C. Waves or particles?

For this section, you might want to help your concentration by brewing some strong coffee, at least two cups. You will be imagining waves combining with their up-down-up motions.

Regarding quantum mechanics, the question often comes up, “Does light consist of waves or particles?” as well as “Are electrons composed of waves or particles?” Now, as we know today, light and electrons both possess properties of classical waves and particles, so it makes some sense to speak of a “wave-particle duality”. And yet the synthesis of a wave concept and a particle concept is not even a central idea of quantum mechanics, a theory that eventually overturned our view of reality, a theory that required a new language in order to describe phenomena. Historically, however, it was the synthesis of a wave concept and a particle concept that led to everything else. This is another example of the curious path of scientific progress.

As we discussed in Chapter 10, Section E, in 1900 Max Planck explained the blackbody spectrum (radiation in the interior of a cavity) by assuming that the radiation in the cavity resided in resonators that could hold energy only in given amounts, thus introducing lumps or quanta. In 1905 Albert Einstein showed that he could explain the short-wavelength portion of the blackbody spectrum if he assumed that the electromagnetic field itself comes in quanta.1 (See Figure 10.3, the extreme left-hand side.) These quanta had to wait twenty years for their name, when Gilbert Lewis christened them “photons”.2

In 1924 Louis de Broglie postulated in his Ph.D. thesis that, if light could consist of particles, perhaps electrons could solve a wave equation.3 Clinton Davisson and Lester Germer demonstrated the wave nature of electrons in 1927 by showing that they display interference.4 (We will discuss interference later.) In order to see more clearly the conundrum presented by these findings, we should look at the double-slit experiment.

In a double-slit experiment we place a wall with two slits between a source and a detection screen. See Figure 13.1. The source may produce our idea of classical particles, like pellets, or it may produce waves, or it may produce quantum particles. We will test our intuition against an actual experiment in these cases.

1 Einstein [1905a]2 Lewis [1926]3 De Broglie [1924]4 Davisson and Germer [1927] performed an experiment in which they scattered electrons from a crystal.

In a double-slit experiment a source fires something toward a wall with two slits. On the far side of the wall lies a detection screen.

Figure 13.1In our first experiment let us say that the source is a pitching machine that flings very fast baseballs toward the right. Now the baseballs don’t all come out level to the ground: some come out higher and some lower. Most hit the wall and bounce off. Let’s say that they are traveling fast enough that they travel approximately along straight paths. When the first ball makes it through the wall, the screen detects it, as we see in Figure 13.2.

In the first double-slit experiment we think of baseballs fired at the wall. After the first detection of a baseball at the screen, we might see the above figure.

Figure 13.2

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After many detections, we might see something that looks like Figure 13.3.

In this first experiment, after many baseballs pass through the wall and are detected, we see a pattern on the detection screen.

Figure 13.3

We don’t see any mystery here.

What can we say about baseballs? They conform to our ideas of classical particles. By “classical particles” we mean two things: 1) “particles” in the minds of physicists in the 1800s and 2) “particles” as we might commonly understand the word.

1. Classical particles are, well, particulate, that is, we can have 0, 1, 2, or 3 particles (or whatever), but we cannot have ⅓ of a particle. Two baseballs produce twice as much baseball fun as one baseball, but one-third of a baseball produces no baseball fun at all. We’ve ruined it.

2. A particle has a well-defined location.3. A particle has a well-defined momentum. Classically, momentum is

mass times velocity.5

Now let’s think about a tray filled with water, and we are viewing it from the top. Or else we could be viewing a very calm bay from the sky. Again we have a wall or obstruction with two slits or holes and a source on the left. In this scenario the source produces waves, some of which travel toward the wall. Most of the waves are blocked by the wall, but some pass through the holes and form the interference pattern at the screen, as shown in Figure 13.4.

5 The expression for momentum in special relativity is somewhat different. If p is momentum, E is the total energy, m is the rest mass, and c is the speed of light, then we have E2 = (pc)2 + (mc2)2.

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In the second double-slit experiment the source produces waves. Hole 1 and Hole 2 each produce a train of waves, and these combine to make an interference pattern at the screen.

Figure 13.4

Let’s think about the waves that show up at the detection screen. A water skater named Alice (Aquarius remigis) sits on the surface of the water at point A of the detection screen. If the Hole 2 were closed and only Hole 1 were open, then waves would travel from S to 1 to A, and the Alice would experience up-down-up-down-up. If only Hole 2 were open, then waves would travel from S to 2 to A, and Alice would experience up-down-up-down-up. With both holes open, water skater Alice experiences the sum, thus UP-DOWN-UP-DOWN-UP, yielding a large wave. Both wave trains are necessary for this result.

Now water skater Bob sits on the surface of the water at point B. If only Hole 1 were open, then waves would travel from S to 1 to B, and Bob would experience up-down-up-down-up. If only Hole 2 were open, then waves would travel from S to 2 to B, a greater distance. These waves are delayed, and so they arrive down-up-down-up-down. With both holes open, Bob the water skater experiences the sum zero-zero-zero-zero-zero. Again, both waves and both paths are necessary for this result.

Water skater Carlos sits at point C. Carlos experiences waves from S to 1 to C that are up-down-up-down-up, and he experiences waves from S to 2 to C that are quite delayed, up-down-up-down-up. With both holes open, Carlos experiences UP-DOWN-UP-DOWN-UP, again a large wave.

