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Now finally, we come to the heart of the matter of quantum interference, as seen from the perspective of in 1920’s quantum physics. (We’ll deal with quantum field theory later this year.)

Last time I looked at some cases of two particle states in which the particles’ behavior is independent — uncorrelated. In the jargon, the particles are said to be “unentangled”. In this situation, and only in this situation, the wave function of the two particles can be written as a product of two wave functions, one per particle. As a result, any quantum interference can be ascribed to one particle or the other, and is visible in measurements of either one particle or the other. (More precisely, it is observable in repeated experiments, in which we do the same measurement over and over.)

In this situation, because each particle’s position can be studied independent of the other’s, we can be led to think any interference associated with particle 1 happens near where particle 1 is located, and similarly for interference involving the second particle.

But this line of reasoning only works when the two particles are uncorrelated. Once this isn’t true — once the particles are entangled — it can easily break down. We saw indications of this in an example that appeared at the ends of my last two posts (here and here), which I’m about to review. The question for today is: what happens to interference in such a case?

Correlation: When “Where” Breaks Down

Let me now review the example of my recent posts. The pre-quantum system looks like this

Figure 1: An example of a superposition, in a pre-quantum view, where the two particles are correlated and where interference will occur that involves both particles together.

Notice the particles are correlated; either both particles are moving to the left OR both particles are moving to the right. (The two particles are said to be “entangled”, because the behavior of one depends upon the behavior of the other.) As a result, the wave function cannot be factored (in contrast to most examples in my last post) and we cannot understand the behavior of particle 1 without simultaneously considering the behavior of particle 2. Compare this to Fig. 2, an example from my last post in which the particles are independent; the behavior of particle 2 is the same in both parts of the superposition, independent of what particle 1 is doing.

Figure 2: Unlike Fig. 1, here the two particles are uncorrelated; the behavior of particle 2 is the same whether particle 1 is moving left OR right. As a result, interference can occur for particle 1 separately from any behavior of particle 2, as shown in this post.

Let’s return now to Fig. 1. The wave function for the corresponding quantum system, shown as a graph of its absolute value squared on the space of possibilities, behaves as in Fig. 3.

Figure 3: The absolute-value-squared of the wave function for the system in Fig, 1, showing interference as the peaks cross. Note the interference fringes are diagonal relative to the x1 and x2 axes.

But as shown last time in Fig. 19, at the moment where the interference in Fig. 3 is at its largest, if we measure particle 1 we see no interference effect. More precisely, if we do the experiment many times and measure particle 1 each time, as depicted in Fig. 4, we see no interference pattern.

Figure 4: The result of repeated experiments in which we measure particle 1, at the moment of maximal interference, in the system of Fig. 3. Each new experiment is shown as an orange dot; results of past experiments are shown in blue. No interference effect is seen.

We see something analogous if we measure particle 2.

Yet the interference is plain as day in Fig. 3. It’s obvious when we look at the full two-dimensional space of possibilities, even though it is invisible in Fig. 4 for particle 1 and in the analogous experiment for particle 2. So what measurements, if any, can we make that can reveal it?

The clue comes from the fact that the interference fringes lie at a 45 degree angle, perpendicular neither to the x1 axis nor to the x2 axis but instead to the axis for the variable 1/2(x1 + x2), the average of the positions of particle 1 and 2. It’s that average position that we need to measure if we are to observe the interference.

But doing so requires we that we measure both particles’ positions. We have to measure them both every time we repeat the experiment. Only then can we start making a plot of the average of their positions.

When we do this, we will find what is shown in Fig 5.

• The top row shows measurements of particle 1.
• The bottom row shows measurements of particle 2.
• And the middle row shows a quantity that we infer from these measurements: their average.

For each measurement, I’ve drawn a straight orange line between the measurement of x1 and the measurement of x2; the center of this line lies at the average position 1/2(x1+x2). The actual averages are then recorded in a different color, to remind you that we don’t measure them directly; we infer them from the actual measurements of the two particles’ positions.

Figure 5: As in Fig. 4, the result of repeated experiments in which we measure both particles’ positions at the moment of maximal interference in Fig. 3. Top and bottom rows show the position measurements of particles 1 and 2; the middle row shows their average. Each new experiment is shown as two orange dots, they are connected by an orange line, at whose midpoint a new yellow dot is placed. Results of past experiments are shown in blue. No interference effect is seen in the individual particle positions, yet one appears in their average.

In short, the interference is not associated with either particle separately — none is seen in either the top or bottom rows. Instead, it is found within the correlation between the two particles’ positions. This is something that neither particle can tell us on its own.

And where is the interference? It certainly lies near 1/2(x1+x2)=0. But this should worry you. Is that really a point in physical space?

You could imagine a more extreme example of this experiment in which Fig. 5 shows particle 1 located in Boston and particle 2 located in New York City. This would put their average position within appropriately-named Middletown, Connecticut. (I kid you not; check for yourself.) Would we really want to say that the interference itself is located in Middletown, even though it’s a quiet bystander, unaware of the existence of two correlated particles that lie in opposite directions 90 miles (150 km) away?

After all, the interference appears in the relationship between the particles’ positions in physical space, not in the positions themselves. Its location in the space of possibilities (Fig. 3) is clear. Its location in physical space (Fig. 5) is anything but.

Still, I can imagine you pondering whether it might somehow make sense to assign the interference to poor, unsuspecting Middletown. For that reason, I’m going to make things even worse, and take Middletown out of the middle.

A Second System with No Where

Here’s another system with interference, whose pre-quantum version is shown in Figs. 6a and 6b:

Figure 6a: Another system in a superposition with entangled particles, shown in its pre-quantum version in physical space. In part A of the superposition both particles are stationary, while in part B they move oppositely. Figure 6b: The same system as in Fig. 6a, depicted in the space of possibilities with its two initial possibilities labeled as stars. Possibility A remains where it is, while possibility B moves toward and intersects with possibility A, leading us to expect interference in the quantum wave function.

The corresponding wave function is shown in Fig. 7. Now the interference fringes are oriented diagonally the other way compared to Fig. 3. How are we to measure them this time?

Figure 7: The absolute-value-squared of the wave function for the system shown in Fig. 6. The interference fringes lie on the opposite diagonal from those of Fig. 3.

The average position 1/2(x1+x2) won’t do; we’ll see nothing interesting there. Instead the fringes are near (x1-x2)=4 — that is, they occur when the particles, no matter where they are in physical space, are at a distance of four units. We therefore expect interference near 1/2(x1-x2)=2. Is it there?

In Fig. 8 I’ve shown the analogue of Figs. 4 and 5, depicting

• the measurements of the two particle positions x1 and x2, along with
• their average 1/2(x1+x2) plotted between them (in yellow)
• (half) their difference 1/2(x1-x2) plotted below them (in green).

That quantity 1/2(x1-x2) is half the horizontal length of the orange line. Hidden in its behavior over many measurements is an interference pattern, seen in the bottom row, where the 1/2(x1-x2) measurements are plotted. [Note also that there is no interference pattern in the measurements of 1/2(x1+x2), in contrast to Fig. 4.]

Figure 8: For the system of Figs. 6-7, repeated experiments in which the measurement of the position of particle 1 is plotted in the top row (upper blue points), that of particle 2 is plotted in the third row (lower blue points), their average is plotted between (yellow points), and half their difference is plotted below them (green points.) Each new set of measurements is shown as orange points connected by an orange line, as in Fig. 5. An interference pattern is seen only in the difference.

Now the question of the hour: where is the interference in this case? It is found near 1/2(x1-x2)=2 — but that certainly is not to be identified with a legitimate position in physical space, such as the point x=2.

First of all, making such an identification in Fig. 8 would be like saying that one particle is in New York and the other is in Boston, while the interference is 150 kilometers offshore in the ocean. But second and much worse, I could change Fig. 8 by moving both particles 10 units to the left and repeating the experiment. This would cause x1, x2, and 1/2(x1-x2) in Fig. 8 to all shift left by 10 units, moving them off your computer screen, while leaving 1/2(x1-x2) unchanged at 2. In short, all the orange and blue and yellow points would move out of your view, while the green points would remain exactly where they are. The difference of positions — a distance — is not a position.

If 10 units isn’t enough to convince you, let’s move the two particles to the other side of the Sun, or to the other side of the galaxy. The interference pattern stubbornly remains at 1/2(x1-x2)=2. The interference pattern is in a difference of positions, so it doesn’t care whether the two particles are in France, Antarctica, or Mars.

We can move the particles anywhere in the universe, as long as we take them together with their average distance remaining the same, and the interference pattern remains exactly the same. So there’s no way we can identify the interference as being located at a particular value of x, the coordinate of physical space. Trying to do so creates nonsense.

This is totally unlike interference in water waves and sound waves. That kind of interference happens in a someplace; we can say where the waves are, how big they are at a particular location, and where their peaks and valleys are in physical space. Quantum interference is not at all like this. It’s something more general, more subtle, and more troubling to our intuition.

[By the way, there’s nothing special about the two combinations 1/2(x1+x2) and 1/2(x1-x2), the average or the difference. It’s easy to find systems where the intereference arises in the combination x1+2x2, or 3x1-x2, or any other one you like. In none of these is there a natural way to say “where” the interference is located.]

The Profound Lesson

From these examples, we can begin to learn a central lesson of modern physics, one that a century of experimental and theoretical physics have been teaching us repeatedly, with ever greater subtlety. Imagining reality as many of us are inclined to do, as made of localized objects positioned in and moving through physical space — the one-dimensional x-axis in my simple examples, and the three-dimensional physical space that we take for granted when we specify our latitude, longitude and altitude — is simply not going to work in a quantum universe. The correlations among objects have observable consequences, and those correlations cannot simply be ascribed locations in physical space. To make sense of them, it seems we need to expand our conception of reality.

In the process of recognizing this challenge, we have had to confront the giant, unwieldy space of possibilities, which we can only visualize for a single particle moving in up to three dimensions, or for two or three particles moving in just one dimension. In realistic circumstances, especially those of quantum field theory, the space of possibilities has a huge number of dimensions, rendering it horrendously unimaginable. Whether this gargantuan space should be understood as real — perhaps even more real than physical space — continues to be debated.

Indeed, the lessons of quantum interference are ones that physicists and philosophers have been coping with for a hundred years, and their efforts to make sense of them continue to this day. I hope this series of posts has helped you understand these issues, and to appreciate their depth and difficulty.

Looking ahead, we’ll soon take these lessons, and other lessons from recent posts, back to the double-slit experiment. With fresher, better-informed eyes, we’ll examine its puzzles again.

The quantum double-slit experiment, in which objects are sent toward and through a pair of slits in a wall,and are recorded on a screen behind the slits, clearly shows an interference pattern. It’s natural to ask, “where does the interference occur?”

The problem is that there is a hidden assumption in this way of framing the question — a very natural assumption, based on our experience with waves in water or in sound. In those cases, we can explicitly see (Fig. 1) how interference builds up between the slits and the screen.

Figure 1: How water waves or sound waves interfere after passing through two slits.

But when we dig deep into quantum physics, this way of thinking runs into trouble. Asking “where” is not as straightforward as it seems. In the next post we’ll see why. Today we’ll lay the groundwork.

Independence and Interference

From my long list of examples with and without interference (we saw last time what distinguishes the two classes), let’s pick a superposition whose pre-quantum version is shown in Fig. 2.

Figure 2: A pre-quantum view of a superposition in which particle 1 is moving left OR right, and particle 2 is stationary at x=3.

Here we have

• particle 1 going from left to right, with particle 2 stationary at x=+3, OR
• particle 1 going from right to left, with particle 2 again stationary at x=+3.

In Fig. 3 is what the wave function Ψ(x1,x2) [where x1 is the position of particle 1 and x2 is the position of particle 2] looks like when its absolute-value squared is graphed on the space of possibilities. Both peaks have x2=+3, representing the fact that particle 2 is stationary. They move in opposite directions and pass through each other horizontally as particle 1 moves to the right OR to the left.

Figure 3: The graph of the absolute-value-squared of the wave function for the quantum version of the system in Fig. 2.

This looks remarkably similar to what we would have if particle 2 weren’t there at all! The interference fringes run parallel to the x2 axis, meaning the locations of the interference peaks and valleys depend on x1 but not on x2. In fact, if we measure particle 1, ignoring particle 2, we’ll see the same interference pattern that we see when a single particle is in the superposition of Fig. 1 with particle 2 removed (Fig. 4):

Figure 4a: The square of the absolute value of the wave function for a particle in a superposition of the form shown in Fig. 2 but with the second particle removed. Figure 4b: A closeup of the interference pattern that occurs at the moment when the two peaks in Fig. 4a perfectly overlap. The real and imaginary parts of the wave function are shown in red and blue, while its square is drawn in black.

We can confirm this in a simple way. If we measure the position of particle 1, ignoring particle 2, the probability of finding that particle at a specific position x1 is given by projecting the wave function, shown above as a function of x1 and x2, onto the x1 axis. [More mathematically, this is done by integrating over x2 to leave a function of x1 only.] Sometimes (not always!) this is essentially equivalent to viewing the graph of the wave function from one side, as in Figs. 5-6.

Figure 5: Projecting the wave function of Fig. 3, at the moment of maximum interference, onto the x1 axis. Compare with the black curve in Fig. 4b.

Because the interference ridges in Fig. 3 are parallel to the x2 axis and thus independent of particle 2’s exact position, we do indeed find, when we project onto the x1 axis as in Fig. 5, that the familiar interference pattern of Fig. 4b reappears.

Meanwhile, if at that same moment we measure particle 2’s position, we will find results centered around x2=+3, with no interference, as seen in Fig. 6 where we project the wave function of Fig. 3 onto the x2 axis.

Figure 6: Projecting the wave function of Fig. 3, at the moment of maximum interference, onto the x2 axis. The position of particle 2 is thus close to x2=3, with no interference pattern.

Why is this case so simple, with the one-particle case in Fig. 4 and the two-particle case in Figs. 3 and 5 so closely resembling each other?

The Cause

It has nothing specifically to do with the fact that particle 2 is stationary. Another example I gave had particle 2 stationary in both parts of the superposition, but located in two different places. In Figs. 7a and 7b, the pre-quantum version of that system is shown both in physical space and in the space of possibilities [where I have, for the first time, put stars for the two possibilities onto the same graph.]

Figure 7a: A similar system to that of Fig. 2, drawn in its pre-quantum version in physical space. Figure 7b: Same as Fig. 7a, but drawn in the space of possibilities.

You can see that the two stars’ paths will not intersect, since one remains at x2=+3 and the other remains at x2=-3. Thus there should be no interference — and indeed, none is seen in Fig. 8, where the time evolution of the full quantum wave function is shown. The two peaks miss each other, and so no interference occurs.

Figure 8: The absolute-value-squared of the wave function corresponding to Figs. 7a-7b.

If we project the wave function of Fig. 8 onto the x1 axis at the moment when the two peaks are at x1=0, we see (Fig. 9) a single peak (because the two peaks, with different values of x2, are projected onto each other). No interference fringes are seen.

Figure 9: At the moment when the first particle is near x1=0, the probability of finding particle 1 as a function of x1 shows a featureless peak, with no interference effects.

Instead the resemblance between Figs. 3-5 has to do with the fact that particle 2 is doing exactly the same thing in each part of the superposition. For instance, as in Fig. 10, suppose particle 2 is moving to the left in both possibilities.

Figure 10: A system similar to that of Fig. 2, but with particle 2 (orange) moving to the left in both parts of the superposition.

(In the top possibility, particles 1 and 2 will encounter one another; but we have been assuming for simplicity that they don’t interact, so they can safely pass right through each other.)

The resulting wave function is shown in Fig. 11:

Figure 11: The absolute-value-squared of the wave function corresponding to Fig.10.

The two peaks cross paths when x1=0 and x2=2. The wave function again shows interference at that location, with fringes that are independent of x2. If we project the wave function onto the x1=0 axis, we’ll get exactly the same thing we saw in Fig. 5, even though the behavior of the wave function in x2 is different.

This makes the pattern clear: if, in each part of the superposition, particle 2 behaves identically, then particle 1 will be subject to the same pattern of interference as if particle 2 were absent. Said another way, if the behavior of particle 1 is independent of particle 2 (and vice versa), then any interference effects involving one particle will be as though the other particle wasn’t even there.

Said yet another way, the two particles in Figs. 2 and 10 are uncorrelated, meaning that we can understand what either particle is doing without having to know what the other is doing.

Importantly, the examples studied in the previous post did not have this feature. That’s crucial in understanding why the interference seen at the end of that post wasn’t so simple.

Independence and Factoring

What we are seeing in Figs. 2 and 10 has an analogy in algebra. If we have an algebraic expression such as

• (a c + b c),

in which c is common to both terms, then we can factor it into

• (a+b)c.

The same is true of the kinds of physical processes we’ve been looking at. In Fig. 10 the two particles’ behavior is uncorrelated, so we can “factor” the pre-quantum system as follows.

Figure 12: The “factored” form of the superposition in Fig. 10.

What we see here is that factoring involves an AND, while superposition is an OR: the figure above says that (particle 1 is moving from left to right OR from right to left) AND (particle 2 is moving from right to left, no matter what particle 1 is doing.)

And in the quantum context, if (and only if) two particles’ behaviors are completely uncorrelated, we can literally factor the wave function into a product of two functions, one for each particle:

• Ψ(x1,x2)=Ψ1(x1)Ψ2(x2)

In this specific case of Fig. 12, where the first particle is in a superposition whose parts I’ve labeled A and B, we can write Ψ1(x1) as a sum of two terms:

• Ψ1(x1)=ΨA(x1) + ΨB(x1)

Specifically, ΨA(x1) describes particle 1 moving left to right — giving one peak in Fig. 11 — and ΨB(x1) describes particle 2 moving right to left, giving the other peak.

