass and energy are equivalent—”

“E = mc
²
!”

“Exactly. Since mass is just another form of energy, you can convert energy into mass and mass into energy. Mass has uncertainty just like all the other forms of energy, and that uncertainty is related to how long the mass stuck around.”

“So a bunny made of cheese would have an uncertain mass?”

“Right. If it was around for only a short time, the uncertainty could be very large—for something like a top quark, which sticks around for only 10
^-25
seconds, the quantum uncertainty due to that lifetime is close to 1% of the total mass.”

“But if it was around for a long time, the mass uncertainty would be small? I want a bunny made of cheese with a small mass uncertainty!”

“Good luck with that. A bunny made of cheese would have an awfully short lifetime around you.”

“Ooh. Good point.”

How does this get us bunnies made of cheese? Well, let’s think about applying this uncertainty principle to empty space. If we look at some small region over a long period of time, we can be quite confident that it is empty. Over a short interval, though, we can’t say for certain that it
isn’t
empty. The space might contain some particles, and in quantum mechanics, that means it will.

Uncertainty about the emptiness of space isn’t as strange as it may seem at first. If a physicist or a stage magician gives a
dog a box to inspect at leisure, she can conclusively state that the box is empty. She can sniff in all the corners, check for false bottoms, and make absolutely sure that there’s nothing hiding in some little recess. If she’s allowed only a brief peek or a quick sniff inside the box, though, she can’t be as confident that the box is empty. There might be something tucked into a corner that she wasn’t able to detect in that short time.

The amount of time needed to determine whether the box is empty also depends on the size of the thing you might expect to find. You don’t need to look for very long to determine whether the box contains Professor Schrödinger’s famous cat, but if you’re attempting to rule out the presence of a much smaller object—a crumb of a dog treat, say—a more thorough inspection is required, and that takes time.
*

The same idea applies to empty space in quantum physics, via the energy-time uncertainty relationship. When we look at an empty box over a long period of time, we can measure its energy content with a small uncertainty, and know that there is only zero-point energy—no particles are in the box. If we look over only a short interval, however, the uncertainty in the energy can be quite large. Since energy is equivalent to mass through Einstein’s famous
E = mc
²
,
this means that we can’t be certain that the box
doesn’t
contain any particles. And as with Schrödinger’s cat, if we don’t know the exact state of what’s in the box, it’s in a superposition of all the allowed states. The cat is both alive and dead, and the box is both empty and full of all manner of particles,
at the same time
.