How to Teach Physics to Your Dog (10 page)

When we’re dealing with classical light waves, it’s easy to understand how part of a wave can pass through the filter. When we talk about light in terms of photons, though, the filter is an all-or-nothing proposition. Any given photon either makes it through, or gets absorbed by the filter. There are no “parts” of photons.

We handle the interaction between photons and polarizing filters by saying that each photon has a probability of passing through the filter that is equal to the fraction of the total wave that makes it through in the classical model. If a beam of light with a polarization at 60° from vertical encounters a vertical polarizing filter, the beam on the far side will be one-fourth as bright, meaning that it has one-fourth as many photons. That means that each individual photon has only one chance in four of making it through the polarizing filter.

Each photon making it through the filter will also have its polarization determined by the filter. Only one photon in four may make it through a vertically oriented polarizing filter, but every one of those photons will pass through a second vertical filter, and none of them will pass through a horizontal filter.
“Vertical” and “horizontal” are then the allowed states of the single photon’s polarization—when we measure the polarization using a filter, we will find the photon in one of those two states (either passing through the vertical filter, or being absorbed by it), and not anywhere in between.

Polarized photons thus provide an excellent system for looking at the core principles of quantum mechanics. Each individual photon can be described in terms of a
wavefunction
, with two parts corresponding to the two
allowed states
, horizontal and vertical polarization. That wavefunction gives you the
probability
of the photon passing through a polarizing filter, and after you make a
measurement
of the polarization with a filter, the photons are in only one of the allowed states. A single photon passing through a polarizing filter demonstrates all the essential features of quantum physics. As a result, polarized photons have been used in many experiments demonstrating quantum phenomena.

“So, let me get this straight. A photon at an angle between horizontal and vertical is in a superposition state? And sending it through a polarizing filter is the same as measuring it?”

“Yes. You get all the features of quantum superposition and measurement—wavefunctions, allowed states, probability, and measurement—using single polarized photons.”

“But I thought you said all this stuff worked the same way when you talked about light as a classical wave?”

“Well, yeah. The end result is the same as the classical polarized wave description.”

“What’s the big deal, then? I mean, your big example of quantum weirdness is something that just reproduces classical results?”

“Well, no. I mean, that’s not my big example. The big example of quantum weirdness is in the next section.”

“Oh. Well, carry on, then.”

One of the best demonstrations of the weirdness of quantum superpositions is an experiment called a quantum eraser. The quantum eraser encapsulates everything that’s strange about single-particle quantum physics in a single experiment: particle-wave duality, superposition states, and the active nature of measurement. If you can understand the quantum eraser, you’ve understood the essential elements of quantum physics.

Many different variants of quantum-eraser experiments have been done over the years,
*
but the simplest starts with a variant of Young’s double-slit experiment (page 18). If we send a beam of photons at a pair of narrow slits, we will see an interference pattern on the far side of the slits, built up out of single photons detected at particular points (as shown in the figure on the next page). We can see the pattern only because light passes through both slits at the same time. If we block one slit, the interference pattern will disappear, and we’ll see only a broad scattering of photons due to the light passing through the unblocked slit.

The interference pattern that we see indicates that the photons are in a superposition state: the wavefunction describing each photon has two parts, one for the photon passing through the left slit, and the other for the photon passing through the right slit. Each photon has passed through
both
slits, at the same time, and the interference between those two components is what produces the pattern we see. Interference patterns always turn up when you have a two-part wavefunction. When we block one slit, we only have a one-part wavefunction, destroying the superposition, and there is no interference pattern.

Interference pattern built up from single photons. Left to right, 1/30 second, 1 second, 100 seconds. Images by Lyman Page at Princteon University, reprinted with permission.

• • •

“Wait, I thought the interference was between two different photons—one that went through the left slit, and one that went through the right slit?”

“That’s an easy thing to think, since we usually send in lots of photons at the same time. We can show that that’s not the case, though, by sending light at the slits one photon at a time.”

“How does a single photon give you an interference pattern?”

“It doesn’t. Each individual photon shows up as a single spot, at a particular position on the screen, and where any individual photon turns up is random.”

“There’s the probability thing again.”

“Exactly. The individual photons are random, but if you repeat the experiment over and over again, and keep track of all the photons, you’ll see them add up to form an interference pattern. There are some places where you’re very likely to find a photon, and other places where there’s absolutely no chance of finding one. The overall pattern is determined by the probability
distribution you get from the wavefunction for each individual photon interfering with itself.”

“So it’s one particle, but it goes through both slits, and then ends up at one place on the other side?”

“Exactly.”

“That’s just weird.”

“That’s quantum physics.”

Instead of blocking one slit, though, let’s imagine covering the two slits with two different polarizing filters, one vertical and one horizontal. We put a filter on the left slit that will pass only horizontally polarized light, and we put a filter on the right slit that will pass only vertically polarized light. If we send in light polarized at an angle of 45° to the vertical, it has a 50% chance of going through a horizontal polarizing filter, and a 50% chance of going through a vertical polarizing filter, so we get some light through each slit.