But this kind of factoring is rare, and not possible in general. None of the examples in the previous post (or of this post, excepting that of its Fig. 5) can be factored. That’s because in these examples, the particles are correlated: the behavior of one depends on the behavior of the other.

Superposition AND Superposition

If the particles are truly uncorrelated, we should be able to put both particles into superpositions of two possibilities. As a pre-quantum system, that would give us (particle 1 in state A OR state B) AND (particle 2 in state C OR state D) in Fig. 13.

Figure 13: The two particles are uncorrelated, and so their behavior can be factored. The first particle is in a superposition of states A and B, the second in a superposition of states C and D.

The corresponding factored wave function, in which (particle 1 moves left to right OR right to left) AND (particle 2 moves left to right OR right to left), can be written as a product of two superpositions:

• Ψ(x1,x2)=Ψ1(x1)Ψ2(x2) = [ΨA(x1)+ΨB(x1)] [ΨC(x2)+ΨD(x2)]

In algebra, we can expand a similar product

• (a+b)(c+d)=ac+ad+bc+bd

giving us four terms. In the same way we can expand the above wave function into four terms

• Ψ(x1,x2)=ΨA(x1)ΨC(x2)+ΨB(x1)ΨC(x2)+ΨA(x1)ΨD(x2)+ΨB(x1)ΨD(x2)

whose pre-quantum version gives us the four possibilities shown in Fig. 14.

Figure 14: The product in Fig. 13 is expanded into its four distinct possibilities.

The wave function therefore has four peaks, one for each term. The wave function behaves as shown in Fig. 15.

Figure 15: The wave function for the system in Fig. 14 shows interference of two pairs of possibilities, first for particle 1 and later for particle 2.

The four peaks interfere in pairs. The top two and the bottom two interfere when particle 1 reaches x1=0, creating fringes that run parallel to the x2 axis and thus are independent of x2. Notice that even though there are two sets of interference fringes when particle 1 reaches x1=0 in all the superpositions, we do not observe this if we only measure particle 1. When we project the wave function onto the x1 axis, the two sets of interference fringes line up, and we see the same single-particle interference pattern that we’ve seen so many times (Figs. 3-5). That’s all because particles 1 and 2 are uncorrelated.

Figure 16: The first instance of interference, seen in two peaks in Fig. 15 is reduced, when projected on to the x1 axis, to the same interference pattern as seen in Figs. 3-5; the measurement of particle 1’s position will show the same interference pattern in each case, because particles 1 and 2 are uncorrelated.

If at the same moment we measure particle 2 ignoring particle 1, we find (Fig. 17) that particle 2 has equal probability of being near x=2.5 or x=-0.5, with no interference effects.

Figure 17: The first instance of interference, seen in two peaks in Fig. 15, shows two peaks with no interference when projected on to the x2 axis. Thus measurements of particle 2’s position show no interference at this moment.

Meanwhile, the left two and the right two peaks in Fig. 15 subsequently interfere when particle 2 reaches x2=1, creating fringes that run parallel to the x1 axis, and thus are independent of x1; these will show up near x=1 in measurements of particle 2’s position. This is shown (Fig. 18) by projecting the wave function at that moment onto the x2 axis.

Figure 18: During the second instance of interference in Fig. 15, the projection of the wave function onto the x2 axis. Locating the Interference?

So far, in all these examples, it seems that we can say where the interference occurs in physical space. For instance, in this last example, it appears that particle 1 shows interference around x=0, and slightly later particle 2 shows interference around x=1.

But if we look back at the end of the last post, we can see that something is off. In the examples considered there, the particles are correlated and the wave function cannot be factored. And in the last example in Fig. 12 of that post, we saw interference patterns whose ridges are parallel neither to the x1 axis nor to the x2 axis. . .an effect that a factored wave function cannot produce. [Fun exercise: prove this last statement.]

As a result, projecting the wave function of that example onto the x1 axis hides the interference pattern, as shown in Fig. 19. The same is true when projecting onto the x2 axis.

Figure 19: Alhough Fig. 12 of the previous post shows an interference pattern, it is hidden when the wave function is projected onto the x1 axis, leaving only a boring bump. The observable consequences are shown in Fig. 13 of that same post.

Consequently, neither measurements of particle 1’s position nor measurements of particle 2’s position can reveal the interference effect. (This is shown, for particle 1, in the previous post’s Fig. 13.) This leaves it unclear where the interference is, or even how to measure it.

But in fact it can be measured, and next time we’ll see how. We’ll also see that in a general superposition, where the two particles are correlated, interference effects often cannot be said to have a location in physical space. And that will lead us to a first glimpse of one of the most shocking lessons of quantum physics.

One More Uncorrelated Example, Just for Fun

To close, I’ll leave you with one more uncorrelated example, merely because it looks cool. In pre-quantum language, the setup is shown in Fig. 20.

Figure 20: Another uncorrelated superposition with four possibilities.

Now all four peaks interfere simultaneously, near (x1,x2)=(1,-1).

Figure 21: The four peaks simultaneously interfere, generating a grid pattern.

The grid pattern in the interference assures that the usual interference effects can be seen for both particles at the same time, with the interference for particle 1 near x1=1 and that for particle 2 near x2=-1. Here are the projections onto the two axes at the moment of maximal interference.

Figure 22a: At the moment of maximum interference, the projection of the wave function onto the x1 axis shows interference near x1=1. Figure 22b: At the moment of maximum interference, the projection of the wave function onto the x2 axis shows interference near x2=-1.

In my last post and the previous one, I put one or two particles in various sorts of quantum superpositions, and claimed that some cases display quantum interference and some do not. Today we’ll start looking at these examples in detail to see why interference does or does not occur. We’ll also encounter a difficulty asking where the interference occurs — a difficulty which will lead us eventually to deeper understanding.

First, a lightning review of interference for one particle. Take a single particle in a superposition that gives it equal probability of being right of center and moving to the left OR being left of center and moving to the right. Its wave function is given in Fig. 1.

Figure 1: The wave function of a single particle in a superposition of moving left from the right OR moving right from the left. The black curve represents the absolute-value-squared of the wave function, which gives the probability of finding the particle at that location. Red and blue curves show the wave function’s real and imaginary parts.

Then, at the moment and location where the two peaks in the wave function cross, a strong interference effect is observed, the same sort as is seen in the famous double slit-experiment.

Figure 2: A closeup of the interference pattern that occurs at the moment when the two peaks in Fig. 1 perfectly overlap. An animation is shown here.

The simplest way to analyze this is to approach it as a 19th century physicist might have done. In this pre-quantum version of the problem, shown in Fig. 3, the particle has a definite location and speed (and no wave function), with

• a 50 percent chance of being left of center and moving right, and
• a 50 percent chance of being right of center and moving left.

Figure 3: A pre-quantum view of Fig. 1, showing a single particle with equal probability of moving right or moving left. The particle will reach x=0 in both possibilities at the same time, but in pre-quantum physics, nothing special happens then.

Nothing interesting, in either possibility, happens when the particle reaches the center. Either it reaches the center from the left and keeps on going OR it reaches the center from the right and keeps on going. There is certainly no collision, and, in pre-quantum physics, there is also no interference effect.

Still, something abstractly interesting happens there. Before the particle reaches the center, the top and bottom of Fig. 3 are different. But just when the particle is at x=0, the two possibilities in the superposition describe the same object in the same place. In a sense, the two possibilities meet. In the corresponding quantum problem, this is the precise moment where the quantum interference effect is largest. That is a clue.

Two Particles, Two Orderings

So now let’s look in Fig. 4 at the example that I gave as a puzzle, a sort of doubling of the single particle example in Fig. 1.

Figure 4: Two particles in a superposition of moving left or moving right — a sort of doubling of Fig. 3.

Here we have two particles moving from left to right OR from right to left, with 50% probability for each of the two possibilities. I haven’t drawn the corresponding quantum wave function for this yet, but I will in a moment.

We might think something interesting would happen when particle 1 reaches x=0 in both possibilities (Fig. 5a), just as something interesting happens when the particle in Figs. 1-3 reaches x=0 in both of its possibilities. But in fact, there is no interference. Nor does anything interesting happen when the blue particle at the top and the orange particle at the bottom arrive at x=1 (Fig. 5b). Similarly, no interference happens when particle 2 reaches x=0 in both possibilities (Fig. 5c). These “events” are really non-events, as far as quantum physics is concerned. Why is this?

Figure 5a: After the two particles in Fig. 4 have moved slightly, the blue particle is at the same point in both halves of the superposition. Yet in the quantum version of this picture, no interference occurs. Figure 5b: As in Fig. 5a, but slightly later; again no interference occurs. Figure 5c: As in Fig. 5a; yet again there is no quantum interference. The Puzzle’s Puzzling Lack of Interference

To understand why interference never occurs in this case, we have to look at the system’s wave function and how it evolves with time.

Before we start, let’s make sure we avoid a couple of misconceptions:

• First, we don’t have two wave functions (one for each particle);
• Second, the wave function is not defined on physical space (the x axis).

Instead we have a single wave function Ψ(x1,x2), defined on the space of possibilities, which has an x1-axis, (which I will draw horizontal), giving the position of particle 1 (the blue one), and an x2 axis (which I will draw vertical) giving the position of particle 2 (the orange one). The square of the wave function’s absolute value at a specific possibility (x1,x2) tells us the probability of simultaneously finding particle 1 at position x1 and particle 2 at position x2.

In Fig. 6, I have shown the absolute-value-squared of the initial wave function, corresponding to Fig. 4.

Figure 6: Graph of the squared absolute value of the initial wave function, |Ψ(x1,x2)|2, corresponding to Fig. 4. The function is shown dark where it is large and white where it is very small. The two peaks are located at (x1,x2)=(-1,-3) and at (x1,x2)=(+1,+3).

In the first possibility in Fig. 4, we have x1=-1 and x2=-3. One peak of the wave function is located at that position, at lower left in Fig. 6. The other peak of Fig. 6, corresponding to the second possibility in Fig. 4, is located at the position x1=+1 and x2=+3, exactly opposite the first peak.

Fig. 7 now shows the exact solution to the Schrodinger equation, which shows how the wave function of Fig. 6 evolves with time.

Figure 7: How the wave function starting from Fig. 6 evolves over time; there is no interference.

What do we see? The two peaks move generally toward each other, but they miss. They never overlap, so they cannot interfere. This is what makes this case different from Fig. 1; the wave function’s peaks in Fig. 1 do meet, and that is why they interfere.

Why, conceptually speaking, do the two peaks miss? We can understand this using the pre-quantum method, drawing the system not in physical space, as in Fig. 4, but in the space of possibilities. The top possibility in Fig. 4 first puts the system at the star in Fig. 8a, moving up and to the right over time. Because the two particles have equal speeds, every change in x1 is matched with an equal change in x2, which means the star moves on a line whose slope is 1 (i.e. it makes a 45 degree angle to the horizontal.) Similarly, the bottom possibility puts the system at the star in Fig. 8b, moving down and to the left.

Figure 8a: In the space of possibilities, the pre-quantum system in the top possibility of Fig. 4 is initially located at the star, and changes with time by moving along the arrow. Figure 8b: Same as Fig. 8a, but for the bottom possibility in Fig. 4.

If the two stars ever did find themselves at the same point, then what is happening in the first possibility would be exactly the same as what is happening in the second possibility. In other words, the two possibilities would cross paths. But this does not happen here; the paths of the stars do not intersect, reflecting the fact that the top possibility and bottom possibility in Fig. 4 never look the same at any time.

Quantum physics combines these two pre-quantum possibilities into the single wave function of Fig. 7. The two peaks follow the arrows of Figs. 8a and 8b, and so they never overlap.

The three (non-)events shown in Figs. 5a-5c above correspond to the following:

• At the time of Fig. 5a, the two peaks in Fig. 7 are on the same vertical line (they have the same x1)
• At the time of Fig. 5b, the two peaks are aligned along the diagonal from lower right to upper left.
• At the time of Fig. 5c, the two peaks are on the same horizontal line (they have the same x2).

The Flipped Order

Let’s now compare this with the next example I gave you in my previous post. It is much like Fig. 4, except that in the second possibility we switch the two particles.

Figure 9: As in Fig. 4, except with the two particles switching places in the bottom part of the superposition.

This case does have interference. How can we see this?

The top possibility is unaltered, and so Fig. 10a is the same as Fig. 8a. But in Fig. 10b, things have changed; the star that was at x1=+1 and x2=+3 in Fig. 8b is now moved to the point x1=+3 and x2=+1. The corresponding arrow, however, still points in the same direction, since the particles’ motions are the same as before (toward more negative x1 and x2.)

Figure 10a: Same as Fig. 8a, except with the point (x1,x2)=(+1,-1) circled. Figure 10b: A new version of Fig. 8b with particles 1 and 2 having switched places. The system now reaches the circled point (x1,x2)=(+1,-1) at the same moment that it does in Fig.10a.

Now the two arrows do cross paths, and the stars meet at the circled location. At that moment, the pre-quantum system appears in physical space as shown in Fig. 11.

Fig. 11: Quantum interference occurs when, in the pre-quantum analogue, the two possibilities put all their particles in the same place.

In both possibilities, the two particles are in the same locations. And so, in the quantum wave function, the two peaks will cross paths and overlap one another, causing interference. The exact wave function is shown in Fig. 12, and its peaks move just like the stars in Fig. 10a-10b, resulting in a striking interference pattern.

Figure 12: The wave function corresponding to Figs. 9-11, showing interference when the peaks overlap. Profound Lessons

What are the lessons that we can draw from this pair of examples?

First, quantum interference occurs in the space of possibilities, not in physical space. It has effects that can be observed in physical space, but we will not be able to visualize or comprehend the interference effect completely using only physical space, whose coordinate in this case is simply x. If we try, we will lose some of its essence. The full effect is only understandable using the space of possibilities, here two-dimensional and spanned by x1 and x2. (In somewhat the same way, we cannot learn the full three-dimensional layout of a room having only a photograph; some information about the room can be inferred, but some part is inevitably lost.)

Second, starting from a pre-quantum point of view, we see that quantum interference is expected when the pre-quantum paths of two or more possibilities intersect. As an exercise, go back to the last post where I gave you multiple examples. In every case with interference, this intersection happens: there is a moment where the top possibility looks exactly like the bottom possibility, as in Fig. 11.

Third, quantum interference is generally not about a particle interfering with itself — or at least, if we try to use that language, we can’t explain when interference does and doesn’t happen. At best, we might say that the system of two particles is interfering with itself — or fails to interfere with itself — based on its peaks, their motions and their potential intersections in the space of possibilities. When the system consists of only one particle, it’s easy to confuse these two notions, because the system interfering looks the same as the particle interfering. More generally, it is very easy to be misled when the space of possibilities has the same number of dimensions as the relevant physical space. But with two or more particles, this confusion is eliminated. For significant interference to occur, at least two possibilities in a superposition must align perfectly, with each and every particle in matching locations. Whether this is possible or not depends on the superposition’s details.

How Do We Observe the Interference?

But now let’s raise the following question. When there is interference, “where” is it? We can see where it is in the space of possibilities; it’s clear as day in Fig. 12. But you and I live in physical space. If quantum interference is really about interfering waves, just like those of water or sound, then the interference pattern should be located somewhere, shouldn’t it? Where is it?

Well, here’s something to think about. The double-slit-like interference pattern in Fig. 2, for one particle in a superposition, produces a real, observable effect just like that of the double-slit experiment. In Fig. 12 we see a similar case at the moment where wave function’s two peaks overlap. How can we observe this interference effect?

An obvious first guess is to measure the position of one of the particles. The result of doing so for particle 1, and repeating the whole experiment many times (just as we always do for the double-slit experiment) is shown in Fig. 13.

Figure 13: If we measure the position of particle 1 at the moment of maximum interference in Fig. 12, and repeat the experiment many times, we will see random dots centered near x=+1, with no interference pattern. (Each new measurement is an orange dot; previous measurements are blue dots.)

There are no interference peaks and valleys at all, in contrast to the case of Fig. 1, which we examined here (in that post’s Fig. 8). Particle 1 always shows up near x1=+1, which is its location where the two peaks intersect (see Figs. 10-12). No interesting structure within or around that peak is observed.

Not surprisingly, if we do the same thing for particle 2, we find the same sort of thing. No interference features appear; there’s just a blob near its pre-quantum location in Fig. 11, x2=-1.

And yet, the quantum interference is plain to see in Fig. 12. If we can’t observe it by measuring either particle’s position, what other options do we have? Where — if anywhere — will we find it? Is it actually observable, or is it just an abstraction?

Last time, I showed you that a simple quantum system, consisting of a single particle in a superposition of traveling from the left OR from the right, leads to a striking quantum interference effect. It can then produce the same kind of result as the famous double-slit experiment.

The pre-quantum version of this system, in which (like a 19th century scientist) I draw the particle as though it actually has a definite position and motion in each half of the superposition, looks like Fig. 1. The interference occurs when the particle in both halves of the superposition reaches the point at center, x=0.

Figure 1: A case where interference does occur.

Then I posed a puzzle. I put a system of two [distinguishable] particles into a superposition which, in pre-quantum language, looks like Fig. 2.

Figure 2: Two particles in a superposition of both particles moving right (starting from left of center) or both moving left (from right of center.) Their speeds are equal.

with all particles traveling at the same speed and passing each other without incident if they meet. And I pointed out three events that would happen in quick succession, shown in Figs. 2a-2c.

Figure 2.1: Event 1 at x=0. Figure 2.2: Event 2a at x=+1 and event 2b at x=-1. Figure 2.3: Event 3 at x=0.

And I asked the Big Question: in the quantum version of Fig. 2, when will we see quantum interference?

1. Will we see interference during events 1, 2a, 2b, and 3?
2. Will we see interference during events 1 and 3 only?
3. Will we see interference during events 2a and 2b only?
4. Will we see interference from the beginning of event 1 to the end of event 3?
5. Will we see interference during event 1 only?
6. Will we see no interference?
7. Will we see interference at some time other than events 1, 2a, 2b or 3?
8. Something else altogether?

