How to Teach Physics to Your Dog (6 page)

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.
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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 with 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.

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.
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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.

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
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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: