How to Teach Physics to Your Dog (7 page)

The dashed curves in the top graph show the wavefunctions for the two different wavelengths (shifted up so you can see them clearly). The solid curve shows the sum of the two wavefunctions. The bottom graph shows the probability distribution resulting from adding them together (the square of the solid curve in the top graph).

The center part of this probability distribution looks an awful lot like the wave packet we want. There’s a region where we have a good probability of finding the bunny, and in that region, we see a wavelength associated with its motion. Outside that region, the probability goes to zero, meaning that there are places where we have no hope of seeing a bunny at all.

Of course, this two-wave wavefunction isn’t exactly what we want, because the no-bunny zone is very narrow and followed
immediately by another lump. But we can improve the situation by adding more waves:

The bottom graph is the probability distribution for a single frequency wave, with two-, three-, and five-frequency graphs above it.

If we add together three different waves, the region where we have a good probability of seeing the bunny gets narrower, and with five different waves, it’s narrower still. As we add more and more waves, the regions of high probability get narrower, and the spaces between them become wider and flatter. What we end up with starts to looks like a long chain of wave packets.

“So, now we’ve got a long chain of bunnies? I thought we were only talking about one bunny.”

“We are talking about one bunny. What we’ve got is a chain of different ‘wave packets,’ which each describe a different place where we might find the single bunny. If we want to narrow that down to just one position for the bunny, we do it by adding together a continuous distribution of wavelengths, not just a set of regularly spaced single wavelengths.”

“Wouldn’t that mean adding together an infinite number of different wavelengths, though?”

“Well, yeah, but that’s what we have calculus for.”

“Oh. I’m not so good at calculus.”

“Very few dogs are. Just take my word for it—we can make a single wave packet by summing a continuous distribution of wavelengths, with different probabilities for different wavelengths.”

“And we end up with an infinite number of bunnies!”

“Sorry, but no. It’s still just one bunny, with an infinite number of possible velocities that are very close to one another.”

“Darn. I still want more bunnies.”

“Well, the infinite-sum wavefunction does define the position reasonably well, so at least you know where the one bunny is.”

“True. And if I know where it is, I can catch it!”

What does it mean to add together lots of different waves with different wavelengths in this way? Well, each wave corresponds to a particular momentum—a different velocity for the (single) bunny moving through the yard. When we add them all together, what we’re doing is saying that there’s a chance of finding the bunny in each of those different states (we’ll talk more about this in
chapter 3
).

Adding these states together is the origin of the uncertainty principle. If we want a narrow and well-defined wave packet, so that we know the position of the bunny very well, we need to add together a great many waves to do that. Each wave corresponds to a possible momentum for the bunny, though, which gives a large uncertainty in the momentum—it could be moving at any one of a large number of different speeds.

On the other hand, if we want to know the momentum very
well, we can use a small number of different wavelengths, but this gives us a very broad wave packet, with a large uncertainty in the position. The bunny can only have a few possible speeds, but we can no longer say where it is with much confidence.

We can’t produce a wave packet with a single well-defined position without using an infinitely wide distribution of wavelengths, and we can’t produce a wave packet with a single well-defined momentum without having it extend over all of space. The best we can hope to do is a single wave packet like we drew at the beginning, with a small uncertainty in the momentum and a small uncertainty in the position. When we go through the mathematical details, we find that the smallest possible product of the uncertainties satisfies the famous Heisenberg relationship
*
:

Δ
x
Δ
p = h/4π

The uncertainty in the position (Δ
x
) multiplied by the uncertainty in the momentum (Δ
p
) is equal to Planck’s constant divided by 4
π
. Any other type of wave packet (and there are lots of different shapes found in nature) will have a larger uncertainty product, so the relationship is usually written with a greater-than-or-equal-to sign:

Δ
x
Δ
p ≥ h/4π

The important result remains the same, though: no matter what you do, there’s no way to make both Δ
x
and Δ
p
zero—as you make one smaller, the other necessarily gets larger, and the product remains above the Heisenberg limit.

Looked at in terms of wavefunctions, then, we can see that this relationship is much more than just a practical limit due to our inability to measure a system without disturbing it. Instead, it’s a deep statement about the limits of reality. We saw in
chapter 1
that quantum particles behave like particles—photons have momentum and collide with electrons in the Compton effect (page 25). We also saw that quantum particles behave like waves—electrons, atoms, and molecules diffract around obstacles and form interference patterns. The price we pay for having both of these sets of properties at the same time is that position and momentum must always be uncertain. The meaning of the uncertainty principle is not just that it’s impossible to measure the position and momentum, it’s that these quantities do not exist in an absolute sense.

The uncertainty principle forces us to completely rethink our understanding of how the universe works. Not only does it change the way we look at single moving particles, but it has profound consequences for the structure of matter at the microscopic level.

Most humans, and even many dogs, picture atoms as tiny little solar systems, with negatively charged electrons orbiting a positively charged nucleus. This picture originated with Niels Bohr in 1913, when he proposed the first quantum model of the hydrogen atom.

