by CHAD ORZEL
* Sir Isaac Newton, of the falling apple story, set forth three laws of motion that govern the behavior of moving objects. The first law is the principle of inertia, that objects at rest tend to remain at rest, and objects in motion tend to remain in motion unless acted on by an external force. The second law quantifies the first, and is usually written as the equation F = ma, force equals mass times acceleration. The third law says that for every action there is an equal and opposite reaction—a force of equal strength in the opposite direction. These three laws describe the motion of macroscopic objects at everyday speeds, and form the core of classical physics.
* You might wonder why you can’t put together two low-energy photons to provide enough energy to free an electron. This would require two photons to hit the same electron at the same instant, and that almost never happens.
* Millikan thought the Einstein model lacked “any sort of satisfactory theoretical foundation,” and described its success as “purely empirical,” which is pretty nasty by physics standards. Ironically, those quotes are from the first paragraph of the paper in which he conclusively confirms the predictions of the theory.
* A nanometer is 10-9m, or one billionth of a meter (0.000000001 m).
* A few die-hard theorists still resisted the idea of photons, because even the Compton effect can be explained without photons, though it’s very complicated. The last resistance collapsed in 1977, when incontrovertible proof of the existence of photons was provided in an experiment by Kimble, Dagenais, and Mandel that looked at the light emitted by single atoms. The seventy-two-year gap between Einstein’s proposal and its final acceptance tells you something about the stubbornness of physicists confronted with a new idea. It can be as difficult to separate a physicist from a cherished model as it is to drag a dog away from a well-chewed bone.
* The proper pronunciation of Louis de Broglie’s surname (his collection of names reflects his aristocratic background—he was the 7th Duc de Broglie) is the source of much confusion among American physicists. I’ve heard “de-BRO-lee,” “de-BRO-glee,” and “de-BROY-lee,” among others. The correct French pronunciation is apparently something close to “de-BROY,” only with a gargly sort of sound to the vowel that you need to be French to make.
* “Crystal,” to a physicist, refers to any solid with a regular and orderly arrangement of atoms in it. This includes the clear and sparkly things that we normally associate with the word, but also a lot of metals and other substances.
† Ironically, Davisson and Germer succeeded only because they broke a piece of their apparatus. They didn’t see any diffraction in the first experiments they did, because their nickel target was made up of many small crystals, each producing a different interference pattern, and the bright spots from the different patterns ran together. Then they accidentally let air into their vacuum system. In the process of repairing the damage, they melted the target, which recrystallized into a single large crystal, producing a single, clear diffraction pattern. Sometimes, the luckiest thing a physicist can do is to break something important.
* In one of the great bits of Nobel trivia, Thomson’s father, J. J. Thomson of Cambridge, won the 1906 Nobel Prize in Physics for demonstrating the particle nature of the electron. This presumably led to some interesting dinner-table conversation in the Thomson household.
CHAPTER 2
Where’s My Bone? The Heisenberg Uncertainty Principle
I’m grading papers on the couch when Emmy comes into the room, looking concerned. “What’s the matter?” I ask.
“I can’t find my bone,” she says. “Do you know where my bone is?”
“I have no idea where your bone is,” I say, “but I can tell you exactly how fast it’s moving.”
There’s silence in response, and when I look up, she’s staring at me blankly.
“It’s a physics joke,” I explain, because that always makes things funnier. “You know, Heisenberg’s uncertainty principle? The uncertainty in the position of an object multiplied by the uncertainty in the momentum is greater than Planck’s constant over four pi? Which means that when one uncertainty is small, the other must be very large.”
Now she’s glaring at me, almost growling. “Stop doing that!” she says.
“What? It’s not all that funny, but it wasn’t that bad.”
“It’s your fault that I can’t find my bone.”
“How is it my fault?”
“You went and measured how fast it’s moving, and the position got all uncertain. And now I can’t find my bone.”
“That’s not what happened,” I say. “The uncertainty principle doesn’t work like that.”
“Yes it does. You just said. You know how fast my bone is moving, and now I can’t find it.”
“First of all, that was a joke. I didn’t really measure the velocity of your bone. Second, that’s a slightly mistaken view of the uncertainty principle. It’s not just that measurement changes the state of the system, it’s that what we can measure is limited by the fact that position and momentum are undefined until we measure them.”
She looks puzzled. “I don’t see the difference.”
“Well, in the picture where you attribute everything to the effects of measurement, you implicitly assume that whatever you’re measuring has some definite and well-defined properties, and the uncertainty in those values arises only from perturbations that occur through the act of measuring them. That’s not what happens, though—in quantum theory, there are no definite values for those quantities. They’re not uncertain because of limits on your measurement, they’re uncertain because they are not defined, and they can’t be defined, due to the quantum nature of reality.”
“Oh.” She looks thoughtful for a moment, then resumes glaring. “I think you lost my bone, and you’re just trying to weasel out of this by being all confusing.”
“No, that’s really how the theory works. It’s a moot point, though, since even if I had perturbed the position of your bone by measuring its velocity, there’s no way that would’ve prevented you from finding it.”
“Yeah? Why not?”
“Well, because the uncertainty involved would be tiny. I mean, your bone has a mass of a couple hundred grams, and if I measured its velocity to within one millimeter per second, that would give an uncertainty in position of only about 10-31 meters. That’s a trillionth of the size of a proton—you’d never even notice that.”
“Yeah? Well, where’s my bone, smart guy?”
“I don’t know. Did you look under the TV cabinet? Sometimes it gets kicked under there.”
She trots over to the TV, and sticks her nose under the cabinet. “Oooh! Here’s my bone!” She paws at it for a minute, and eventually succeeds in knocking it out from under the cabinet. “I have a bone!” she announces proudly, and begins chewing it noisily, the uncertainty principle forgotten.
The Heisenberg uncertainty principle is probably the second most famous result from modern physics, after Einstein’s E = mc2 (the most famous result from relativity). Most people wouldn’t know a wavefunction if they tripped over one, but almost everyone has heard of the uncertainty principle: it is impossible to know both the position and the momentum of an object perfectly at the same time. If you make a better measurement of the position, you necessarily lose information about its momentum, and vice versa.
In this chapter, we’ll describe how the uncertainty principle arises from the particle-wave duality we’ve already discussed. The uncertainty principle is often presented as a statement that a measurement of a system changes the state of that system, and in this form, references to quantum uncertainty turn up in all sorts of places, from politics to pop culture to sports.* Ultimately, though, uncertainty has very little to do with the details of the measurement process. Quantum uncertainty is a fundamental limit on what can be known, arising from the fact that quantum objects have both particle and wave properties.
Uncertainty is also the first place where quantum physics collides wi
th philosophy. The idea of fundamental limits to measurement runs directly counter to the goals and foundations of classical physics. Dealing with quantum uncertainty requires a complete rethinking of the basis of physics, and leads directly to the issues of measurement and interpretation in chapters 3 and 4.
HEISENBERG’S MICROSCOPE: SEMICLASSICAL ARGUMENTS
The traditional description of uncertainty as the act of measurement changing the state of the system is essentially based in classical physics, and was developed in the 1920s and ’30s in order to convince classically trained physicists that quantum uncertainty needed to be taken seriously. This is what physicists call a semiclassical argument—the physics used is classical, with a few modern ideas added on. It’s not the full picture, but it has the advantage of being readily comprehensible.
The idea behind the semiclassical treatment of uncertainty is familiar to any dog. Imagine you have a bunny in the yard whose position and velocity you would like to know very well. When you attempt to make a better determination of its position (by getting closer to it), you inevitably change its velocity by making it run away. No matter how slowly you creep up on it, sooner or later, it always takes off, and you never really have a good idea of both the position and the velocity.
An electron isn’t a sentient being like a bunny, so it can’t run off of its own accord, but a similar process takes place.
An incoming photon bounces off a stationary electron, and is collected by a microscope lens in order to measure the electron’s position. In the collision, though, the electron acquires some momentum, leading to uncertainty in its momentum.
To measure the position of an electron, you need to do something to make it visible, such as bouncing a photon of light off it and viewing the scattered light through a microscope. But the photon carries momentum (as we saw in chapter 1 [page 24]), and when it bounces off the electron, it changes the momentum of the electron. The electron’s momentum after the collision is uncertain, because the microscope lens collects photons over some range of angles, so you can’t tell exactly which way it went.
You can make the momentum change smaller by increasing the wavelength of the light (decreasing the momentum that the photon has available to give to the electron), but when you increase the wavelength, you decrease the resolution of your microscope, and lose information about the position.* If you want to know the position well, you need to use light with a short wavelength, which has a lot of momentum, and changes the electron’s momentum by a large amount. You can’t determine the position precisely without losing information about the momentum, and vice versa.
The real meaning of the uncertainty principle is deeper than that, though. In the microscope thought experiment illustrated above, the electron has a definite position and a definite velocity before you start trying to measure it, and still has a definite position and velocity after the measurement. You don’t know what the position and velocity are, but they have definite values. In quantum theory, however, these quantities are not defined. Uncertainty is not a statement about the limits of measurement, it’s a statement about the limits of reality. Asking for the precise position and momentum of a particle doesn’t even make sense, because those quantities do not exist.
