How to Teach Physics to Your Dog (9 page)

Quantum randomness is a tremendously disturbing idea for people raised on classical physics, where if you know the starting conditions of your experiment well enough, you can predict the outcome with absolute certainty: you know that the dog will be in the kitchen, and looking just confirms what you already knew. Quantum mechanics doesn’t work that way, though: identically prepared experiments can give completely different results, and all you can predict are probabilities. This randomness is the philosophical issue that led Einstein to make a variety of comments that have been rendered as “God does not play dice with the universe.”
*

• • •

“Physicists are silly.”

“Why do you say that?”

“Well, what’s disturbing about randomness? I never know the outcome of anything for sure before it happens, and I’m fine.”

“Well, you’re a dog, not a physicist. But you do make a good point—any responsible practical treatment of classical physics has to include some element of probability in its predictions, just because you can never account for all the little perturbations that might affect the outcome of an experiment.”

“Like that butterfly in Brazil, causing all this weather.”

“Exactly. That’s the usual metaphor: a butterfly flaps its wings in the Amazon, and a week later, there’s a storm in Schenectady. It’s the classic example of chaos theory, which shows that probability is unavoidable even in classical physics, because you can never account for every single butterfly that might affect the weather.”

“Stupid chaos butterflies.”

“The thing is, quantum probability is a different game altogether. The probabilities we end up with in classical physics are a practical limitation. If, by some miracle, you really
could
keep track of every butterfly in the world, then you would be able to predict the weather with certainty, at least for a while. Quantum physics doesn’t allow that.”

“You mean the butterflies are covered by the uncertainty principle, so you don’t know where they are?”

“Only partly—it’s deeper than that. In quantum physics, even if you perform the same experiment twice under identical conditions—down to the very last butterfly wing-flap—you still won’t be able to predict the exact outcome of the second experiment, only the probability of getting various outcomes. Two identical experiments can and will give you different results.”

“Oh. You know what? That is pretty disturbing. Maybe you’re not so silly, after all.”

“Thanks for the vote of confidence.”

The final principle of quantum theory is the idea of
measurement
. In quantum mechanics, measurement is an active process. The act of measuring something creates the reality that we observe.
*

To give a concrete example, let’s imagine that you have a dog treat in one of two boxes. The boxes are sealed, soundproof (so you can’t hear the treat rattling), and airtight (so you can’t sniff it out): you can’t tell which box the treat is in without opening one of the boxes.

If we want to describe this as a quantum mechanical object, we need to write down a wavefunction with two parts, one part describing the probability of finding the treat in the box on the left, and the other describing the probability of finding the treat in the box on the right. We can do this by adding together the wavefunctions for the treat being in the left-hand box only and the right-hand box only, just as we did in the preceding chapter (page 42) when we made a wave packet by adding together bunny states.

Now, imagine that you open one of the boxes, and find the treat, then close the box back up. You still have one treat and two boxes, but you’ve measured the position of the treat. What does the wavefunction look like?

The wavefunction now has only one part—the piece describing a treat in the left-hand box—because we know exactly where the treat is. If you found it in the left-hand box, the next time you open that box, there’s a 100% chance that it will be there, and
there’s a 0% chance of finding the treat in the right-hand box. The other part that was there before you opened the box, giving the probability of being in the right-hand box, is gone, due to the measurement you made.

Now throw away those boxes, take two new boxes prepared in the same manner as the first pair, and you’ll have a two-part wavefunction again. The result of opening the first box won’t necessarily be the same, though. You might very well find the treat in the right-hand box this time. If you do, and keep closing and reopening that set of boxes, you’ll always find the treat in the right-hand box. Again, you go from a two-part wavefunction to a one-part wavefunction.

So, what’s the big deal? After all, that’s just how probabilities work, right? In the first experiment, the treat was in the left-hand box all along, but you just didn’t know it, and in the second experiment, the treat was in the right-hand box. The state of the treat didn’t change, but your knowledge about the state of the treat did.

Quantum probabilities don’t work that way. When we have a two-part wavefunction (a “superposition state”), it doesn’t mean that the object is in one of the two states, it means that the object is in
both
states
at the same time
. The dog treat isn’t in the left-hand box all along, it’s simultaneously in both left and right boxes until after you open the box, and find it in one or the other.

“That’s pretty strange. Why should we believe it?”

“Well, we can demonstrate the weird features of quantum mechanics with an experiment called a quantum eraser.”

“Oooh! I like that! Let’s erase some cats!”

“It doesn’t work on macroscopic objects. It uses polarized light, which I have to explain first.”

“Awww . . . Why can’t we just erase stuff?”

“I’ll keep it as short as I can, but this is important stuff. Polarized light is the best system around for giving concrete examples
of quantum effects. We’ll need it for this chapter, and also chapters 7 and 8.”

“Oh, all right. As long as I can erase stuff later.”

“We’ll see what we can do.”

We can show both the existence of superposition states and the effects of measurement using the polarization of light. Polarized photons are extremely useful for testing the predictions of quantum mechanics, and will show up again and again in coming chapters, so we need to take a little time to discuss polarization of light, which comes from the idea of light as a wave.