In Bohr’s model, the one electron of a hydrogen atom can orbit the nucleus only in certain very specific orbits, with particular
well-defined values of energy. These orbits are the “allowed states” of hydrogen, and an electron in an allowed state will happily remain there. Electrons can never be found in an orbit with an in-between energy. Physicists often talk about these states as if they were steps on a staircase, and the electrons were dogs looking for a place to sleep. The dog can rest comfortably on the ground floor, or on one of the steps, but any attempt to lie down halfway between two steps will end badly.

Bohr’s model works brilliantly to describe the characteristic colors of light emitted and absorbed by hydrogen. Electrons can move between the allowed states by absorbing or emitting photons of light, with the frequency of the emitted light corresponding to the difference between the energies of the two states. The Bohr model thus solved a problem that had stymied physicists for years.

When Bohr proposed the model, it was a bold break with prior physics. Unfortunately, it’s a cobbled-together mix of classical and quantum ideas, with no solid theoretical justification. Louis de Broglie’s wave model of the electron provided the missing theoretical basis, but while particle-wave duality justifies the idea of allowed states, it requires us to discard the image of electrons orbiting the nucleus like planets orbiting the sun.

The fundamental problem with this picture is the same issue that leads to uncertainty. For the solar-system model to be accurate, the electron must have a well-defined position somewhere along the allowed orbit, and a well-defined momentum moving it along that orbit. But this can’t possibly work—if we try to define the electron’s position well enough to locate it along a planetary orbit, it must have a huge uncertainty in momentum, meaning that we can’t say where it’s going. If we try to define the electron’s momentum well enough to place it on an orbital track, it must have a huge uncertainty in position, meaning that we can’t even be sure it’s near the nucleus it’s supposed to be orbiting.

When we account for the wave nature of the electron, we
are forced to discard the whole idea of electrons as planets. Instead, the electron hovers around the nucleus in a fuzzy sort of “cloud,” with a position that is uncertain, but confined to a region near the nucleus, and a momentum that is uncertain, but limited to values that keep it near the nucleus. Bohr’s idea of allowed energy states still applies—the electron will always have one of the limited number of energy values predicted by Bohr’s theory—but these states no longer correspond to electrons moving in particular orbits.

“So, wait—the electron isn’t in a particular place, it’s just kind of near the atom?”

“That’s right. The different energy states correspond to different probabilities of finding the electrons at particular positions, and higher-energy states will give you a better chance of finding the electron farther from the nucleus than lower-energy states. But for any of the allowed states, the electron could be at just about any point within a few nanometers of the nucleus.”

“But what happens if you have two atoms close together?”

“Well, if you bring two atoms close enough together, an electron that starts out attached to one atom can end up on the other atom, because of this quantum uncertainty in the position. We’ll talk a little more about this in
chapter 6
, when we talk about tunneling.”

“Okay.”

“You can also get situations where an electron is sort of ‘shared’ between two atoms. That’s how chemical bonds form. And if you get a bunch of atoms together in a solid, one electron can be shared among the whole solid. That’s the basis for the quantum theory of solids, which lets us understand how metals conduct electricity and how to make semiconductor computer chips. It’s all because electrons extend beyond specific planetary orbits.”

“Uncertain electrons are weird.”

“Strictly speaking, it’s not just electrons. Everything in the universe is subject to the uncertainty principle, and has an uncertain position and velocity.”

“That can’t be right. I mean, I can see my bone right over there, and it has a definite position, and a velocity of zero.”

“Ah, but the
quantum
uncertainty associated with your bone is dwarfed by the
practical
uncertainty involved in measuring it. If you look at it really carefully, you might be able to specify its position to within a millimeter or so—”

“I
always
look at my bone carefully.”

“—and with heroic effort, you might bring that down to a hundred nanometers. In that case, the velocity uncertainty of your hundred-gram bone would be only 10
^-27
m/s. So, the velocity would be zero, plus or minus 10
^-27
m/s.”

“That’s pretty slow.”

“Yeah, you could say that. At that speed, it would take the age of the universe to cross the thickness of a single atom.”

“Okay, that’s
really
slow.”

“We don’t see quantum uncertainty associated with everyday objects because they’re just too big. We only see uncertainty directly when we look at very small particles confined to very small spaces.”

“Like electrons near atoms!”

“Exactly.”

Uncertainty has another, even more profound effect on the structure of atoms. Electrons must always have uncertainty in both their position and momentum, and that means that the energy of an electron in an atom can never be zero. To have zero energy while still being part of an atom, an electron would need to be not moving, sitting right on top of the nucleus. This is impossible, as we’ve already seen—the closest we can come is to make a narrow electron wave packet centered on the nucleus, which will include lots of different states with nonzero momentum.
Even the lowest-energy-allowed state of hydrogen, then, has some energy.

This is a general phenomenon, and applies to any confined quantum particle. If we know that a particle is in some particular region of space, that limits the uncertainty in the position, and increases the uncertainty in the momentum. Confined quantum particles are never at rest—they’re like puppies in a basket, always squirming and wiggling and shifting around, even when they’re asleep.

This tiny residual motion is called zero-point energy, which is the minimum quantum energy associated with a particle due to its confinement. Zero-point energy provides an absolute lower limit to the energy a confined particle can have—no matter how carefully you prepare the system, the particles in that system will always be in motion, with small random fluctuations constantly changing the magnitude and direction of their velocity.