So? Well? What’s the correct answer?

The correct answer is … 6. No interference occurs — not in any of the three events in Figs. 2.1-2.3, or at any other time.

• But wait. . . how can that make sense? How can it be that particle 1 interferes with itself in the case of Fig. 1 and does not interfere with itself in the case of Fig. 2?!

How, indeed?

Perhaps thinking of the particle as interfering with itself is . . . problematic.

Perhaps imagining individual particles interfering with themselves might not be sufficient to capture the range of quantum phenomena. Perhaps we will need to focus more on systems of particles, not individual particles — or more generally, to consider physical systems as a whole, and not necessarily in parts.

Intuition From Other Examples

To start to gain some intuition, consider some other examples. Some have interference, some do not. What distinguishes one class from the other?

For example, the case of Fig. 4 looks almost like Fig. 2, except that the two particles in the bottom part of the superposition are switched. Is there interference in this case?

Figure 4: Similar to Fig. 2, but with the twoparticles reversed in the bottom part of the superposition.

Yes.

How about Fig. 5. In this case, the orange particle is stationary in both parts of the superposition. Is there interference?

Figure 5: In this case, the blue particle is moving (horizontal arrow), but the orange one is stationary in both cases (vertical arrow).

Yes, there is.

And Fig. 6? Again the orange particle is stationary in either part of the superposition.

Figure 6: Similar to Fig. 5, in that the orange particle is again stationary.

No interference this time.

What about Fig. 7 and Fig. 8?

Figure 7: Now the particles in each part of the superposition move in opposite directions. Figure 8: As in Fig. 7, but with the two particles switched in the bottom part of the superposition.

Yes, interference in both cases. And Figs. 9 and 10?

Figure 9: The blue particle is stationary in both parts of the superposition. Figure 10: Similar to Fig. 9, except that now the orange particle is stationary in the bottom part of the superposition.

There is interference in the example of Fig. 10, but not that of Fig. 9.

To understand the twists and turns of the double-slit experiment and its many variants, one must be crystal clear about why the above examples do or do not generate interference. We’ll spend several posts exploring them.

What’s Happening (and Where)?

Let’s focus on the cases where interference does occur: Figs. 1, 4, 5, 7, 8, and 10. First, can you identify what they have in common that the cases without interference (Figs. 2, 6 and 9) lack? And second — bringing back the bonus question from last time, which now comes to the fore — in the cases that show interference, exactly when does it happen, and how can we observe it?

Next time we will start the process of going through the examples in Fig. 2 and Figs. 4-10, to see in each case

• How does the wave function actually behave?
• Why is there (or is there not) interference?
• If there is interference,
  • where does it occur?
  • how exactly can it be observed?

From what we learn, we will try to extract some deep lessons.

If you are truly motivated to understand our quantum world, I promise you that this tour of basic quantum phenomena will be well worth your time.

A very curious thing about quantum physics, 1920’s style, is that it can create observable interference patterns that are characteristic of overlapping waves. It’s especially curious because 1920’s quantum physics (“quantum mechanics”) is not a quantum theory of waves. Instead it is a quantum theory of particles — of objects with position and motion (even though one can’t precisely know the position and the motion simultaneously.)

(This is in contrast to quantum field theory of the 1950s, which [in its simplest forms] really is a quantum theory of waves. This distinction is one I’ve touched on, and we’ll go into more depth soon — but not today.)

In 1920s quantum physics, the only wave in sight is the wave function, which is useful in one method for describing the quantum physics of these particles. But the wave function exists outside of physical space, and instead exists in the abstract space of possibilities. So how do we get interference effects that are observable in physical space from waves in a weird, abstract space?

However it works, the apparent similarity between interference in 1920s quantum physics and the interference observed in water waves is misleading. Conceptually speaking, they are quite different. And appreciating this point is essential for comprehending quantum physics, including the famous double slit experiment (which I reviewed here.)

But I don’t want to address the double-slit experiment yet, because it is far more complicated than necessary. The complications obscure what it is really going on. We can make things much easier with a simpler experimental design, one that allows us to visualize all the details, and to explore why and how and where interference occurs and what its impacts are in the real world.

Once we’ve understood this simpler experiment fully, we’ll be able to discard all sorts of misleading and wrong statements about the double-slit experiment, and return to it with much clearer heads. A puzzle will still remain, but its true nature will be far more transparent without the distracting cloud of misguided clutter.

The Incoming Superposition Experiment

We’ve already discussed what can happen to a particle in a superposition of moving to the left or to the right, using a wave function like that in Fig. 1. The particle is outgoing from the center, with equal probability of going in one direction or the other. At each location, the square of the wave function’s absolute value (shown in black) tells us the probability of finding the particle at that location… so we are most likely to find it under one of the two peaks.

Figure 1: The wave function of a single particle in a superposition of moving outward from the center to the left or right. The wave function’s real and imaginary parts are shown in red and blue; its absolute-value squared in shown in black.

But now let’s turn this around; let’s look at a superposition in which the particle is incoming, with a wave function shown in Fig. 2. This is just the time-reversal of the wave function in Fig. 1. (We could create this superposition in a number of ways. I have described one of them previously — but let’s not worry today about how we got here, and keep our attention on what will happen when the two peaks in the wave function meet.)

Figure 2: The wave function of a single particle in a superposition of moving left or right toward the center. This is just Fig. 1 with time running in the opposite direction.

Important Caution! Despite what you may intuitively guess, the two peaks will not collide and interrupt each others’ motion. Objects that meet in physical space might collide, with significant impact on their motion — or they might pass by each other unscathed. But the peaks in Fig. 2 aren’t objects; the figure is a graph of a probability wave — a wave function — describing a single object. There’s no other object for our single object to collide with, and so it will move steadily and unencumbered at all times.

This is also clear when we use my standard technique of first viewing the system from a pre-quantum point of view, in which case the superposition translates into the two possibilities shown in Fig. 3: either the particle is moving to the right OR it is moving to the left. In neither possibility is there a second object to collide with, so no collision can take place.

Figure 3: In the pre-quantum version of the superposition in Fig. 2, the particle is initially to the left of center and moving to the right OR it is to the right of center and moving to the left.

The wave function for the particle, Ψ(x1), is a function of the particle’s possible position x1. It changes over time, and to find out how it behaves, we need to solve the famous Schrödinger equation. When we do so, we find Ψ(x1) evolves as depicted in Figs. 4a-4c, in which I’ve shown a close-up of the two peaks in Fig. 2 as they cross paths, using three different visualizations. These are the same three approaches to visualization shown in this post, each of which has its pros and cons; take your pick. [Note that there are no approximations in Fig. 4; it shows an exact solution to the Schrödinger equation.]

Figure 4a: A close-up look at the wave function of Fig. 2 as its two peaks approach, cross, and retreat. In black is the absolute-value-squared of the wave function; in red and blue are the wave function’s real and imaginary parts. Figure 4b: Same as Fig. 4a, with the curve showing the absolute value of the wave function, and with color representing the wave function’s argument [or “phase”]. Figure 4c: Same as Fig. 4a. The wave function’s absolute-value-squared is indicated in gray scale, with larger values corresponding to darker shading.

The wave function’s most remarkable features are seen at the “moment of crossing,” which is when our pre-quantum system has the particle reaching x=0 in both parts of the superposition (Fig. 5.)

Figure 5: The pre-quantum system at the moment of crossing, when the particle is at x=0 in both parts of the superposition.

At the exact moment of crossing, the wave function takes the form shown in Figs. 6a-c.

Figure 6a: Graph of the wave function Ψ(x1) at the crossing moment; in black is the absolute-value-squared of the wave function; in red and blue are the wave function’s real and imaginary parts. Figure 6b: Graph of the absolute value |Ψ(x1)| of the wave function at the crossing moment; the color represents the wave function’s argument [or “phase”].
Figure 6c: The absolute-value-squared of the wave function at the crossing moment, indicated in gray scale; larger values of |Ψ(x1)|2 are shown darker, with |Ψ(x1)|2=0 shown in white.

The wiggles in the wave function are a sign of interference. Something is interfering with something else. The pattern superficially resembles that of overlapping ripples in a pond, as in Fig. 7.

Figure 7: The overlap of two sets of ripples caused by an insect’s hind legs creates a visible interference pattern. Credit: Robert Cubitt.

If this pattern reminds you of the one seen in the double-slit experiment, that’s for a very good reason. What we have here is a simpler version of exactly the same effect (as briefly discussed here; we’ll return to this soon.)

These wiggles have a consequence. The quantity |Ψ(x1)|2, the absolute-value-squared of the wave function, tells us the probability of finding this one particle at this particular location x1 in the space of possibilities. (|Ψ(x1)|2 is represented as the black curve in Fig. 6a, as the square of the curve in Fig. 6b, and as the gray-scale value shown in Fig. 6c.) If |Ψ(x1)|2 is large at a particular value of x1, there is a substantial probability of measuring the particle to have position x1. Conversely, If |Ψ(x1)|2=0 at a particular value of x1, then we will not find the particle there.

[Note: I have repeated asserted this relationship between the wave function and the probable results of measurements, but we haven’t actually checked that it is true. Stay tuned; we will check it some weeks from now.]

So if we measure the particle’s position x1 at precisely the moment when the wave function looks like Fig. 5, we will never find it at the grid of points where the wave function is zero.

More generally, suppose we repeat this experiment many times in exactly the same way, setting up particle after particle in the initial superposition state of Fig. 2, measuring its position at the moment of crossing, and recording the result of the measurement. Then, since the particles are most probably found where |Ψ(x1)|2 is large and not where it is small, we will find the distribution of measured locations follows the interference pattern in Figs. 6a-6c, but only appearing one particle at a time, as in Fig. 8.

Figure 8: The experiment is repeated with particle after particle, with each particle’s position measured at the crossing moment. Each new measurement is shown as an orange dot; previous measurements are shown as blue dots. As more and more particles are observed, the interference pattern seen in Figs. 6a-6c gradually appears.

This gradual particle-by-particle appearance of an interference pattern is similar to what is seen in the double-slit experiment; it follows the same rules and has the same conceptual origin. But here everything is so simple that we can address basic questions. Most importantly, in this 1920’s quantum physics context, what is interfering with what, and where, and how?

• Is each particle interfering with itself?
  • Is it sometimes acting like a particle and sometimes acting like a wave?
  • Is it simultaneously a wave and a particle?
  • Is it something in between wave and particle?
• Is each particle interfering with other particles that came before it, and/or with others that will come after it?
• Is the wave function doing the interfering, as a result of the two parts of the superposition for particle 1 meeting in physical space?
• Or is it something else that’s going on?

Well, to approach these questions, let’s use our by now familiar trick of considering two particles rather than one. I’ll set up a scenario and pose a question for you to think about, and in a future post I’ll answer it and start addressing this set of questions.

Checking How Quantum Interference Works

Let’s put a system of two [distinguishable] particles into a superposition state that is roughly a doubling of the one we had before. The superposition again includes two parts. Rather than draw the wave function, I’ll draw the pre-quantum version (see Fig. 3 and compare to Fig. 2.) The pre-quantum version of the quantum system of interest looks like Fig. 9.

Figure 9: Two particles in a superposition of both particles moving right (starting from left of center) or both moving left (from right of center.) Their speeds are equal.

Roughly speaking, this is just a doubling of Fig. 3. In one part of the superposition, particles 1 and 2 are traveling to the right, while in the other they travel to the left. To keep things as simple as possible, let’s say

• all particles in all situations travel at the same speed; and
• if particles meet, they just pass through each other (much as photons or neutrinos would), so we don’t have to worry about collisions or any other interactions.

In this scenario, several interesting events happen in quick succession as the top particles move right and the bottom particles move left.

Event 1 (whose pre-quantum version is shown in Fig. 10a): at x=0, particle 1 arrives from the left in the top option and from the right in the bottom option.

Figure 10a: The pre-quantum system when event 1 occurs.

Events 2a and 2b: (whose pre-quantum versions is shown in Fig. 10b):

• at x=+1, particle 1 arrives from the left in the top option while particle 2 arrives from the right in the bottom option
• at x=-1, particle 2 arrives from the left in the top option while particle 1 arrives from the right in the bottom option

Figure 10b: The pre-quantum system when events 2a and 2b occur.

Event 3 (whose pre-quantum version is shown in Fig. 10c): at x=0, particle 2 arrives from the left in the top option and from the right in the bottom option.

Figure 10c: The pre-quantum system when event 3 occurs.

So now, here is The Big Question. In this full quantum version of this set-up, with the full quantum wave function in action, when will we see interference?

1. Will we see interference during events 1, 2a, 2b, and 3?
2. Will we see interference during events 1 and 3 only?
3. Will we see interference during events 2a and 2b only?
4. Will we see interference from the beginning of event 1 to the end of event 3?
5. Will we see interference during event 1 only?
6. Will we see no interference?
7. Will we see interference at some time other than events 1, 2a, 2b or 3?
8. Something else altogether?

And a bonus question: in any events where we see interference, where will the interference occur, and what roughly will it look like? (I.e. will it look like Fig. 6, where we had a simple interference pattern centered around x=0, or will it look somewhat different?)

What’s your vote? Make your educated guesses, silently or in the comments as you prefer. I’ll give you some time to think about it.

A pause from my quantum series to announce a new interview on YouTube, this one on the Blackbird Physics channel, hosted by UMichigan graduate student and experimental particle physicist Ibrahim Chahrour. Unlike my recent interview with Alan Alda, which is for a general audience, this one is geared toward physics undergraduate students and graduate students. A lot of the topics are related to my book, but at a somewhat more advanced level. If you’ve had a first-year university physics class, or have done a lot of reading about the subject, give it a shot! Ibrahim asked great questions, and you may find many of the answers intriguing.

Here’s the list of the topics we covered, with timestamps.

• 00:00 Intro
• 00:40 Why did you write “Waves in an Impossible Sea”?
• 03:50 What is mass?
• 09:03 What is Relativistic Mass? Is it a useful concept?
• 17:50 Why Quantum Field Theory (QFT) is necessary
• 23:50 Electromagnetic Field, Photons, and Quantum Electrodynamics (QED)
• 36:17 Particles are ripples in their Fields
• 38:47 Fields with zero-mass particles vs. ones whose particles have mass?
• 46:49 The Standard Model of Particle Physics
• 52:08 What was the motivation/history behind the Higgs field?
• 1:02:05 How the Higgs field works
• 1:05:33 The Higgs field’s “Vacuum Expectation Value”
• 1:12:02 The hierarchy problem
• 1:24:18 The current goals of the Large Hadron Collider

What is really going on in the quantum double-slit experiment? The question raised in this post’s title seems to lie at the heart of the matter. In this experiment, which I recently reviewed here, particles of some sort are aimed, one at a time, at a wall with two slits, and their arrival is recorded on a screen behind the wall. As a parade of particles proceeds, one by one, past the wall, an interference pattern somehow appears, emerging gradually like a spectre on the screen.

Interference is a familiar effect, commonly seen in water waves and sound waves. If water waves passed through a pair of slits in a wall, interference would be observed and no one would be surprised. But here we have one particle passing through the wall at a time; it’s not at all the same thing. How can we explain the interference effect in this case?

It’s natural to imagine that somehow either

• each particle acts like a wave, goes through both slits, and interferes with itself, or
• the quantum wave function that describes each particle (or all the particles [?]) goes through both slits and interferes with itself.

So… which is it? Did the particle go through both slits, or did the wave function?

In 1920s quantum physics, there is a very simple answer to this question.

The answer is,…

“No.“

No — neither the particle nor the wave function [not its wavy pattern or its peak(s) or any other part of it] goes through the two slits.

• What!? Then how can there be interference?

That question I will answer in a later post, probably next week or the following. But first, let’s confront the title of this post in a simpler context, so that we can see clearly why — in 1920s quantum physics — the answer to its question is “neither one”.

The Double Door Experiment

Key to understanding the double-slit experiment is to simplify it down to its bare essence. Having a two-dimensional problem where particles are going through slits in a wall is more complicated than necessary. Instead, let’s take a one-dimensional problem that we’ve already looked at, where an object is in a superposition state of going to the left OR going to the right. We already saw that this object is not to be viewed as both going to the left AND going to the right. By setting up measurements on both sides, we saw that it can only be measured to be doing one or the other, and never both. Superposition is an OR, not an AND. And a true particle can only have one position at a time.

The Particle and the Doors: A First Look

In this context, let’s ask the question: can a particle simultaneously go through two doors on opposite sides of a room? This is the same question as the two-slit question, because I can turn one into the other using tubes behind the slits, as in Fig. 1.

Figure 1: An object can be in a superposition of passing through one slit OR the other; by attaching tubes to the slits we can obtain a superposition of the object going to the right OR to the left. The doors that lie ahead are marked in orange.

By sending a particle toward two slits, we can arrange for its wave function to be in the superposition state we want (Fig. 2)

Figure 2: The wave function of a single particle in a superposition of traveling to the left towards one door or traveling to the right towards the other door.

and then we can ask whether we can observe it going through both doors.

Well, in a recent post we put two balls in the same locations that we now want to place the doors, and we asked if a particle in this very same superposition state can hit both balls simultaneously. The answer was “no”. The same argument applies here; the particle cannot be observed to pass through both doors simultaneously. It can only go through one or the other.

Why? A particle, which has a position and a momentum (even if unknown to us), cannot have two positions. If it starts between the two doors, it can move through one door or the other, but it cannot do both, because then it would have two positions at once.

Maybe you’re not immediately convinced. If not, stay tuned, as I’ll come back to this again later.

The Wave Function and the Doors: A First Look

But for now, let’s turn to the other question arising from my post’s title. Why can’t the particle’s wave function move through both doors, just like a water wave or sound wave does?