This arrangement of filters gives us a way of measuring which slit the light went through. If we put a vertical polarizer in front of our detector, we will only see light that went through the right-hand slit, and if we put a horizontal polarizer in front of our detector, we will only see light that went through the left-hand slit. The polarizer in front of the detector lets us tell which slit the photon went through, just as if we had put a detector right next to the slit and measured the position directly.

What happens when we do this? When we look at the light with the filters over the two slits, we don’t see any sign of an interference pattern. When we measure the polarization of the light, we measure which slit the light went through, and that takes us from a two-part wavefunction, which produces an interference pattern, to a one-part wavefunction, which does not. The act of measuring which slit the photon went through destroys the component of the wavefunction describing the photon going through the other slit, just as the act of opening one
of the boxes destroyed the component for the treat being in the other box.

We don’t even need to put a polarizing filter on the detector—by putting the polarizers over the slits, we have “tagged” each photon, and the mere fact that we
can
measure which slit it went through is enough to destroy the pattern. In the treats-in-boxes example, this is like someone writing “Treat” on the outside of the box containing the treat—we no longer need to open the box to destroy the superposition.

The disappearing pattern is pretty weird in its own right, but things get weirder: we can undo the measurement after the fact by using a 45° polarizer instead of a horizontal or vertical polarizer to look at the light after the slits. If we do this, we see an interference pattern again! A 45° polarizer will pass either horizontal or vertical polarization, each with a 50% probability, which means any light we detect after the polarizer could have gone through either of the two slits, or even both at once. The third filter “erases” the information we had gained by tagging the photon, like somebody removing the label from the box containing the treat. Inserting the extra polarizer makes it as if we had never made the measurement at all. The second part of the wavefunction isn’t destroyed after all, and we can see interference.

The quantum-eraser experiment encapsulates everything that is strange about the core principles of quantum mechanics. The appearance of the interference pattern shows the superposition of quantum states, as each photon goes through both slits at the same time, and the disappearance and reappearance of the pattern when we add polarizing filters shows the active nature of quantum measurement. Just the fact that it is possible to measure which slit the particle went through is enough to completely change the results of the experiment.

These four ideas—wavefunctions, allowed states, probability, and measurement—are the central elements of quantum theory. The interference patterns we see in experiments with photons
*
confirm that quantum particles really do occupy multiple states at the same time. The disappearance and reappearance of the pattern in the quantum-eraser experiment confirms that measurement is an active process and determines what happens in subsequent experiments.

We still have a problem, though, because there is no mathematical process for describing how to get from a probability to the result of a measurement. We use the Schrödinger equation to calculate the wavefunctions for the allowed states of a physical object, and we use the wavefunction to calculate the probability distribution, but we cannot use the probability distribution to predict the exact result of an individual measurement. Something mysterious happens in the process of making a measurement.

This “measurement problem” is the origin of the competing interpretations of quantum mechanics, and the point where physics is forced to become philosophy. All interpretations use the same methods to calculate probabilities for the outcomes of repeated measurements. They differ only in how they explain the step from a quantum superposition state, where the wave-function consists of two (or more) states at the same time, to the classical result of a single measurement, where the object is found in one and only one state.

The first interpretation put forward for quantum theory was developed by Niels Bohr and coworkers at his institute in Denmark,
and is thus known as the Copenhagen interpretation. The Copenhagen interpretation is a very ad hoc approach to the problem of measurement in quantum mechanics (which is in some ways typical of Bohr’s approach
*
).

The Copenhagen interpretation tries to avoid the problems of superposition and measurement by drawing a strict line between microscopic and macroscopic physics. Microscopic objects—photons, electrons, atoms, and molecules—are governed by the rules of quantum mechanics, but macroscopic objects—dogs, physicists, and measurement apparatus—are governed by classical physics. There’s an absolute separation between the two, and you will never see a macroscopic object behaving in a quantum manner.

Quantum measurement involves the interaction of a macroscopic measurement apparatus with a microscopic object, and that interaction changes the state of the microscopic object. The usual description is that the wavefunction “collapses” into a single state. This “collapse,” in the Copenhagen interpretation, is an actual change of the wavefunction from a spread-out quantum state with multiple possible measurement outcomes to a state with a single measured value.
^†

In the most extreme variants of the Copenhagen interpretation, the collapse requires not only a macroscopic measurement
apparatus, but also a conscious observer to note the measurement. In this view, a tree that falls in the forest hasn’t really fallen until some person (or dog) comes along and observes it.

“So, wait, doesn’t that mean rejecting the entire idea of an objective physical reality?”

“In its most extreme forms, yes. Werner Heisenberg was probably the most radical of the Copenhagen crowd, and he insisted quite strongly that it was a mistake to talk about electrons having an independent reality. In Heisenberg’s view, the only things we can really talk about are the outcomes of specific measurements. He rejected the whole notion of talking about what the electrons were doing between measurements.”