A wave, such as a beam of light, is defined by five properties. We have already talked about four of these: the wavelength (distance between crests in the wave pattern), frequency (how many times the wave oscillates per second at a given point), amplitude (the distance between the top of a crest and the bottom of a trough), and the direction in which the wave moves. The fifth is the polarization, which is basically the direction along which the wave oscillates. An impatient dog owner out for a walk can attempt to get his dog’s attention by shaking the leash up and down, which makes a vertically polarized wave in the leash, or by shaking the leash from side to side, which makes a horizontally polarized wave.

Like a shaken leash, a classical light wave has a direction of oscillation associated with it. The oscillation is always at right angles to the direction of motion, but can point in any direction around that (that is, left, right, up, or down, relative to the direction the light is moving). Physicists typically represent the polarization state of a beam of light by an arrow pointing along the direction of oscillation—a vertically polarized beam of light is represented by an arrow pointing up, and a horizontally polarized
beam of light is represented by an arrow pointing to the right, as seen in the figure below.

Left: vertical polarization, represented as an up arrow. Middle: horizontal polarization, represented as a right arrow. Right: polarization between vertical and horizontal, which can be thought of as a sum of horizontal and vertical components.

• • •

“Wait, what are these pictures, again?”

“Imagine that you’re right behind the beam of light, and looking down the direction of motion. The arrow indicates the direction of the oscillation of the wave. An up arrow means that you’ll see the wave moving up and down; a right arrow means that you’ll see it moving side to side.”

“So . . . an up arrow is like chasing a bunny that bounds up and down, while a right arrow is like chasing a squirrel that zigzags back and forth?”

“Sure, that works.”

“Are up and to the right the only options?”

“You can have arrows in other directions, too. An arrow to the left also indicates a side-to-side oscillation, but it’s out of phase with the arrow to the right.”

“So, a right arrow is a squirrel that zigs to the right first, and a left arrow is a squirrel that zags to the left first?”

“Yeah. If you insist on examples involving prey animals.”

“I like prey animals!”

• • •

The polarization of a wave can be horizontal or vertical, but also any angle in between. We can think of the in-between angles as being made up of a horizontal part and a vertical part, as shown in the figure above. Each of these components is less intense (that is, it has a smaller amplitude, indicated in the figure by the length of the arrow) than the total wave, but they add together to give the same final intensity at some angle. You can think of this addition as a combination of steps, just like the way that we can get from one point to another by either taking three steps east followed by four steps north, or by taking five steps in a direction about 37° east of due north.

“So, an in-between angle is like a bunny that’s zigzagging left and right, while also hopping up and down?”

“Yes, that’s right.”

“Or a squirrel that’s jumping up and down while it zigzags left and right?”

“I think that’s about enough prey examples for now.”

“You’re no fun.”

Thinking of in-between polarizations as a sum of horizontal and vertical components is a useful trick because it makes it easy to see what happens when light encounters a polarizing filter. Polarizing filters are devices that will allow light polarized at a particular angle—vertical, say—to pass through unimpeded, while light polarized at an angle 90° away—horizontal—will be completely absorbed. You can understand the effect by imagining a dog on a leash that passes through a picket fence. If you shake the leash up and down, the wave will pass right through, but side-to-side shaking will be blocked by the boards of the fence.

When light at an angle between vertical and horizontal strikes a vertically oriented polarizing filter, only the vertical component of the light will pass through. This lowers the intensity of
the light on the other side, by an amount that depends on the angle. For small angles, most of the light makes it through—at an angle of 30°, the beam on the far side is three-fourths as bright as the initial beam—while for larger angles, most of the beam is blocked—at 60° from vertical, the beam on the far side is only one-fourth as bright as the initial beam. At an angle of 45°, midway between horizontal and vertical, exactly half of the light will pass through the filter.

The light on the far side of the filter is polarized at the angle of the filter, no matter what angle it started at. For this reason, polarizing filters are commonly called polarizers: light passing through a vertically oriented polarizing filter will emerge as vertically polarized light, whether it started with vertical polarization or at some other angle. The overall amount of light will be different, but the polarization will be the same. All of the light passing through a vertically oriented filter will pass through a second vertical filter, and all of it will be blocked by a horizontally oriented filter.

“What is all this good for, anyway?”

“Other than helping demonstrate quantum physics? Plenty. Light polarization is an extremely useful thing. Digital displays on watches, cell phones, and televisions use a polarizer in front of a light source to vary the amount of light that gets through. And polarizing filters are also used to make sunglasses.”

“Sunglasses?”

“Yeah, those sunglasses that I wear when I take you for walks are actually polarizing filters. The light from the sun is unpolarized—it’s as likely to be horizontal as vertical—but when light reflects off a surface, it tends to become slightly polarized. Light reflecting off the road out in front of us when we’re walking has more horizontal polarization than vertical, so by wearing vertical polarizers as sunglasses, I can block most of that light.”

“What’s the point of that? Doesn’t it make it harder to see?”

“Actually, it reduces the glare off the road, and makes it easier to see things up ahead.”

“Things . . . Like bunnies in the road?”

“For example, yes.”

“Can I have some polarized sunglasses so I can see bunnies?”

“The ones I have won’t fit on your ears, but we’ll look into it. Later. First, I have to talk about quantum measurement with polarized light.”

“Oh, yeah. Quantum physics. Right.”

How does all this apply to light as a particle, though? We spent a good chunk of
chapter 1
describing how a beam of light is both a stream of photons and a smooth wave. The last few pages have been discussing polarization in classical terms. How do we handle light polarization in quantum physics?