Actually, since wave functions don’t move (they just describe particles that do) what we really want to know is slightly different. The initial wave function has a wavy pattern; does this wavy pattern go through both doors?

Certainly water waves and sound waves could go through both doors. They are waves in physical space. So are the doors (or slits) they they can pass through.

But the wave function is a wave in the space of possibilities, and not in physical space. Conversely, the doors do not exist in the space of probabilities; doors are physical objects. Therefore the wave function (and its wavy patterns) cannot pass through the doors at all!

The very idea is nonsensical, the sort of thing that René Magritte would have enjoyed painting. Having the wave function (or its pattern) pass through physical doors would be akin to you entering into Shakespeare’s Romeo and Juliet to save the lovers from their fates, or enjoying the taste of an apple painted by Rembrandt, or walking through a giant hole in a physicist’s argument. Physical space and the space of possibilities are conceptually different; they have distinct meanings. At best one space merely represents what is happening in the other. And so the objects that exist in one don’t exist as objects in the other. (Even when these spaces have the same shape, which sometimes they do, they represent different things, as indicated in the fact that they have different axes.)

To convince you further of these statements, let’s take a look at a simpler example. Consider a system where there is just a single door, but there are two particles. Let’s see why the wave function of these particles (and its wave pattern) can’t even pass through one door, much less two.

The Wave Function of Two Particles and a Single Door

We’ll put the door on the right in physical space. Superposition states aren’t needed here, so instead we’ll send both particles rightward toward the door, in simple wave-packet states. These two particles will be given the same near-definite momentum, but their poorly known positions are shifted apart, so that they are separated in physical space. In pre-quantum language, the set-up is shown in Fig. 3.

Figure 3: The pre-quantum picture of our system: two particles travel at the same speed to the right toward an open door (orange).

What wave function do we need to describe this? It’s simplest to put the two particles in wave packet states with near-definite momentum, somewhat separated in space but with similar motion. You might first think the wave function for such a system would roughly look like Fig. 4:

Figure 4: The wrong wave function! Even though it appears as though this wave function shows two particles, one trailing the other, similar to Fig. 3, it instead shows a single particle with definite speed but a superposition of two different locations (i.e. here OR there.)

But no! That’s a trap that’s super-easy to fall into; such a wave function describes one particle in a superposition of two locations, not two particles.

Instead, because the first particle has position x1 and the second has position x2 respectively, their wave function, a function that exists in the two dimensional space of possibilities with axes x1 and x2, takes the form ψ(x1,x2). If we start with x1 near 2 and x2 near 0, as in Fig. 3, then the wave function for these two particles looks like Fig. 5: it has a peak near x1=2 and x2=0. (The dashed black lines are just there to guide your eyes.) The peak indicates that x1 near 2 and x2 near 0 are the most probable values for the two particles’ positions.

Figure 5: The initial wave function of our system, with particle 1 located near x1=2 and particle 2 located near x2=0. The dashed lines are there to guide the eye. The vertical axis is the absolute value of the wave function, whose square at a certain point gives the probability for the corresponding possibility. The colors represent the wave function’s complex argument [or “phase”].

Now, where’s the door that we want to try to make the wave function go through? Great question. In physical space it is located at x=+4. Let’s now draw that door in the space of possibilities, whose axes are x1 and x2. How should we do that?

Think it over…

Unsure? Confused?

That’s fine; there’s no reason not to be confused the first time you think about it. Here’s the answer.

Particle 1 goes through the door when x1=+4, which is the vertical blue dashed line in Fig. 6. Meanwhile, particle 2 goes through it when x2=+4, so that’s the horizontal blue dashed line.

Figure 6: The door (or rather, the locations where the particles meet the door) in the space of possibilities. Particle 1 is located at the door if the two-particle system lies on the vertical blue line; particle 2 is located at the door if the system is on the horizontal blue line.

Does that look like a door to you? Certainly it doesn’t look like the door in physical space. And that’s because in the space of possibilities, the dashed lines are not a door, with mass and thickiness and a material make-up. Instead the lines represent a certain set of possibilities, namely that one of the particles is at the location of the physical door.

In fact, there’s a special point, the intersection at x1=x2=4 where the lines cross, that represents the possibility that both particles are simultaneously at the location of the door. No such intersection of lines exists in physical space. This is an intersection of two classes of possibilities, and such a thing can only exist in the space of possibilities.

The lines divide the space into four regions, representing four more general classes of possibilities, shown in Fig. 7. In the lower left region, both particles are to the left of the door. At far right, particle 1 is to the right of the door while particle 2 is to the left; the reverse is true in the upper left region. Finally, at upper right is the region where both particles are to the right of the door.

Figure 7: How the dashed lines divide the space of possibilities into four general regions, classified by where the two particles are located relative to the door.

The wave function’s pattern moves in the two dimensions that are spanned by the x1 and x2. axes. Both particles are moving to the right in physical space, at approximately the same speed. Consequently, the wave function, as it evolves, carries the most probable state of the system across three of the regions:

• initially both particles are to the left of the door
• then particle 1 is to the right of the door while particle 2 remains to the left
• and finally both particles are to the right of the door.

In pre-quantum physics the path traversed by the two-particle system would look like Fig. 8:

Figure 8: How the pre-quantum two-particle system of Fig. 3 crosses the space of possibilities. The system’s initial configuration is represented as the star. Over time, the first and then the second particle pass through the door. Compare to Fig. 7.

The wave function evolves as shown in Fig. 9, very similarly to Fig. 8.

Fig. 9: The absolute value of the wave function, with its argument (or phase) indicated by the colors. The blue dashed lines are those of Figs. 6-8, and the path of the peak is similar to that shown in Fig. 8.

Now, is the wave function going through the door? Again, there is no door here; there are simply lines that tell us when one particle is coincident with the door, as well as a point where both particles are coincident with the door. It’s true that the wavy pattern and the peak of the wave function (which indicates the possibilities where the system is most likely to be found are passing across the lines. But can we say the wave function (or its wavy pattern) passes through the lines?

No: moving through a door involves moving in physical space, along the x-axis, through a gap in a door frame. This is something the wave function does not — cannot — do.

The Wave Function of Two Particles and Two Doors

If you’re still not yet entirely convinced, consider what happens if we have two doors, one on each side. Let’s put our two particles each in a superposition state of moving leftward or moving rightward. Again we’ll set one particle off before the other, so that the first will reach the doors well before the second does.

Our pre-quantum view of such as system is that it now has four possibilities: particle 1 can be going right or left, and particle 2 can be going right or left, as shown in Fig. 10. Only the upper-right option appeared in Fig. 3.

Figure 10: Now with two doors and both particles in left/right superpositions, the four pre-quantum possibilities generalizing Fig. 3.

This requires a wave function that initially looks like Fig. 11, with four peaks, one for each general possibility sketched in Fig. 10.

Figure 11: The initial wave function has four peaks, corresponding to the four possibilities in Fig. 10; compare to Fig. 5.

But where are the two doors? They appear in the space of possibilities on four lines; in addition to the blue lines we had in Figs. 6-9, corresponding to particles 1 or 2 being at the righthand door, we now have two more, shown in green in Fig. 12, corresponding to one or the other particle being at the lefthand door.

Figure 12: The generalization of Fig. 6 to the case of two doors. The locations where the particles are at the left door are marked as green dashed lines; the blue dashed lines indicate where the particles are at the right door.

Notice the two intersections between the blue and green lines! What are these?! No such intersections between the doors can possibly occur in physical space. So this makes it even clearer that these lines cannot be identified with the doors.

What do these two intersections actually represent? The upper left intersection combines the possibility that particle 1 is at the left door and simultaneously particle 2 is at the right door. It’s the other way around at the lower right intersection.

Note there is no intersection, or any point at all, corresponding to particle 1 being at the left door and simultaneously being at the right door. Indeed, such a point would have x1=-4 AND x1=+4, which is impossible; every point in the space of possibilities has a unique value of x1.

How does the wave function behave over time? It behaves as shown in Fig. 13:

Figure 13: How the wave function of Fig. 10 evolves over time; each peak takes a path similar to that of Fig. 9.

What are the four peaks, and what are they doing? They correspond to the four possibilities shown in Fig. 10. Clockwise from the rightmost peak (which appeared in Fig. 9,)

1. particle 1 goes through the right door, followed by particle 2.
2. particle 1 goes through the right door, after which particle 2 goes through the left door.
3. particle 1 goes through the left door, followed by particle 2.
4. particle 1 goes through the left door, after which particle 2 goes through the right door.

You can tell which is which by looking at which colored line is crossed first, and which is crossed second, by each peak.

Any one of these four things may happen. Two, or more, may not. And not one of them includes the possibility that either particle goes through both doors simultaneously.

Variation on the Theme

As another instructive variation, suppose we send particles 1 and 2 off at exactly the same moment. Then the wave function, shown in Fig. 13, looks very similar to Fig. 12, except that every peak goes through an intersection of the dashed lines, because the two particles arrive at the doors simultaneously.

Figure 13: As in Fig. 12, but with the two balls no longer shifted relative to one another, so that they arrive simultaneously at one door or the other.

Again there are four possibilities,

1. both particles can arrive at the right door simultaneously,
2. both can arrive at the left door simultaneously,
3. particle 1 can arrive at the left door just as the particle 2 arrives at the right door
4. or vice versa.

And so it certainly is possible for particle 1 to go through the left door and particle 2 to go through the right door simultaneously. No problem with that.

However, it is simply impossible — illogical, in fact — for particle 1 to go through both doors simultaneously. There’s no spot in the space of possibilities that could even represent that inconsistent scenario.

Lessons for the Double Slit Experiment

The lesson? Many fascinating things happen in quantum physics once we have superpositions and multiple particles and multiple doors. But two things that cannot happen (in 1920’s quantum physics) include the following:

• No particle can go through two doors at once; it can have only one position at a time.
• No wave function, living as it does in the space of possibilities, can pass through any door in physical space.

The same applies for the two slits in the quantum double-slit experiment. One cannot make sense of that experiment without learning this lesson: from the viewpoint of 1920’s quantum physics, the interference effects do not arise from any object, whether particle or wave function or pattern in the wave function, physically moving through the physical slits in physical space.

So where do the interference effects come from? The wave function’s pattern can travel across regions of possibility space that are associated with the slits. We need to work out the consequences of this observation, and interpret it properly. Stay tuned to this channel; the answer is near at hand.

So far, in the context of 1920s quantum physics, I’ve given you a sense for what an ultra-microscopic measurement consists of, and how one can make a permanent record of it. [Modern (post-1950s) quantum field theory has a somewhat different picture; please keep that in mind. We’ll get to it later.] Along the way I’ve kept the object being measured very simple: just an incoming projectile with a fairly definite motion and moderately definite position, moving steadily in one direction. But now it’s time to consider objects in more interesting quantum situations, and what it means to measure them.

The question for today is: what is a quantum superposition?

I will show you that a quantum superposition of two possibilities, in which the wave function of a system contains one possibility AND another at the same time, does not mean that both possibilities occur; it means that one OR the other may occur.

Instead of a projectile that has a near definite motion, as we’ve considered in recent posts, let’s consider a projectile that is in a quantum superposition of two possible near-definite motions:

• maybe it is moving to the left at a near-definite speed, or
• maybe it is moving to the right at a similar near-definite speed.

This motion is along the x-axis, the coordinate of a one-dimensional physical space. If the projectile is isolated from the rest of the world, we can write a wave function for it alone, which might initially look like

Fig. 1: The wave function of the projectile at the initial time, with two peaks about to head in opposite directions; see Fig. 2.

in which case its evolution over time will look like this:

Fig. 2: The evolution of the isolated projectile’s wave function.

Again I emphasize this is not the wave function of two particles, despite what you might intuitively guess. This is the wave function of a single particle in a superposition of two possible behaviors. For a similar example that we’ll return to in a few weeks, see this post.

Because the height and speed of the two peaks is the same, there is a left-to-right symmetry between them. We can therefore conclude, before we even start, that there’s a 50-50 chance of the particle going right versus going left. More generally, whatever we observe to the left (x<0) will happen with the same probability as what we observe to the right (x>0).

Today I will show you that even though the wave function has one peak moving to the left AND one peak moving to the right, nevertheless this wave function does not describe a projectile that is moving to the left AND moving to the right. Instead, it means that the projectile is moving to the left OR moving to the right. Superposition is an OR, not an AND. In other words, in pre-quantum language, we have either

Fig. 4: The pre-quantum view of the wave function in Figs. 1 and 2; either possibility may occur.

We never have both.

But don’t take my word for it. Let’s see how quantum physics actually works.

First Measurement: A Ball to the Left

Our first goal: to detect the projectile if it is moving to the left.

Let’s start by doing almost the same thing we did in this post, which you may want to read first in order to understand the pictures and the strategy that I’ll present below. To do this, we’ll put a measurement ball on the left, which the projectile will strike if it is moving to the left.

Since we now have a system of two objects rather than one, the space of possibilities for the system now has to be two-dimensional, to include both the position x1 of the projectile and the position x2 of the ball. This now requires us to consider a wave function for not just the projectile alone, as we did in Figs. 1 and 2, but for the projectile and the ball together. This wave function will give us probabilities for each possible arrangement of the projectile and ball — for each choice of x1 and x2.

We’ll put the ball at x2 = -1 initially — to the left of the projectile initially — so that the initial wave function looks like Fig. 4, which shows its absolute value squared as a function of x1 and x2.

Figure 4: The absolute square of the wave function for the projectile (with position x1 near zero) in a superposition of states as in Fig. 1, and the ball which stands ready at position x2=-1 (to the projectile’s left in physical space.)

This wave function has the same shape in x1 as the wave function in Fig. 1, but now centered on the line x2=-1. A collision between projectile and ball will become likely when a peak of the wave function approaches the point x1=x2=-1.

As usual, let’s try to think about this in a pre-quantum language first. If I’m right about wave functions, we have two options:

• The projectile is heading to the left and the measurement ball will react OR
• The projectile is heading to the right and the measurement ball will not react.

Since our wave function is left-to-right symmetric, each option is equally likely, and so if we do this experiment repeatedly, we should see the ball react about half the time.

Here are the two pre-quantum options shown in the usual way, with

• a picture in physical space on the left, and
• a picture of the same event in the space of possibilities on the right.

In the first possibility (Fig. 5a), the projectile moves left, strikes the ball, and the ball recoils to the left. As the ball moves to the left in physical space, the system moves down (toward more negative x2) in the space of possibilities.

Figure 5a: As viewed from physical space (left) and the space of possibilities (right), the projectile moves left and strikes the ball, after which the ball moves left. The ball thus measures the leftward motion of the projectile. The dashed orange line indicates where a collision can occur.

OR

Figure 5a: As viewed from physical space (left) and the space of possibilities (right), the projectile moves right, leaving the ball unscathed. The ball thus measures the rightward motion of the projectile. The dashed orange line indicates where a collision can occur.

In the second possibility (Fig. 5b), the projectile moves right and the ball remains unscathed; in this case, viewed in the space of possibilities, x2 remains at -1 during the entire process while x1 changes steadily toward more positive values.

What about in quantum physics? The wave function should include both options in Figs. 5a and 5b.

Here is an actual solution to the Schrödinger wave equation, showing that this is exactly what happens (and it has more details than the sketches I’ve been doing in my measurement posts, such as this one or this one.) The two peaks spread out more quickly than in my sketches (and I have consequently adjusted the vertical axis as time goes on so that the two bumps remain easily visible.) But the basic prediction is correct: there are indeed two peaks, one moving like the pre-quantum system in Fig 5a, changing direction and moving toward more negative x2, and the other moving like the pre-quantum system in Fig. 5b, moving steadily toward more positive x1.

Figure 6: Actual solution to Schrödinger’s wave equation, showing the absolute square of the wave function beginning with Fig. 4. Notice how the right-moving peak travels steadily toward more positive x1, as in Fig. 5b, while the left-moving peak shows signs of the collision and the subsequent motion of the system toward more negative x2, as in Fig. 5a.

Importantly, even though the system’s wave function displays both possibilities to us at the same time, there is no sense in which the system itself can be in both possibilities at the same time. The system has a near-50% probability of being observed to be within the first peak, near-50% probability of being observed to be within the second, and exactly 0% probability of being observed within both.

Second Measurement: A Ball to the Right

Now let’s put a ball to the right instead, at x=+1. This is a different ball from the previous (we’ll use both of them in a moment) so I’ll color it differently and call its position x3. The pre-quantum behaviors are the same as before, but with x2 replaced with x3 and with the collision happening at positive values of x1 and x3 instead of negative values of x1 and x2.

Figure 7a: As in Figure 5a, but with the orientation reversed.

OR

Figure 7b: As in Figure 5b, but with the orientation reversed.

The quantum version is just a 180-degree rotation of Fig. 6 with x2 replaced with x3.

Figure 8: The evolution of the absolute-value squared of the wave function in this case; compare to Fig. 6 and to Figs. 7a and 7b. Third Measurement: A Ball on Both Sides

But what happens if we put a ball on the left and a ball on the right? Initially the balls are at x2=-1 and x3=+1. What happens later?

Now there are four logical possibilities for what might happen:

1. The ball on the left responds while the ball on the right does not
2. The ball on the right responds while the ball on the left does not
3. Neither ball responds
4. Both balls respond

Where in the space of possibilities do these four options lie? The four logical possibilities listed above would put the ball’s positions in these four possible places:

• Option 1: x2 < -1 and x3 = +1 (and x1 negative, as in Fig. 5a)
• Option 2: x2 = -1 and x3 > +1 (and x1 positive, as in Fig. 7a)
• Option 3: x2 = -1 and x3 = +1 (and x1 is ???)
• Option 4: x2 < -1 and x3 > +1 (and x1 is ???)