This fundamental uncertainty is a consequence of the dual nature of quantum particles. As we saw in the previous chapter, experiments have shown that light and matter have both particle-like and wavelike properties. If we’re going to describe quantum particles mathematically—and physics is all about mathematical description of reality—we need to find some way of talking about these objects that allows them to have both particle and wave properties at the same time. We’ll find that the only way is to have both the position and the momentum of the quantum particles be uncertain.
BUILDING A QUANTUM PARTICLE: PROBABILITY WAVES
The usual way of describing particles mathematically, dating from the late 1920s, is through quantum wavefunctions. The wavefunction for a particular object is a mathematical function that has some value at every point in the universe, and that value squared gives the probability of finding a particle at a given position at a given time. So the question we need to ask is, What sort of wavefunction gives a probability distribution that has both particle and wave properties?
Constructing a probability distribution for a classical particle is easy, and the result looks something like this:
The probability of finding the object—say, that pesky bunny in the backyard—is zero everywhere except right at the well-defined position of the object. As you look across the yard, you see nothing, nothing, nothing, BUNNY!, nothing, nothing, nothing.
This wavefunction doesn’t meet our requirements, though: it has a well-defined position, but it’s just a single spike, and a spike does not have a wavelength. Remember, the wavelength corresponds to the momentum of the bunny, which is one of the quantities we’re trying to describe, so it needs to have some value.
Well, then, how do we draw a probability distribution with an obvious wavelength? That’s also easy to do, and it looks like this:
Here, the probability of finding the bunny at a given position oscillates: bunny, Bunny, BUNNY, Bunny, bunny, Bunny, BUNNY, Bunny, bunny, and so on.
This wavefunction doesn’t meet our requirements, either. The wavelength is easy to define—just measure the distance between two points where the probability is largest—so we have a well-defined momentum, but we can’t identify a specific position for the bunny. The bunny is spread out over the entire yard, with a good probability of finding it at lots of different places. There are places where the probability of seeing a bunny is low, but they don’t account for much space.
What we need is a “wave packet,” a wavefunction that combines particle and wave properties in a single probability distribution, like this:
This wavefunction is what we’re after: nothing, nothing, bunny, Bunny, BUNNY, Bunny, bunny, nothing, nothing. The bunny is very likely to be found in a small region of space, and the probability of finding it outside that region drops off to zero. Inside that region, we see oscillations in the probability, which allow us to measure a wavelength, and thus the momentum.
This wave packet has the particle and wave properties that we’re looking for. As a consequence, it also has some uncertainty in both the position and momentum of the particle.
The uncertainty in position is immediately obvious on looking at the wave packet. The bunny can’t be pinned down to a specific location, but there are several different positions within a small range where the probability of finding it is reasonably good. The bunny is most likely to be found right in the center of the wave packet, but there’s a good chance of finding it a little bit to the left, or a little bit to the right. The position as described by this wave packet is necessarily uncertain.
The uncertainty in the wavelength is not as obvious, but it’s there because this wave packet is actually a combination of a great many waves, each with a slightly different momentum. Each of these waves represents a particular possible momentum for the bunny, so just as there are several different positions where the bunny might be found, there are also several different possible values of the momentum. The momentum of the bunny described by this wave packet thus has some uncertainty.
How do we get a wave packet by combining many waves? Well, let’s start with two simple waves, one corresponding to a bunny casually hopping across the yard, and another one with a shorter wavelength (the graph below shows 20 full oscillations of one, in the same space as 18 of the other), corresponding to a bunny moving faster, perhaps because it knows there’s a dog nearby. Now let’s add those two wavefunctions together.
“Wait a minute—now we have two bunnies?”
“No, each wavefunction describes a bunny with a particular momentum, but it’s the same bunny both times.”
“But doesn’t adding them together mean that you have two bunnies?”
“No, in this case, it just means that there are two different states*
you might find the single bunny in. When you look out into the yard, there’s some probability of finding the bunny moving slowly, and some probability of finding it moving a little faster. The way we account for that mathematically is by adding the two waves together.”
“Oh. Darn. I was hoping for more bunnies.”
• • •
When we add these two waves together, we find that there are some places where they are in phase, and add up to give a bigger wave. In other places, they’re out of phase, and cancel each other out. The wavefunction we get from adding them together (the solid line in the figure) has lumps in it—there are places where we see waves, and places where we see nothing. When we square that to get the probability distribution, we get the bottom graph:
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.