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At the detection screen we see a series of maxima and minima, a pattern physicists call an interference pattern. Such a pattern is characteristic of waves.

Let’s think about “classical waves”, as physicists in the 1800s might conceive them or as we might understand the word today. What can we say about classical waves?

1. Waves interfere, that is, they add. For example, up-down-up + down-up-down = zero-zero-zero. They form interference patterns.

2. Waves are not particulate. We can always split a wave into two parts; for example, we can split one light beam into two beams using a beam splitter. Physicists call it a “beam splitter”, but you might be more familiar with “beam-splitter glass” in the one-way mirrors that police use to separate a witness from possible suspects in police line-ups. The suspects stand in a well lit room and the witness stands in a darkened room, the two rooms separated by a one-way mirror. The light from the suspects’ faces travels to the one-way-mirror glass and gets split, half going through to the witness and half reflected back to the suspects. Anyway, waves can always be divided.

Although interference patterns are fascinating, we still don’t see a mystery here.

Now we could perform the double-slit experiment with light. Under certain conditions6 we will see an interference pattern, since light is a wave phenomenon. The experiment becomes interesting when the source sends through one photon at a time. Let’s perform a double-slit experiment by sending through one electron at a time. We need to enclose the apparatus and create a vacuum. The source propels electrons toward the wall one at a time. After a detection of one electron, we might see something like Figure 13.2. After the detection of five electrons, we might see something like Figure 13.5.

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In the third double-slit experiment the source produces electrons one at a time. The detection screen here has detected five electrons.

Figure 13.5

We do not see any discernible pattern. After many electrons have shown up at the screen, a pattern does emerge, as we see in Figure 13.6.7

In the third double-slit experiment, an interference pattern emerges over time. Each single electron creates one detection on the screen, but the pattern emerges after the screen has detected many electrons.

Figure 13.6

At this point, several questions arise:

1. How does a single electron “know” never to show up at point B?

Even when only one electron is in the chamber, Nature is somehow performing a calculation involving the two paths S 1 B and S 2 B. That explains the cancellation up-down-up + down-up-down = zero-zero-zero at B.

2. Consider an electron that shows up at point C. Did that electron go through Slit 1 or through Slit 2?

We are tempted to say that it went through Slit 1, but we cannot. We would at least like to say, “The electron went through Slit 1 or Slit 2, but we do not know which,” but most physicists resist this statement. Nature seems to make a calculation involving both slits while the electron is in the apparatus. At any rate, quantum mechanics involves a calculation, and we can see the calculation represented pictorially in Figure 13.7. The probability of finding the electron at point C is related to the amplitude of the wave at C, and physicists calculate that amplitude by adding both waves shown in the figure. Physicists have mathematical language to describe the electron in the apparatus, but we do not have English language to describe the electron after it is emitted and before it is detected. If the calculation involves a sum over a path through Slit 1 and a path through

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Slit 2, surely the underlying reality would not have an electron going through Slit 1 or Slit 2.8

Figure 13.7In order to calculate the probability that an electron will show up at point C, physicists calculate the amplitude of the wave at C by adding the amplitudes of the waves from the two paths S-1-C and S-2-C. The square of the magnitude of the amplitude yields the probability.

Some physicists state, “The electron passes through both slits.” More conservative physicists take exception with this statement, issuing their seemingly wordsmithed statement, “While the electron is in the chamber, it is represented theoretically by an expression that takes two paths into account. We can calculate a meaningful number only when we calculate, for example, the probability that the electron will show up at a given place on the screen.”

If we want a single answer to the question, “Which slit?”, we will not get it. When the electron is in the chamber, it literally has no position. To ask, “Where is the electron?” or “Was it here, or was it there?” makes as much sense as to ask, “How does the electron feel about Keynesian economics?”

Once we concede that electrons do not have well-defined locations, we may as well concede that they do not have well-defined velocities and momenta as well. We need a more compelling argument in quantum mechanics, but it turns out that electrons do not have well-defined momenta. Summing up, we can say the following about electrons (and about photons).

1. Electrons are particulate.2. Electrons display interference patterns.3. Electrons do NOT have well-defined locations.4. Electrons do NOT have well-defined momenta.

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Because of these properties we can say also add the following:

5. Electrons do NOT follow well-defined paths.

Note the irony here. The pillar of classical science was the ability to predict the paths, or trajectories, of particles, and in this experiment we discover not that we cannot predict those trajectories but that the trajectories do not even exist. We can calculate, as a consolation prize, perhaps, the probabilities of detecting particles at certain locations, or the probabilities of making other detections, depending on the experimental apparatus. A poor consolation, indeed!

This excerpt has been taken from a book by Garrett Biehle.The book has a working title The Struggle for the Truth: How Scientists Determine Reality.

6 In order to see an interference pattern, the size and separation of the slits must be approximately the same as the wavelength of the light.

7 Jönsson [1961] explicitly performed this experiment with electrons. Taylor [1909] first performed this experiment with photons, shown through the apparatus one at a time.

8 The Bohm-de Broglie interpretation of quantum mechanics holds precisely this position, to wit, that the probability of finding the electron can be calculated by a wave equation, and yet we can posit that a real, physical particle passes through one slit of the other. Hardy [1992] points out a contradiction in this paradigm, particularly showing that such an interpretation cannot be Lorentz invariant.