The fact that it is not obvious where to put x1 in the last two options should already make you suscpicious; but just setting their x1 to zero for now, let’s draw where these four options occur in the space of possibilities. In Fig. 9 I’ve drawn the lines x2=-1 and x3=+1 across the box, with option 3 at their crossing point. Option 1 lies below down and to the left of option 3; option 2 is found to the rigt of option 3; and option 4 is found down and to the right.

Figure 9: Where the four options are located, roughly speaking. The lines cross at the location x2=-1, x3=+1. If I’m right, only the two cases where one ball moves will have any substantial probability.

What does the wave function actually do? Can the simple two-humped superposition at the start, analogous to Fig. 4, end up four-humped?

Not in this case, anyway. Fig. 10, which depicts the peaks of the absoulte-value-squared of the wave function only, shows the output of the Schrödinger equation. Compare the result to Fig. 9; there are peaks only for options 1 and 2, in which one ball moves and the other does not.

Figure 10: A plot showing where the absolute-value squared of the wave function is largest as the wave function evolves. The axes are as in Fig. 9. Initially the two peaks move in opposite directions parallel to the x1 axis; then, after the projectile collides with one ball or the other, one peak moves down (to more negative x2) and the other to the right (more positive x3). These correspond to the expected options when one and only one ball moves; see Fig. 9.

With balls on either side of it, the projectile cannot avoid hitting one of them, whether it goes right or left, which rules out option 3. And the wave function does not put a peak at option 4, showing there’s no way the projectile can cause both balls to move. The two peaks in the wave function move only in the x1 direction as the projectile goes left OR right; then the projectile collides with one ball OR the other; then the ball with which it collided moves, meaning that the system moves to more negative x2 (i.e. down in Fig. 10) OR to more positive x3 (i.e. to the right in Fig. 10), just as expected from Fig. 9.

Actually it’s not difficult to get the third option — but we don’t need quantum physics for that!

We simply change the original wave function to contain three possibilities: the projectile moves left, or it moves right, or it doesn’t move at all. If it doesn’t move at all, then neither ball will react, a third option even in pre-quantum physics:

If the projectile were isolated, we would encode this notion in a wave function which looks like this:

and when we include the two balls we would see the wave function with three peaks, one sitting still at the point marked “Neither Ball Moves” in Fig. 9. But this isn’t particularly exciting or surprising, since it’s intuitively obvious that a stationary projectile won’t hit either ball.

Every Which Way

There simply is no wave function you can choose — no initial superposition for the single projectile — which can cause the projectile to collide with both balls. The equations will never let this happen, no matter what initial wave function you feed into them. It’s impossible… because a superposition is an OR, not an AND. There is no way to make the projectile go left AND right — not if it’s a particle in 1920s quantum physics, anyway.

Yes, the wave function itself can have peaks that appear at to be in several places at the same time within the space of possibilities, as in Figs. 6, 8, and 10. But the wave function is not the physical system. The wave function tells us about the probabilities for the system’s possibilities; its peaks are just indicating what the most likely possibilities are.

The system itself can only realize one of the many possibilities — it can only be found (through a later measurement) in one place within the space of possibilities. This is always true, even though the wave function for the system highlights all the most probable possibilities simultaneously.

A particle, in the strict sense of the term, is an object with a position and a momentum, even though we cannot know both perfectly at any moment, thanks to Heisenberg’s uncertainty principle. It can only be measured to be in one place, or can only be measured to be traveling in one direction, at a time. In 1920s quantum physics, these statements apply to an electron, which is viewed as a strict particle, and so it cannot go in two directions at once, nor can it be in two places at once. The fact that we are always somewhat ignorant of where an electron is and/or where it is going, and the fact that quantum physics puts ultimate limitations on our ability to know both simultaneously, do not change these basic conceptual lessons… the lessons of (and for) the 1920s.

As part of my post last week about measurement and measurement devices, I provided a very simple example of a measuring device. It consists of a ball sitting in a dip on a hill (Fig. 1a), or, as a microscopic version of the same, a microsopic ball, made out of only a small number of atoms, in a magnetic trap (Fig. 1b). Either object, if struck hard by an incoming projectile, can escape and never return, and so the absence of the ball from the dip (or trap) serves to confirm that a projectile has come by. The measurement is crude — it only tells us whether there was a projectile or not — but it is reasonably definitive.

Fig. 1a: A ball in a dimple on the side of the hill will be easily and permanently removed from its perch if struck by a passing object. Fig. 1b: Similarly to Fig. 1a, a microscopic ball in a trap made from electric and/or magnetic field may easily escape the trap if struck.

In fact, we could learn more about the projectile with a bit more work. If we measured the ball’s position and speed (approximately, to the degree allowed by the quantum uncertainty principle), we would get an estimate of the energy carried by the projectile and the time when the collision occurred. But how definitive would these measurements be?

With a macroscopic ball, we’d be pretty safe in drawing conclusions. However, if the objects being measured and the measurement device are ultra-microscopic — something approaching atomic size or even smaller — then the measurement evidence is fragile. Our efforts to learn something from the microscopic ball will be in vain if the ball suffers additional collisions before we get to study it. Indeed, if a tiny ball interacts with any other object, microscopic or macroscopic, there is a risk that the detailed information about its collision with the projectile will be lost, long before we are able to obtain it.

Amplify Quickly

The best way to keep this from happening is to quickly translate the information from the collision, as captured in the microscopic ball’s behavior, into some kind of macroscopic effect. Once the information is stored macroscopically, it is far harder to erase.

For instance, while a large meteor striking the Earth might leave a pond-sized crater, a subatomic particle striking a metal table might leave a hole only an atom wide. It doesn’t take much to fill in an atom-sized hole in the blink of an eye, but a crater that you could swim in isn’t going to disappear overnight. So if we want to know about the subatomic particle’s arrival, it would be good if we could quickly cause the hole to grow much larger.

This is why almost all microscopic measurements include a step of amplification — the conversion of a microscopic effect into a macroscopic one. Finding new, clever and precise ways of doing this is part of the creativity and artistry of experimental physicists who study atoms, atomic nuclei, or elementary particles.

There are various methods of amplification, but most methods can be thought of, in a sort of cartoon view, as a chain of ever more stable measurements, such as this:

• a first measurement using a microscopic device, such as our tiny ball in a trap;
• a second measurement that measures the device itself, using a more stable device;
• a third measurement that measures the second device, using an even more stable one;
• and so on in a chain until the last device is so stable that its information cannot easily or quickly be erased.

Amplification in Experiments The Geiger-Müller Counter

A classic and simple device that uses amplification is a Geiger counter (or Geiger-Müller counter). (Hans Geiger, while a postdoctoral researcher for Ernest Rutherford, performed a key set of experiments that Rutherford eventually interpreted as evidence that atoms have tiny nuclei.) This counter, like our microscopic ball in Fig. 1b, simply records the arrival of high-energy subatomic projectiles. It does so by turning the passage of a single ultra-microscopic object into a measurable electric current. (Often it is designed to make a concurrent audible electronic “click” for ease of use.)

How does this device turn a single particle, with a lot of energy relative to a typical atomic energy level but very little relative to human activity, into something powerful enough to create a substantial, measurable electric current? The trick is to use the electric field to create a chain reaction.

The Electric Field

The electric field is present throughout the universe (like all cosmic fields). But usually, between the molecules of air or out in deep space, it is zero or quite small. However, when it is strong, as when you have just taken off a wool hat in winter, or just before a lightning strike, it can make your hair stand on end.

More generally, a strong electric field exerts a powerful pull on electrically charged objects, such as electrons or atomic nuclei. Positively charged objects will accelerate in one direction, while negatively charged objects will accelerate in the other. That means that a strong electric field will

• separate positively charged objects from negatively charged objects
• cause both types of objects to speed up, albeit in opposite direction,

Meanwhile electrically neutral objects are largely left alone.

The Strategy

So here’s the strategy behind the Geiger-Müller counter. Start with a gas of atoms, sitting inside of a closed tube in a region with a strong electric field. Atoms are electrically neutral, so they aren’t much affected by the electric field.

But the atoms will serve as our initial measurement devices. If a high-energy subatomic particle comes flying through the gas, it will strike some of the gas atoms and “ionize” them — that is, it will strip an electron off the atom. In doing so it breaks the electrically neutral atom into a negatively charged electron and a positively charged leftover, called an “ion.”

If it weren’t for the strong electric field, the story would remain microscopic; the relatively few ions and electrons would quickly find their way back together, and all evidence of the atomic-scale measurements would be lost. But instead, the powerful electric field causes the ions to move in one direction and the electrons to move in the opposite direction, so that they cannot simply rejoin each other. Not only that, the field causes these subatomic objects to speed up as they separate.

This is especially significant for the electrons, which pick up so much speed that they are able to ionize even more atoms — our secondary measurement devices. Now the number of electrons freed from their atoms has become much larger.

The effect is an chain reaction, with more and more electrons stripped off their atoms, accelerated by the electric field to high speed, allowing them in their turn to ionize yet more atoms. The resulting cascade, or “avalanche,” is called a Townsend discharge; it was discovered in the late 1890s. In a tiny fraction of a second, the small number of electrons liberated by the passage of a single subatomic particle has been multiplied exceedingly, and a crowd of electrons now moves through the gas.

The chain reaction continues until this electron mob arrives at a wire in the center of the counter — the final measurement device in the long chain from microscopic to macroscopic. The inflow of a huge number of the electrons onto the wire, combined with the flow of the ions onto the wall of the device, causes an electrical current to flow. Thanks to the amplification, this current is large enough to be easily detected, and in response a separate signal is sent to the device’s sound speaker, causing it to make a “click!”

Broader Lessons

It’s worth noting that the strategy behind the Geiger-Müller counter requires an input of energy from outside the device, supplied by a battery or the electrical grid. When you think about it, this is not surprising. After the initial step there are rather few moving electrons, and their total motion-energy is still rather low; but by the end of the avalanche, the motion-energy of the tremendous number of moving electrons is far greater. Since energy is conserved, that energy has to have come from somewhere.

Said another way, to keep the electric field strong amid all these charged particles, which would tend to cancel the field out, requires the maintenance of high voltage between the outer wall and inner wire of the counter. Doing so requires a powerful source of energy.

Without this added energy and the resulting amplification, the current from the few initially ionized atoms would be extremely small, and the information about the passing high-energy particle could easily be lost due to ordinary microscopic processes. But the chain reaction’s amplification of the number of electrons and their total amount of energy dramatically increases the current and reduces the risk of losing the information.

Many devices, such as the photomultiplier tube for the detection of photons [particles of light], are like the Geiger-Müller counter in using an external source of energy to boost a microscopic effect. Other devices (like the cloud chamber) use natural forms of amplification that can occur in unstable systems. (The basic principle is similar to what happens with unstable snow on a steep slope: as any off-piste skier will warn you, under the correct circumstances a minor disturbance can cause a mountain-wide snow avalanche.) If these issues interest you, I suggest you read more about the various detectors and subdetectors at ongoing particle experiments, such as those at the Large Hadron Collider.

Amplification in a Simplified Setting

I’ve described the Geiger-Müller counter without any explicit reference to quantum physics. Is there any hope that we could understand how this process really takes place using quantum language, complete with a wave function?

Not in practice: the chain reaction is far, far too complicated. A quantum system’s wave function does not exist in the physical space we live in; it exists in the space of possibilities. Amplification involving hordes of electrons and ions forces us to consider a gigantic space of possibilities; for instance, a million particles moving in our familiar three spatial dimensions would correspond to a space of possibilities that has three million dimensions. Neither you nor I nor the world’s most expert mathematical physicist can visualize that.

Nevertheless, we can gain intuition about the basic idea behind this device by simplifying the chain reaction into a minimal form, one that involves just three objects moving in one dimension, and three stages:

• an initial measurement involving something microscopic
• addition of energy to the microscopic measurement device
• transfer of the information by a second measurement to something less microscopic and more stable.

You can think of these as the first steps of a chain reaction.

So let’s explore this simplified idea. As I often do, I’ll start with a pre-quantum viewpoint, and use that to understand what is happening in a corresponding quantum wave function.

The Pre-Quantum View

The pre-quantum viewpoint differs from that in my last post (which you should read if you haven’t already) in that we have two steps in the measurement rather than just one:

• a projectile is measured by a microscopic ball (the “microball”),
• the microball is similarly measured by a larger device, which I’ll refer to as the “macroball”.

The projectile, microball and macroball will be colored purple, blue and orange, and their positions along the x-axis of physical space will be referred to as x1, x2 and x3. Our space of possibilities then is a three-dimensional space consisting of all possible values of x1, x2 and x3.

The two-step measurement process really involves four stages:

• The projectile approaches the stationary balls from the left.
• The projectile collides with the microball and (in a small change from the last post, for convenience) bounces off to the left, leaving the microball moving to the right.
• The microball is then subject to a force that greatly accelerates it, so that it soon carries a great deal of motion-energy.
• The highly energetic microball now bounces off the macroball, sending the latter into motion.

The view of this process in physical space is shown on the left side of Fig 2. Notice the acceleration of the microball between the two collisions.

Figure 2: (Left) In physical space, the projectile travels to the right and strikes the stationary microball, causing the latter to move; the microball is then accelerated to high speed and strikes the macroball, which recoils in response. The information from the initial collision has been transferred to the more stable macroball. (Right) The same process seen in the space of possibilities; note the labels on the axes. The system is marked by a red dot, with a gray trail showing its history. Note the two collisions and the acceleration between them. At the end, the system’s x3 is increasing, reflecting the macroball’s motion

On the right side of Fig. 2, the motion of the three-object system within the space of possibilities is shown by the moving red dot. To make it easier to see how the red dot moves acrossthe space of possibilities, I’ve plotted its trail across that space as a gray line. Notice there are two collisions, the first one when the projectile and microball collide (x1=x2) and the second where the two balls collide (x2=x3), resulting in two sudden changes in the motion of the dot. Notice also the rapid acceleration between the first collision and the second, as the microball gains sufficient energy to give the macroball appreciable speed.

The Quantum View

In quantum physics, the idea is the same, where the dot representing the system’s value of (x1, x2, x3) is replaced by the peak of a spread-out wave function. It’s difficult to plot a wave function in three dimensions, but I can at least mark out the region where its absolute value is large — where the probability to find the system is highest. I’ve sketched this in Fig. 3. Not surprisingly if follows the same path as the system in Fig. 2.

Figure 3: Sketch of the wave function for this system (compare to Fig. 2a), showing only the location of the highest peak of the wave function (the region where we are most likely to find the system.)

In the pre-quantum case of Fig. 2, the red dot asserts certainty; if we were to measure x1, x2 and/or x3, we would find exactly the values of these quantities corresponding to the location of the dot. In quantum physics of Fig. 3, the peak of the wave function asserts high probability but not certainty. The wave function is spread out; we don’t know exactly what we would find if we directly measured x1, x2 and x3 at any particular moment.

Still, the path of the wave function’s peak is very similar to the path of the red dot, as was also true in the previous post. Generally, in the examples we’ve looked at so far, we haven’t shown much difference between the pre-quantum viewpoint and the quantum viewpoint. You might even be wondering if they’re more similar than people say. But there can be big differences, as we will see very soon.

The Wider View

If I could draw something with more than three dimensions, we could add another stage to our microball and macroball; we could accelerate the macroball and cause it to collide with something even larger, perhaps visible to the naked eye. Or instead of one macroball, we could amplify and transfer the microball’s energy to ten microballs, which in turn could have their energy amplified and transferred to a hundred microballs… and then we would have something akin to a Townsend discharge avalanche and a Geiger-Müller counter. Both in pre-quantum and in quantum physics, this would be impossible to draw; the space of possibilities is far too large. Nevertheless, the simple example in Figs. 2 and 3 provides some intuition for how a longer chain of amplification would work. It shows the basic steps needed to turn a fragile microscopic measurement into a robust macroscopic one, suitable for human scientific research or for our sense perceptions in daily living.

In the articles that will follow, I will generally assume (unless specified otherwise) that each microscopic measurement that I describe is followed by this kind of amplification and conversion to something macroscopic. I won’t be able to draw it, but as we can see in this example, the fundamental underlying idea isn’t that hard to understand.

Nature could be said to be constructed out an immense number of physical processes… indeed, that’s almost the definition of “physics”. But what makes a physical process a measurement? And once we understand that, what makes a measurement in quantum physics, a fraught topic, different from measurements that we typically perform as teenagers in a grade school science class?

We could have a long debate about this. But for now I prefer to just give examples that illustrate some key features of measurements, and to focus attention on perhaps the simplest intuitive measurement device… one that we’ll explore further and put to use in many interesting examples of quantum physics.

Measurements and Devices

We typically think of measurements as something that humans do. But not all measurements are human artifice. A small fraction of physical processes are natural measurements, occuring without human intervention. What distinguishes a measurement from some other garden variety process?

A central element of a measurement is a device, natural or artificial, simple or complicated, that records some aspect of a physical process semi-permanently, so that this record can be read out after the process is over, at least for a little while.

For example, the Earth itself can serve as a measurement device. Meteor Crater in Arizona, USA is the record of a crude measurement of the size, energy and speed of a large rock, as well of how long ago it impacted Earth’s surface. No human set out to make the measurement, but the crater’s details are just as revealing as any human experiment. It’s true that to appreciate and understand this measurement fully requires work by humans: theoretical calculations and additional measurements. But still, it’s the Earth that recorded the event and stored the data, as any measurement device should.

Figure 1: A rock’s energy, measured by the Earth. Meteor Crater, Arizona, USA; National Map Seamless Server – NASA Earth Observatory

The Earth has served as a measurement device in many other ways: its fossils have recorded its life forms, its sedimentary rocks have recorded the presence of its ancient seas, and a layer of iridium and shocked quartz have provided a record of the giant meteor that killed off the dinosaurs (excepting birds) along with many other species. The data from those measurements sat for many millions of years, unknown until human scientists began reading it out.

I’m being superficial here, skipping over all sorts of subtle issues. For instance, when does a measurement start, and when is it over? For instance, did the measurement of the rock that formed Meteor Crater start when the Earth and the future meteor were first created in the early days of the solar system, or only when the rock was within striking distance of our planet? Was it over when Meteor Crater had solidified, or was it complete when the first human measured its size and shape, or was it finished when humans first inferred the size of the rock that made the crater? I don’t want to dwell on these definitional questions today. The point I’m making here is that measurement has nothing intrinsically to do with human beings per se. It has to do with the ability to record a process in such a way that facts about that process can be extracted, long after the process is over.

The measurement device for any particular process has to satisfy some basic requirements.

• Pre-measurement metastability: The device must be fairly stable before the process occurs, so that it doesn’t react or change prematurely, but not so stable that it can’t change when the process takes place.
• Sensitivity: During the interaction between the device and whatever is being measured, the device needs to react or change in some substantial way that is predictable (at least in part).
• Post-measurement stability: The change to the device during the measurement has to be semi-permanent, long-lasting enough that there’s time to detect and interpret it.
• Interpretability: The change to the device has to be substantial and unambiguous enough that it can be used to extract information about what was measured.

Examples of Devices

A simple example: consider a small paper cup as a device for measuring the possible passage of a rubber ball. If the paper cup is sitting on a flat, horizontal table, it is reasonably stable and won’t go anywhere, barring a strong gust of wind. But if a rubber ball goes flying by and hits the cup, the cup will be knocked off the table… and thus the cup is very sensitive to the collision with the ball. The change is also stable and semi-permanent; once the cup is on the floor, it won’t spontaneously jump back up onto the table. And so, after setting a cup on a table in a windowless room near a squash court and returning days later, we can figure out from the position of the cup whether a rubber ball (or something similar) has passed close to the cup while we were away. Of course, this is a very crude measurement, but it captures the main idea.

Incidentally, such a measurement is sometimes referred to as “non-destructive”: the cup is so flimsy that its the effect of the cup on the ball is very limited, and so the ball continues onward almost unaffected. This is in contrast to the measurement of the rock that made Meteor Crater, which most certainly was “destructive” to the rock.

Yet even in this destructive event, all the criteria for a measurement are met. The Earth and its dry surface in Arizona are (and were) pretty stable over millennia, despite erosion. The Earth’s surface is very sensitive to a projectile fifty meters across and moving at ten or more kilometers per second; and the resulting deep, slowly-eroding crater represents a substantial, semi-permanent change that we can interpret roughly 50,000 years later.

In Figure 2 is a very simple and crude device designed to measure disturbances ranging from earthquakes to collisions. It consists of a ball sitting stationary within a dimple (a low spot) on a hill. It will remain there as long as it isn’t jostled — it is reasonably stable. But it is sensitive: if an earthquake occurs, or if something comes flying through the air and strikes the ball, it will pop out of the dimple. Then it will roll down the hill, never to return to the its original perch — thus leaving a long-lasting record of the process that disturbed it. We can later read the ball’s absence from the dimple, or its presence far off to the right, as evidence of some kind of violent disturbance, whereas if it remains in the dimple we may conclude that no such violent disturbance has occurred.

Figure 2: If the ball in the dip is subjected to a disturbance, it will end up rolling off to the right, thus recording the existence of the event that disturbed it.

What about measurement devices in quantum physics? The needs are often the same; a measurement still requires a stable yet sensitive device that can respond to an interaction in a substantial, semi-permanent, interpretable way.

Today we’ll keep things very simple, and limit ourselves to a quantum version of Fig. 2, employed in the simplest of circumstances. But soon we’ll see that when measurements involve quantum physics, surprising and unfamiliar issues quickly arise.

An Simple Device for Quantum Measurement

Here’s an example of a suitable device, a sort of microscopic version of Fig. 2. Imagine a small ball of material, perhaps a few atoms wide, that is gently trapped in place by forces that are strong but not too strong. (These might be of the form of an ion trap or an atom trap; or we might even be speaking of a single atom incorporated into a much larger molecule. The details do not matter here.) This being quantum physics, the trap might not hold the ball in place forever, thanks to the process known as “tunneling“; but it can be arranged to stay in place long enough for our purposes.

Figure 3: A nearly-atomic-sized object in an idealized trap; if jostled sharply, it may move past the dark ring and permanently escape.

If the ball is bumped by an atom or subatomic particle flying by at high speed, it may be knocked out of its trap, following which it will keep moving. So if we look in the trap and discover it empty, or if we find the ball far outside the trap, we will know that some energetic object must have passed through the trap. The ball’s location and motion record the existence of that passing object. (They also potentially record additional information, depending on how we set up the experiment, about the object’s motion-energy and its time of arrival.)

To appreciate a measurement involving quantum physics, it’s often best to first think through what happens in a pre-quantum version of the same scenario. Doing so gives us an opportunity to use two complementary views of the measurement: an intuitive one in physical space and more abstract one in the space of possibilities. This will help us interpret the quantum case, where an understanding of a measurement can only be achieved in the space of possibilities.

A Measurement in Pre-Quantum Physics

We’re going to imagine that an incoming projectile (which I’ll draw in purple) is moving along a straight line (which we’ll call the x-axis) and strikes the measuring device — the ball (which I’ll draw in blue) sitting inside its trap. To keep things simple enough to draw, I’ll assume that any collision that occurs will leave the ball and projectile still moving along the x-axis.

With these two objects restricted to a one-dimensional line, our space of possibilities will be two-dimensional, one dimension representing the possible positions x1 of the projectile, and the other representing the possible positions x2 of the ball. (If you are not at all familiar with the space of possibilities and how to think about it, I recommend you first read this article, which addresses the key ideas, and this article, which gives an example very much relevant to this post.)

Below in Fig. 4 is an animation showing what happens, from two viewpoints, as the projectile strikes the ball, allowing the ball’s motion to measure the passage of the projectile.

The first (at left) is the familiar viewpoint: what would happen before our eyes, in physical space, if these objects were big enough to see. The projectile moves to the right, with the ball stationary; a collision occurs, following which the projectile continues on the right, albeit a bit more slowly, and the ball, having popped out of its trap, moves off the the right.

The second viewpoint (at right) is not something we could see; it happens in the space of possibilities (or “configuration space,”) which we can see only in our minds. In this two-dimensional space, with axes that are the projectile’s and ball’s possible positions x1 and x2, the system — the combination of the projectile and ball — is at any moment sitting at one point. That point is indicated by a star; its location has as its x1 coordinate the projectile’s position at a moment in time, while its x2 coordinate is the ball’s position at that same moment in time.

Figure 4: (Left) In physical space, the projectile travels to the right and strikes the stationary ball, causing the latter to move. (Right) The same process seen in the space of possibilities; note the labels on the axes. On the diagonal line, the two objects would be coincident in physical space, with x1 = x2.

The two animations are synchronized in time. I suggest you spend some time with the animation until it is clear to you what is happening.

• Initially, the star moves horizontally. This indicates that the value of x2 isn’t changing; the ball is stationary. Both x1 and x2 are initially negative, so the star is in the lower-left quadrant.
• Notice the diagonal line, at x1 = x2 ; if the system is on that line, a collision between the two objects is occurring, since they are at the same point. It is when the star reaches this line that the ball begins to move, and the star’s motion is correspondingly no longer horizontal.
• After the collision, both the projectile and ball move to the right, which means the values of x1 and x2 are both increasing. This in turn means that the star moves up and to the right following the collision, eventually reaching the upper-right quadrant where both x1 and x2 are positive.

By contrast, if the measurement device were switched off, so that the projectile and the ball could no longer interact, the projectile would just continue its constant motion to the right, unchanged, and the ball would remain at its initial location, as in Fig. 5. In the space of possibilities, the star would move to the right as the projectile’s position x1 steadily increases, while it would remain at the same vertical level because the ball’s position x2 is never changing.

Figure 5: Same as Fig. 4 except that no collision occurs; the ball remains stationary and the projectile continues on steadily. The Same Measurement in Quantum Physics

Now, how do we describe the measurement in quantum physics? In general we cannot portray what happens in a quantum system using only physical space. Instead, our system of two objects is described by a single wave function, which is a function of the space of possibilities. That is, it is a function of x1 and x2, and also time, since it changes from moment to moment. [Important: the system is not described by two wave functions (i.e., one per object), and the single wave function of the system is not a function of physical space, with its coordinate x. There is one wave function, and it is a function of all possibilities.]

At each moment in time, and for each possible arrangement of the system — for each of the possible locations of the two objects, with the projectile having position x1 and the ball having position x2 — this function gives us a complex number Ψ(x1, x2; t). The absolute value squared of this number gives us the probability of the corresponding possibility — the probability that if we choose to measure the positions of the projectile and ball, we will find the projectile has position x1 and that the ball has position x2.

What I’m going to do now is plot for you this wave function, using a 3d plot, where two of the axes are x1 and x2 and the third axis is the absolute value of Ψ(x1, x2; t). [Not its square, though the difference doesn’t matter much here.] The colors give the argument (or “phase”) of the complex number Ψ(x1, x2; t). As suggested by recent plots where we looked at wave functions for a single particle, the flow of the color bands often conveys the motion of the system across the space of possibilities; you’ll see this in the patterns below.

Going in the reverse order from above, let’s first look at the quantum wave function corresponding to Fig. 5, when no measurement takes place and the projectile passes by the ball unimpeded. You can see that the peak in the wave function, telling us most probable values for the results of measurements of x1 and x2, if carried out at a specific time t, moves along roughly the same path as the star in Fig. 5: the most probable values of x1 increase steadily with time, while those of x2 remain fixed.

Figure 6: The wave function corresponding to a quantum version of Fig. 5, with no measurement carried out; the system is most likely to be to be found where the wave function is largest. The projectile’s most likely position x1 steadily increases while the most likely position x2 of the ball remains constant. Compare to the right-hand panel of Fig. 5.

In this situation, the ball’s behavior has nothing to do with the projectile. We cannot learn anything one way or the other about the projectile from the position or motion of the ball.

What about when a measurement takes place, as in Fig. 4? Again, as seen in Fig. 7, the majority of the wave function follows the path of the star, with the most probable values of x2 beginning to increase around the most likely time of the collision. This change in the most likely value of x2 is an indication of the presence of the projectile and its interaction with the ball. [Note: Fig. 7, unlike other quantum wave functions shown in this series, is a sketch, not a precise solution to the full quantum equations; I simply haven’t yet found a set-up where the equations can be solved numerically with enough precision and speed to get a nice-looking result. I expect I’ll eventually find an example, but it might take some time.]

Figure 7: As in Fig. 6, but including the measurement illustrated in Fig. 4. [Note this is only a sketch, not a full calculation.] The most likely position x2 of the ball is initially constant but begins to increase following the collision, thus recording the observation of the projectile. Compare to the right-hand panel of Fig. 4.

More precisely, because of the collision, the motion of the ball is now correlated with that of the projectile — their motions are logically and physically related. That by itself is not unusual; all interactions between objects lead to some level of correlation between them. But this correlation is stable; as a result of the collision, the ball is highly unlikely to be found back in its initial position. And so, when we later look at the trap and find it empty, this does indeed give us reliable information about the projectile, namely that at some point it passed through the trap. (This type of correlation, both within and beyond the measurement context, will be a major topic in the future.)

So far, this all looks quite straightforward. The motion of the star in Fig. 4 is seen in the motion of the peak of the wave function in Fig. 7. Similar behavior is seen in Figs. 5 and 6. But these are simple cases: where the projectile’s motion is well-known, its location is not too uncertain, and the measurement device is almost perfect. We will soon explore far more complex and interesting quantum examples, using this simple one as our conceptual foundation, and things won’t be so straightforward anymore.

I’ll stop here for today. Please let me know in the comments if there are aspects of this story that you find confusing; we need all to be on the same page before we advance into the more subtle elements of our quantum world.

In my last post, I looked at how 1920’s quantum physics (“Quantum Mechanics”, or QM) conceives of a particle with definite momentum and completely uncertain position. I also began the process of exploring how Quantum Field Theory (QFT) views the same object. I’m going to assume you’ve read that post, though I’ll quickly review some of its main points.

In that post, I invented a simple type of particle called a Bohron that moves around in a physical space in the shape of a one-dimensional line, the x-axis.

• I discussed the wave function in QM corresponding to a Bohron of definite momentum P1, and depicted that function Ψ(x1) (where x1 is the Bohron’s position) in last post’s Fig. 3.
• In QFT, on the other hand, the Bohron is a ripple in the Bohron field, which is a function B(x) that gives a real number for each point x in physical space. That function has the form shown in last post’s Fig. 4.

We then looked at the broad implications of these differences between QM and QFT. But one thing is glaringly missing: we haven’t yet discussed the wave function in QFT for a Bohron of definite momentum P1. That’s what we’ll do today.

The QFT Wave Function

Wave functions tell us the probabilities for various possibilities — specifically, for all the possible ways in which a physical system can be arranged. (That set of all possibilities is called “the space of possibilities“.)

This is a tricky enough idea even when we just have a system of a few particles; for example, if we have N particles moving on a line, then the space of possibilities is an N-dimensional space. In QFT, wave functions can be extremely complicated, because the space of possibilities for a field is infinite dimensional, even when physical space is just a one-dimensional line. Specifically, for any particular shape s(x) that we choose, the wave function for the field is Ψ[s(x)] — a complex number for every function s(x). Its absolute-value-squared is proportional to the probability that the field B(x) takes on that particular shape s(x).

Since there are an infinite number of classes of possible shapes, Ψ in QFT is a function of an infinite number of variables. Said another way, the space of possibilities has an infinite number of dimensions. Ugh! That’s both impossible to draw and impossible to visualize. What are we to do?

Simplifying the Question

By restricting our attention dramatically, we can make some progress. Instead of trying to find the wave function for all possible shapes, let’s try to understand a simplified wave function that ignores most possible shapes but gives us the probabilities for shapes that look like those in Fig. 5 (a variant of Fig. 4 of the last post). This is the simple wavy shape that corresponds to the fixed momentum P1:

where A, the amplitude for this simple wave, can be anything we like. Here’s what that shape looks like for A=1:

Figure 5: The shape A cos(P1 x) for A=1.

If we do this, the wave function for this set of possible shapes is just a function of A; it tells us the probability that A=1 vs. A=-2 vs. A=3.2 vs. A=-4.57, etc. In other words, we’re going to write a restricted wave function Ψ[A] that doesn’t give us all the information we could possibly want about the field, but does tell us the probability for the Bohron field B(x) to take on the shape A cos(P1 x).

This restriction to Ψ[A] is surprisingly useful. That’s because, in comparing the state containing one Bohron with momentum P1 to a state with no Bohrons anywhere — the “vacuum state”, as it is called — the only thing that changes in the wave function is the part of the wave function that is proportional to Ψ[A].

In other words, if we tried to keep all the other information in the wave function, involving all the other possible shapes, we’d be wasting time, because all of that stuff is going to be the same whether there’s a Bohron with momentum P1 present or not.

To properly understand and appreciate Ψ[A] in the presence of a Bohron with momentum P1, we should first explore Ψ[A] in the vacuum state. Once we know the probabilities for A in the absence of a Bohron, we’ll be able to recognize what has changed in the presence of a Bohron.

The Zero Bohron (“Vacuum”) State

In the last post, we examined what the QM wave function looks like that describes a single Bohron with definite momentum (see Fig. 3 of that post). But what is the QM wave function for the vacuum state, the state that has no Bohrons in it?

The answer: it’s a meaningless question. QM is a theory of objects that have positions in space (or other simple properties.) If there are no objects in the theory, then there’s… well… no QM, no wave function, and nothing to discuss.

[You might complain that the Bohron field itself should be thought of as an “object” — but aside from the fact that this is questionable (is air pressure an object?), the QM of a field is QFT, so taking this route would just prove my point.]

In QFT, by contrast, the “vacuum state” is perfectly meaningful and has a wave function. The full vacuum state wave function Ψ[s(x)] is too complicated for us to talk about today. But again, if we keep our focus on the special shapes that look like cos[P1 x], we can easily write the vacuum state’s wave function for that shape’s amplitude, Ψ[A].

Understanding the Vacuum State’s Wave Function

You might have thought, naively, that if a field contains no “particles”, then the field would just be zero; that is, it would have 100% probability to take the form B(x)=0, and 0% probability to have any other shape. This would mean that Ψ[A] would be non-zero only for A=0, forming a spike as shown in Fig. 6. Here, employing a visualization method I use often, I’m showing the wave function’s real part in red and its imaginary part in blue; its absolute-value squared, in black, is mostly hidden behind the red curve.

Figure 6: A naive guess for the vacuum state of the Bohron field would have B(x) = 0 and therefore A=0. But this state would have enormously high energy and would rapidly spread to large values of A.

We’ve seen a similar-looking wave function before in the context of QM. A particle with a definite position also has a wave function in the form of a spike. But as we saw, it doesn’t stay that way: thanks to Heisenberg’s uncertainty principle, the spike instantly spreads out with a speed that reflects the state’s very high energy.

The same issue would afflict the vacuum state of a QFT if its wave function looked like Fig. 6. Just as there’s an uncertainty principle in QM that relates position and motion (changes in position), there’s an uncertainty principle in QFT that relates A and changes in A (and more generally relates B(x) and changes in B(x).) A state with a definite value of position immediately spreads out with a huge amount of energy, and the same is true for a state with a definite value of A; the shape of Ψ[A] in Fig. 6 will immediately spread out dramatically.

In short, a state that momentarily has B(x) = 0, and in particular A=0, won’t remain in this form. Not only will it change rapidly, it will do so with enormous energy. That does not sound healthy for a supposed vacuum state — the state with no Bohrons in it — which ought to be stable and have low energy.

The field’s actual vacuum state therefore has a spread of values for A — and in fact it is a Gaussian wave packet centered around A=0. In QM we have encountered Gaussian wave packets that give a spread-out position; here, in QFT, we need a packet for a spread-out amplitude, shown in Fig. 7 using the representation in which we show the real part, imaginary part, and absolute-value squared of the wave function. In Fig. 7a I’ve made the A-axis horizontal; I’ve then replotted exactly the same thing in Fig. 7b with the A axis vertical, which turns out to be useful as we’ll see in just a moment.

Figure 7a: The real part (red), imaginary part (blue, and zero) and absolute-value-squared of Ψ[A] (the wave function for the amplitude of the shape in Fig. 5) for the vacuum state. Figure 7b: Same as Fig. 7a, turned sideways for better intuition.

Another way to represent this same wave function involves plotting points at a grid of values for A, with each point drawn in gray-scale that reflects the square of the wave function |Ψ(A)|2, as in Fig. 8. Note that the most probable value for A is zero, but it’s also quite likely to be somewhat away from zero.

Figure 8: The value of |Ψ(A)|2 for the vacuum state, expressed in gray-scale, for a grid of choices of A. Note the most probable value of A is zero.

But now we’re going to go a step further, because what we’re really interested in is not the wave function for A but the wave function for the Bohron field. We want to know how that field B(x) is behaving in the vacuum state. To gain intuition for the vacuum state wave function in terms of the Bohron field (remembering that we’ve restricted ourselves to the shape cos[P1 x] shown in Fig. 5), we’ll generalize Fig. 8: instead of one dot for each value of A, we’ll plot the whole shape A cos[P1 x] for a grid of choices of A, using gray-scale that’s proportional to |Ψ(A)|2. This is shown in Fig. 9; in a sense, it is a combination of Fig. 8 with Fig. 5.

Figure 9: For a grid of values of A, the shape Acos[P1 x] is drawn in gray-scale that reflects the magnitude of |Ψ(A)|2, and thus the probability for that value of A. This picture gives us intuition for the probabilities for the shape of the field B(x) in the vacuum state. The Bohron field is generally not zero in this state, even though the possible shapes of B(x) are centered around B(x) = 0.

Remember, this is not showing the probability for the position of a particle, or even that of a “particle”. It is showing the probability in the vacuum state for the field B(x) to take on a certain shape, albeit restricted to shapes proportional to cos[P1 x]. We can see that the most likely value of A is zero, but there is a substantial spread around zero that causes the field’s value to be uncertain.

In the vacuum state, what’s true for a shape with momentum P1 would be true also for any and all shapes of the form cos[P x] for any possible momentum P. In principle, we could combine all of those shapes, for all of the different momenta, together in a much more complicated version of Fig. 9. However, that would make the picture completely unreadable, so I won’t try to do that — although I’ll do something intermediate, with multiple values of P, in later posts.

Oh, and I mustn’t forget to flash a warning: everything I’ve just told you and will tell you for the rest of this post is limited to a child’s version of QFT. I’m only describing what the vacuum state looks like for a “free” (i.e. non-interacting) Bohron field. This field doesn’t do anything except send individual “particles” around that never change or interact with each other. If you want to know more about truly interesting QFTs, such as the ones in the real world — well, expect some things to be recognizable from today’s post, but much of this will, yet again, have to be revisited.

The One-Bohron State

Now that we know the nature of the wave function for the vacuum state, at least when restricted to shapes proportional to cos[P1 x], how does this change in the presence of a single Bohron of momentum P1?

The answer is quite simple: the wave function Ψ(A) changes from to (up to an overall constant of no interest to us here.) Depicting this state in analogy to what we did for the vacuum state in Figs. 7b, 8 and 9, we find Figs. 10, 11 and 12.

Figure 10: As in Fig. 7, but for the one-Bohron state. Note the probability for A=0 is now zero, and the probability (black curve) peaks at non-zero positive and negative values of A. Figure 10: As in Fig. 8, but for the one-Bohron state. Figure 10: As in Fig. 9, but for the one-Bohron state. Note the probability for B(x)=0 is zero in the one-Bohron state with momentum P1, in contrast to the vacuum state.

Notice that the one-Bohron state is clearly distinguishable from the vacuum state; most notably the probability for A=0 is now zero, and its spread is larger, with the most likely values for A now non-zero.

There’s one more difference between these states, which I won’t attempt to prove to you at the moment. The vacuum state doesn’t show any motion; that’s not surprising, because there are no Bohrons there to do any moving. But the one-Bohron state, with its Bohron of definite momentum, will display signs of a definite speed and direction. You should imagine all the wiggles in Fig. 12 moving steadily to the right as time goes by, whereas Fig. 9 is static.

Well, that’s it. That’s what the QFT wave function for a one-Bohron state of definite momentum P1 looks like — when we ignore the additional complexity that comes from the shapes for other possible momenta P, on the grounds that their behavior is the same in this state as it is in the vacuum state.

A Summary of Today’s Steps

That’s more than enough for today, so let me emphasize some key points here. Compare and contrast:

• In QM:
  • The Bohron with definite momentum is a particle with a position, though that position is unknown.
  • The wave function for the Bohron, spread out across the space of the Bohron’s possible positions x1, has a wavelength with respect to x1.
• In QFT:
  • The Bohron “particle” (i.e. wavicle) is intrinsically spread out across physical space [the horizontal x-axis in Figs. 9 and 12] and the Bohron itself has a wavelength with respect to x.
  • Meanwhile the wave function, spread out across the space of possible amplitudes A (the vertical axis in Figs. 7a, 8, 10 and 11) does not contain simply packaged information about how the activity in the Bohron field is spread out across physical space x; both the vacuum state and one-Bohron states are spread out, but you can’t just read off that fact from Figs. 8 and 11.
  • And note that the wave function has nothing simple to say about the position of the Bohron; after all the spread-out “particle” doesn’t even have a clearly defined position!

Just to make sure this is clear, let me say this again slightly differently. While in QM, the Bohron particle with definite momentum has an unknown position, in QFT, the Bohron “particle” with definite momentum does not even have a position, because it is intrinsically spread out. The QFT wave function says nothing about our uncertainty about the Bohron’s location; that uncertainty is already captured in the fact that the real (not complex!) function B(x) is proportional to a cosine function. Indeed physical space, and its coordinate x, don’t even appear directly in Ψ(A). Instead the QFT wave function, in the restricted form we’ve considered, only tells us the probability that B(x) = A cos[P1 x] for a particular value of A — and that those probabilities are different when there is a single Bohron present (Fig. 12) compared to when there is none (Fig. 9).

I hope you can now start to see why I don’t find the word particle helpful in describing a QFT Bohron. The Bohron does have some limited particle-like qualities, most notably its indivisibility, and we’ll explore those soon. But you might already understand why I prefer wavicle.

We are far from done with QFT; this is just the beginning of our explorations. There are many follow-up questions to address, such as

• Can we put our QFT Bohron into a wave packet state similar to last post’s Fig. 2? What would that look like?
• Do these differences between QM and QFT have implications for how we think about experiments, such as the double-slit experiment or Bell’s particle-pair experiment?
• What do QFT wave functions look like if there are two “particles” rather than just one? There are several cases, all of them interesting.
• How do measurements work, and how are they different, in QM versus QFT?
• What about fields more complicated than the Bohron field, such as the electron field or the electromagnetic field?

We’ll deal with these one by one over the coming days and weeks; stay tuned.

Why do I find the word particle so problematic that I keep harping on it, to the point that some may reasonably view me as obsessed with the issue? It has to do with the profound difference between the way an electron is viewed in 1920s quantum physics (“Quantum Mechanics”, or QM for short) as opposed to 1950s relativistic Quantum Field Theory (abbreviated as QFT). [The word “relativistic” means “incorporating Einstein’s special theory of relativity of 1905”.] My goal this week is to explain carefully this difference.

The overarching point:

• in QM, an electron really is a particle, at least to the limited extent that QM can handle that notion;
• in QFT, an electron is at best a “particle”, with the word in quotation marks
  • (and I personally prefer the term wavicle.)

I’ve discussed this to some degree already in my article about how the view of an electron has changed over time, but here I’m going to give you a fuller picture. To complete the story will take two or three posts, but today’s post will already convey one of the most important points.

There are two short readings that you may want to dofirst.

• The first is an article about the distinction between physical space (within which all objects actually are located) and the abstract space of possibilities (which consists of all ways that the objects in a physical system could be arranged in physical space.) This issue, essential in quantum physics of any type, will be crucial below.
• The second is a short section of a blog post that explains how QM views an isolated object in a state of definite momentum — i.e. something whose motion (both speed and direction) is precisely known, and whose position is therefore completely unknown, thanks to Heisenberg’s uncertainty principle.

I’ll will review the main point of the second item, and then I’ll start explaining what an isolated object of definite momentum looks like in QFT.

Removing Everything Extraneous

First, though, let’s make things as simple as possible. Though electrons are familiar, they are more complicated than some of their cousins, thanks to their electric charge and “spin”, and the fact that they are fermions. By contrast, bosons with neither charge nor spin are much simpler. In nature, these include Higgs bosons and electrically-neutral pions, but each of these has some unnecessary baggage. For this reason I’ll frame my discussion in terms of imaginary objects even simpler than a Higgs boson. I’ll call these spinless, chargeless objects “Bohrons” in honor of Niels Bohr (and I’ll leave the many puns to my readers.)

For today we’ll just need one, lonely Bohron, not interacting with anything else, and moving along a line. Using 1920s QM in the style of Schrödinger, we’ll take the following viewpoints.

• A Bohron is a particle and exists in physical space, which we’ll take to be just a line — the set of points arranged along what we’ll call the x-axis.
• The Bohron has a property we call position in physical space. We’ll refer to its position as x1.
• For just one Bohron, the space of possibilities is simply all of its possible positions — all possible values of x1. [See Fig. 1]
• The system of one isolated Bohron has a wave function Ψ(x1), a complex number at each point in the space of possibilities. [Note it is not a function of x, the points in physical space; it is a function of x1, the possible positions of the Bohron.]
• The wave function predicts the probability of finding the Bohron at any selected position x1: it is proportional to |Ψ(x1)|2, the square of the absolute value of the complex number Ψ(x1).

Figure 1: For a Bohron moving along a line, physical space is the x-axis where the Bohron (blue dot) is located. The space of possibilities, the set of all possible arrangements of our one-Bohron system (red star) is the the x1-axis. This subtle but important distinction becomes clearer when we have two or more Bohrons; the physical space is unchanged, but possibility space is totally different. A QM State of Definite Momentum

In a previous post, I described states of definite momentum. But I also described states whose momentum is slightly less definite — a broad Gaussian wave packet state, which is a bit more intutive. The wave function for a Bohron in this state is shown in Fig. 2, using three different representations. You can see intuitively that the Bohron’s motion is quite steady, reflecting near definite momentum, while the wave function’s peak is very broad, reflecting great uncertainty in the Bohron’s position.

• Fig. 2a shows the real and imaginary parts of Ψ(x1) in red and blue, along with its absolute-value squared |Ψ(x1)|2 in black.
• Fig. 2b shows the absolute value |Ψ(x1)| in a color that reflects the argument [i.e. the phase] of Ψ(x1).
• Fig. 2c indicates |Ψ(x1)|2, using grayscale, at a grid of x1 values; the Bohron is more likely to be found at or near dark points than at or near lighter ones.

For more details and examples using these representations, see this post.

Figure 2a: The wave function for a wave packet state with near-definite momentum, showing its real (red) and imaginary (blue) parts and its absolute value squared (black.) Figure 2b: The same wave function, with the curve showing its absolute value and colored by its argument. Figure 2c: The same wave function, showing its absolute value squared using gray-scale values on a grid of x1 points. The Bohron is more likely to be found near dark-shaded points.

To get a Bohron of definite momentum P1, we simply take what is plotted in Fig. 2 and make the broad peak wider and wider, so that the uncertainty in the Bohron’s position becomes infinite. Then (as discussed in this post) the wave function for that state, referred to as |P1>, can be drawn as in Fig. 3:

Figure 3a: As in Fig. 2a, but now for a state |P1> of precisely known momentum to the left. Figure 3b: As in Fig. 2b, but now for a state |P1> of precisely known momentum to the left. Figure 3c: As in Fig. 2c, but now for a state |P1> of precisely known momentum; note the probability of finding the Bohron is equal at every point at all times.

In math, the wave function for the state at some fixed moment in time takes a simple form, such as

where i is the square root of -1. This is a special state, because the absolute-value-squared of this function is just 1 for every value of x1, and so the probability of measuring the Bohron to be at any particular x1 is the same everywhere and at all times. This is seen in Fig. 3c, and reflects the fact that in a state with exactly known momentum, the uncertainty on the Bohron’s position is infinite.

Let’s compare the Bohron (the particle itself) in the state |P1> to the wave function that describes it.

• In the state |P1>, the Bohron’s location is completely unknown. Still, its position is a meaningful concept, in the sense that we could measure it. We can’t predict the outcome of that measurement, but the measurement will give us a definite answer, not a vague indefinite one. That’s because the Bohron is a particle; it is not spread out across physical space, even though we don’t know where it is.
• By contrast, the wave function Ψ(x1) is spread out, as is clear in Fig. 3. But caution: it is not spread out across physical space, the points of the x axis. It is spread out across the space of possibilities — across the range of possible positions x1. See Fig. 1 [and read my article on the space of possibilities if this makes no sense to you.]
• Thus neither the Bohron nor its wave function is spread out in physical space!

We do have waves here, and they have a wavelength; that’s the distance between one crest and the next in Fig. 3a, and the distance beween one red band and the next in Fig. 3b. That wavelength is a property of the wave function, not a property of the Bohron. To have a wavelength, an object has to be wave-like, which our QM Bohron is not.

Conversely, the Bohron has a momentum (which is definite in this state, and is something we can measure). This has real effects; if the Bohron hits another particle, some or all of its momentum will be transferred, and the second particle will recoil from the blow. By contrast, the wave function does not have momentum. It cannot hit anything and make it recoil, because, like any wave function, it sits outside the physical system. It merely describes an object with momentum, and tells us the probable outcomes of measurements of that object.

Keep these details of wavelength (the wave function’s purview) and the momentum (the Bohron’s purview) in mind. This is how 1920’s QM organizes things. But in QFT, things are different.

First Step Toward a QFT State of Definite Momentum

Now let’s move to quantum field theory, and start the process of making a Bohron of definite momentum. We’ll take some initial steps today, and finish up in the next post.

Our Bohron is now a “particle”, in quotation marks. Why? Because our Bohron is no longer a dot, with a measurable (even if unknown) position. It is now a ripple in a field, which we’ll call the Bohron field. That said, there’s still something particle-like about the Bohron, because you can only have an integer number (1, 2, 3, 4, 5, …) of Bohrons, and you can never have a fractional number (1/2, 7/10, 2.46, etc.) of Bohrons. This feature is something we’ll discuss in later posts, but we’ll just accept it for now.

As fields go, the Bohron field is a very simple example. At any given moment, the field takes on a value — a real number — at each point in space. Said another way, it is a function of physical space, of the form B(x).

Very, very important: Do not confuse the Bohron field B(x) with a wave function!!

• This field is a function in physical space (not the space of possibilities). B(x) is a function of physical space points x that make up the x-axis, and is not a function of a particle’s position x1, nor is it a function of any other coordinate that might arise in the space of possibilities.
• I’ve chosen the simplest type of QFT field: B(x) is a real number at each location in physical space. This is in contrast to a QM wave function, which is a complex number for each possibility in the space of possibilities.
• The field itself can carry energy and momentum and transport it from place to place. This is unlike a wave function, which can only describe the energy and momentum that may be carried by physical objects.

Now here’s the key distinction. Whereas the Bohron of QM has a position, the Bohron of QFT does not generally have a position. Instead, it has a shape.

If our Bohron is to have a definite momentum P1, the field must ripple in a simple way, taking on a shape proportional to a sine or cosine function from pre-university math. An example would be:

where A is a real number, called the “amplitude” of the wave, and x is a location in physical space.

At some point soon we’ll consider all possible values of A — a part of the space of possibilities for the field B(x) — so remember that A can vary. To remind you, I’ve plotted this shape for A=1 in Fig. 4a and again for A=-3/2 in Fig 4b.

Figure 4a: The function A cos[P1 x], for the momentum P1 set equal to 1 and the amplitude A set equal to 1. Figure 4b: Same as Fig. 4a, but with A = -3/2 . Initial Comparison of QM and QFT

At first, the plots in Fig. 4 of the QFT Bohron’s shape look very similar to the QM wave function of the Bohron particles, especially as drawn in Fig. 3a. The math formulas for the two look similar, too; compare the formula after Fig. 3 to the one above Fig. 4.

However, appearances are deceiving. In fact, when we look carefully, EVERYTHING IS COMPLETELY DIFFERENT.

• Our QM Bohron with definite momentum has a wave function Ψ(x1), while in QFT it has a shape B(x); they are functions of variables which, though related, are different.
  
  
• On top of that, there’s a wave function in QFT too, which we haven’t drawn yet. When we do, we’ll see that the QFT Bohron’s wave function looks nothing like the QM Bohron’s wave function. That’s because
  • the space of possibilities for the QM wave function is the space of possible positions that the Bohron particle can have, but
  • the space of possibilities for the QFT wave function is the space of all possible shapes that the Bohron field can have.
• The plot in Fig. 4 shows a curve that is both positive and negative but is drawn colorless, in contrast to Fig. 3b, where the curve is positive but colored. That’s because
  • the Bohron field B(x) is a real number with no argument [phase], whereas
  • the QM wave function Ψ(x1) for the state of definite momentum has an always-positive absolute value and a rapidly varying argument [phase].
• The axes in Fig. 4 are labeled differently from the axis in Fig. 3. That’s because (see Fig. 1)
  • the QFT Bohron field B(x) is found in physical space, while
  • the QM wave function Ψ(x1) for the Bohron particle is found in the particle’s space of possibilities.
• The absolute-value-squared of a wave function |Ψ(x1)|2 is interpreted as a probability (specifically, the probability for the particular possibility that the particle is at position x1. There is no such interpretation for the square of the Bohron field |B(x)|2. We will later find a probability interpretation for the QFT wave function, but we are not there yet.
  
  
• Both Fig. 4 and Figs. 3a, 3b show curves with a wavelength, albeit along different axes. But they are very different in every sense
  • In QM, the Bohron has no wavelength; only its wave function has a wavelength — and that involves lengths not in physical space but in the space of possibilities.
  • In QFT,
    • the field ripple corresponding to the QFT Bohron with definite momentum has a physical wavelength;
    • meanwhile the QFT Bohron’s wave function does not have anything resembling a wavelength! The field’s space of possibilities, where the wave function lives, doesn’t even have a recognizable notion of lengths in general, much less wavelengths in particular.

I’ll explain that last statement next time, when we look at the nature of the QFT wave function that corresponds to having a single QFT Bohron.

A Profound Change of Perspective

But before we conclude for the day, let’s take a moment to contemplate the remarkable change of perspective that is coming into our view, as we migrate our thinking from QM of the 1920s to modern QFT. In both cases, our Bohron of definite momentum is certainly associated with a definite wavelength; we can see that both in Fig. 3 and in Fig. 4. The formula for the relation is well-known to scientists; the wavelength λ for a Bohron of momentum P1 is simply

where h is Planck’s famous constant, the mascot of quantum physics. Larger momentum means smaller wavelength, and vice versa. On this, QM and QFT agree.

But compare:

• in QM, this wavelength sits in the wave function, and has nothing to do with waves in physical space;
• in QFT, the wavelength is not found in the field’s wave function; instead it is found in the field itself, and specifically in its ripples, which are waves in physical space.

I’ve summarized this in Table 1.

Table 1: The Bohron with definite momentum has an associated wavelength. In QM, this wavelength appears in the wave function. In QFT it does not; both the wavelength and the momentum are found in the field itself. This has caused no end of confusion.

Let me say that another way. In QM, our Bohron is a particle; it has a position, cannot spread out in physical space, and has no wavelength. In QFT, our Bohron is a “particle”, a wavy object that can spread out in physical space, and can indeed have a wavelength. (This is why I’d rather call it a wavicle.)

[Aside for experts: if anyone thinks I’m spouting nonsense, I encourage the skeptic to simply work out the wave function for phonons (or their counterparts with rest mass) in a QM system of coupled balls and springs, and watch as free QFT and its wave function emerge. Every statement made here is backed up with a long but standard calculation, which I’m happy to show you and discuss.]

I think this little table is deeply revealing both about quantum physics and about its history. It goes a long way toward explaining one of the many reasons why the brilliant founding parents of quantum physics were so utterly confused for a couple of decades. [I’m going to go out on a limb here, because I’m certainly not a historian of physics; if I have parts of the history wrong, please set me straight.]

Based on experiments on photons and electrons and on the theoretical insight of Louis de Broglie, it was intuitively clear to the great physicists of the 1920s that electrons and photons, which they were calling particles, do have a wavelength related to their momentum. And yet, in the late 1920s, when they were just inventing the math of QM and didn’t understand QFT yet, the wavelength was always sitting in the wave function. So that made it seem as though maybe the wave function was the particle, or somehow was an aspect of the particle, or that in any case the wave function must carry momentum and be a real physical thing, or… well, clearly it was very confusing. It still confuses many students and science writers today, and perhaps even some professional scientists and philosophers.

In this context, is it surprising that Bohr was led in the late 1920s to suggest that electrons are both particles and waves, depending on experimental context? And is it any wonder that many physicists today, with the benefit of both hindsight and a deep understanding of QFT, don’t share this perspective?

In addition, physicists already knew, from 19th century research, that electromagnetic waves — ripples in the electromagnetic field, which include radio waves and visible light — have both wavelength and momentum. Learning that wave functions for QM have wavelength and describe particles with momentum, as in Fig. 3, some physicists naturally assumed that fields and wave functions are closely related. This led to the suggestion that to build the math of QFT, you must go through the following steps:

• first you take particles and describe them with a wave function, and then
• second, you make this wave function into a field, and describe it using an even bigger wave function.

(This is where the archaic terms “first quantization” and “second quantization” come from.) But this idea was misguided, arising from early conceptual confusions about wave functions. The error becomes more understandable when you imagine what it must have been like to try to make sense of Table 1 for the very first time.

In the next post, we’ll move on to something novel: images depicting the QFT wave function for a single Bohron. I haven’t seen these images anywhere else, so I suspect they’ll be new to most readers.

Yesterday I posted an animation of a quantum wave function, and as a brain teaser, I asked readers to see if they could interpret it. Here it is again:

Yesterday’s wave function, showing an interesting interference phenomenon.

Admittedly, it’s a classic trap — one I use as a teaching tool in every quantum physics class. The wave function definitely looks, intuitively, as though two particles are colliding. But no. . . the wave function describes only one particle.

And what is this particle doing? It’s actually in the midst of a disguised version of the famous double slit experiment! This version is much simpler than the usual one, and will be super-useful to us going forward. It will make it significantly easier to see how all the puzzles of the double-slit experiment play out, both from the old, outdated but better known perspective of 1920’s quantum physics and from the modern perspective of quantum field theory.

You can read the details about this wave function — why it can’t possibly describe two particles, why it shows interference despite there being only one particle, and why it gives us a simpler version of the double-slit experiment — in an addendum to yesterday’s post.

A scientific brain teaser for readers: here’s a wave function evolving over time, presented in the three different representations that I described in a post earlier this week. [Each animation runs for a short time, goes blank, and then repeats.] Can you interpret what is happening here?

The explanation — and the reasons why this example is particularly useful, informative, and interesting (I promise!) — is coming soon [it will be posted here tomorrow morning Boston time, Friday Feb 21st.]

[Note added on Thursday: I give this example in every quantum mechanics class I teach. No matter how many times I have said, with examples, that a wave function exists in the space of possibilities, not in physical space, it happens every time that 90%-95% thinks this shows two particles. It does not. And that’s why I always give this example.]

What is a wave function in quantum physics?

Such a question generates long and loud debates among philosophers of physics (and more limited debate among most physicists, who tend to prefer to make predictions using wave functions rather than wondering what they are.) I have a foot in both camps, even though I have no real credentials among the former set. But no matter; today I won’t try to answer my own question in any profound way. We can debate the deeper meaning of wave functions another time.

Instead I just want to address the question practically: what is this function for, in what sense does it wave, and how does it sit in the wider context of physics?

Schrödinger’s Picture of the World

Quantum physics was born slowly, in stages, beginning around 1900. The most famous stage is that of 1925, when Heisenberg, along with Born and Jordan, developed one approach, using matrices, and Schrödinger developed another, using a “wave function”. Both methods could predict details of atomic physics and other systems, and Schrödinger soon showed the two approaches were equivalent mathematically. Nevertheless, he (and many others) felt his approach was more intuitive. This is why the wave function approach is emphasized — probably over-emphasized — in many books on quantum physics.

Suppose we want to investigate a physical system, such as a set of interacting subatomic objects that together make up a water molecule. Westart by imagining the system as being in some kind of initial physical state, and then we ask how that state changes over time. In standard first-year undergraduate physics, using the methods of the 17th-19th century, we would view the initial physical state as consisting of the locations and motions of all the objects at some initial time. Armed with that information, we could then calculate precisely what the system would do in the future.

But experimental data on atomic physics revealed that this older method simply doesn’t agree with nature. Some other approach was needed.

In 1920s quantum physics in the style of Schrödinger, the state of the system is (under-)specified by an unfamiliar object: a function on the space of possibilities for the system. This function gives us a complex number for each possibility, whose square tells us the probability for that particular possibility. More precisely, if we measure the system carefully, Schrödinger’s function at the time of the measurement tells us the probability of our measurements giving one outcome versus another.

For instance, suppose the system consists of two particles, and let’s call the possible position of the first particle x1 and that of the second x2. Then Schrödinger’s function will take the form Ψ(x1,x2) — a function giving us a complex number for each of the possible locations of the two particles. (As I’ve emphasized repeatedly, even though we have a system of two particles, there is only one wave function; I’ve given you a couple of examples of what such functions are like here and here.)

If we want to know the probability of finding the first particle at some definite position X1 and the second at a definite position X2 — assuming we do the measurements right now — that probability is proportional to the quantity |Ψ(X1,X2)|2, i.e. the square of the function when the first particle is at X1 and the second is at X2.

If we choose not to make a measurement right away, Schrödinger’s equation tells us how the function changes with time; if the function was initially Ψ(x1,x2; t=0) = Ψ(x1,x2), then after a time T it will have a new form Ψ(x1,x2; t=T) which we can calculate from that equation. If we then measure the positions of the particles, the probabilities for various measurement outcomes will be given by the square of the updated function, |Ψ(x1,x2; t=T)|2.

Schrödinger’s function is usually called a “wave function”. But this comes with a caveat: it’s not always actually a wave…see below. So it is more accurate to call it a “state function.”

Wave Functions Are Not Things

Probably thanks to advanced chemistry classes, in which pictures of atoms are often drawn that suggest that each electron has its own wave function, it is a common error to think that every particle has a wave function, and that wave functions are physical objects that travel through ordinary space and carry energy and momentum from one place to another, much like sound waves and ocean waves do. But this is wrong, in a profound, crucial sense.

If the electrons and atomic nuclei that make up atoms are like characters in a 19th century novel, the wave function is like an omniscient narrator. No matter how many characters appear in the plot, there is only one such narrator. That narrator is not a character in the story. Instead the narrator plays the role of storyteller, with insight into all the characters’ minds and motivations, able to give us many perspectives on what is going on — but with absolutely no ability to change the story by, say, personally entering into a scene and interposing itself between two characters to prevent them from fighting. The narrator exists outside and beyond the story, all-knowing yet powerless.

A wave function describes the objects in a system, giving us information about all the locations, speeds, energies and other properties that they might have, as well as about how they influence one another as they move around in our familiar three-dimensional space. The system’s objects, of which there can be as many as we like, can do interesting things, such as clumping together to form more complex objects such as atoms. As they move around, they can do damage to these clumps; for instance, they can ionize atoms and break apart biological DNA molecules. The system’s wave function, by contrast, does not travel in three-dimensional space and has neither momentum nor energy nor location. It cannot form clumps of objects, nor can it damage them. It is not an object in the way that electrons , photons and neutrinos are objects. Nor is it a field like the electric field, the Higgs field, and the electron field, which exist in three dimensions and whose waves do have momentum, energy, speed, etc. Most important, each system has one, and only one, wave function, no matter how many objects are in the system.

[One might argue that a wave function narrator is less omniscient, thanks to quantum physics, than in a typical novel; but then again, that might depend on the author, no? I leave this to you to debate.]

I wrote the article “Why a Wave Function Can’t Hurt You” to emphasize these crucial points. If you’re still finding this confusing, I encourage you to read that article.

Some Facts About Wave Functions

Here are a few interesting facts about wave functions. I’ll state them mostly without explanation here, though I may go into more details sometime in the future.

• It is widely implied in books and articles that wave functions emerged for the first time in quantum physics — that they were completely absent from pre-quantum physics. But this is not true; wave functions first appeared in the 1830s.
  
  In the “Hamilton-Jacobi” reformulation of Newton’s laws, the evolution of a non-quantum system is described by a wave function (“Hamilton’s characteristic function”) that is a function on the space of possibilities and satisfies a wave equation quite similar to Schrödinger’s equation. However, in contrast to Schrödinger’s function, Hamilton’s function is a real number, not a complex number, at each point in the space of possibilities, and it cannot be interpreted in terms of probabilities. In very simple situations, Hamilton’s function is the argument (or phase) of Schrödinger’s function, but more generally the two functions can be very different.
  
  
• Wave functions are essential in Schrödinger’s approach to quantum physics. But in other approaches, including Heisenberg’s and the later method of Feynman, wave functions and wave equations do not directly appear. (The situation in pre-quantum physics is completely analogous; the wave function of Hamilton appears neither in Newton’s formulation of the laws of motion nor in the reformulation known as the “action principle” of Maupertuis.)
  
  This is an indication that one should be cautious ascribing any fundamental reality to this function, although some serious scientists and philosophers still do so.
  
  
• The relevant space of possibilities of which the wave function is a function is only half as big as you might guess. For instance, in our example of two particles above, even though the function specifies the probabilities for the various possible locations and motions of the objects in the system, it is actually only a function of either the possible locations or the possible motions (more specifically, the particles’ momenta.) If we write it as a function of the possible locations, then the probabilities for the objects’ motions are figured out through a nontrivial mathematical procedure, and vice versa.
  
  The fact that the wave function can only give half the information explicitly, no matter how we write it down, is related to why it is impossible to know objects’ positions and motions precisely at the same time.
  
  
• For objects moving around in a continuous physical space like the space of the room that you are sitting in, waves are a natural phenomenon, and Schrödinger’s function and the equation that governs it are typical of waves. But in many interesting systems, objects do not actually move, and there’s nothing wavy about the function, which is best referred to as a “state function”. As an example, suppose our system consists of two atoms trapped in a crystal, so that they cannot move, but each has a “spin” that can point up or down only. Then
  • the space of possibilities is just the four possible arrangements of the spins: up-up, up-down, down-up, down-down;
  • the wave state function doesn’t look like a wave, and is instead merely a discrete set of four complex numbers, one for each of the four arrangements;
  • the square of the each of these four complex numbers gives us the probabilities for finding the two spins in each of the four possible arrangements;
  • and Schrödinger’s equation for how the state function changes with time is not a wave equation but instead a 4 x 4 matrix equation.

The space of possibilities for two trapped atoms, each with spin that can be up or down, consists only of the above four physical states; Schrödinger’s state function provides a single complex number for each one, and is in no sense wave-like.

• So although the term “wave function” suggests that waves are an intrinsic part of quantum physics, they actually are not. For the design and operation of quantum computers, one often just needs state functions made of a finite set of complex numbers, as in the example I’ve just given you.
  

• Another case where a state function isn’t a wave function in the sense you might imagine is in quantum field theory, widely used both in particle physics and in the study of many materials, such as metals and superconductors. In this context, the state function shows wavelike behavior but not for particle positions, in contrast to 1920s quantum physics. More on this soon.
  
  
• For particle physics, we need relativistic quantum field theory, which incorporates Einstein’s special relativity (with its cosmic speed limit and weird behavior of space and time). But in a theory with special relativity, there’s no unique or universal notion of time. Unfortunately, Schrödinger’s approach requires a wave function defined at an initial moment in time, and his equation tells us how the function changes from the initial time to any later time. This is problematic. Because my definition of time will differ from yours if you are moving relative to me, my form of the wave function will differ from yours, too. This makes the wave function a relative quantity (like speed), not an intrinsic one (like electric charge or rest mass). That means that, as for any relative quantitiy, if we ever want to change perspective from one observer to another, we may have to recalculate the wave function — an unpleasant task if it is complicated.
  
  Despite this, the wave function approach could still be used. But it is far more common for physicists to choose other approaches, such as Feynman’s, which are more directly compatible with Einstein’s relativity.

If you’re of a certain age, you know Alan Alda from his wonderful acting in television shows and in movies. But you may not know of his long-standing interest in science communication and his podcast Clear and Vivid (named for the characteristics that he feels all communication should have.)

Alda and I had a great conversation about the idea that we are made of waves, and what it means for our relationship to the universe. A slimmed-down version of that discussion is now available on his podcast. I hope you enjoy it!

Separately, as promised: to my last post, which covered various ways of depicting and interpreting wave functions, I’ve added explanations of the two quantum wave functions that I placed at the end. Tomorrow I’ll take a step back and consider wave functions from a larger point of view, taking a brief look at what they are (and aren’t), what’s “wavy” (and not) about them, and at their roles in contexts ranging from pre-quantum physics of the 19th century to quantum field theory of the 21st.

Before we knew about quantum physics, humans thought that if we had a system of two small objects, we could always know where they were located — the first at some position x1, the second at some position x2. And after Isaac Newton’s breakthroughs in the late 17th century, we believed that by combining this information with knowledge of the objects’ motions and the forces acting upon them, we could calculate where they would be in the future.

But in our quantum world, this turns out not to be the case. Instead, in Erwin Schrödinger’s 1925 view of quantum physics, our system of two objects has a wave function which, for every possible x1 and x2 that the objects could have, gives us a complex number Ψ(x1, x2). The absolute-value-squared of that number, |Ψ(x1, x2)|2, is proportional to the probability for finding the first object at position x1 and the second at position x2 — if we actually choose to measure their positions right away. If instead we wait, the wave function will change over time, following Schrödinger’s wave equation. The updated wave function’s square will again tell us the probabilities, at that later time, for finding the objects at those particular positions.

The set of all possible object locations x1 and x2 is what I am calling the “space of possibilities” (also known as the “configuration space”), and the wave function Ψ(x1, x2) is a function on that space of possibilities. In fact, the wave function for any system is a function on the space of that system’s possibilities: for any possible arrangement X of the system, the wave function will give us a complex number Ψ(X).

Drawing a wave function can be tricky. I’ve done it in different ways in different contexts. Interpreting a drawing of a wave function can also be tricky. But it’s helpful to learn how to do it. So in today’s post, I’ll give you three different approaches to depicting the wave function for one of the simplest physical systems: a single object moving along a line. In coming weeks, I’ll give you more examples that you can try to interpret. Once you can read a wave function correctly, then you know your understanding of quantum physics has a good foundation.

For now, everything I’ll do today is in the language of 1920s quantum physics, Schrödinger style. But soon we’ll put this same strategy to work on quantum field theory, the modern language of particle physics — and then many things will change. Familiarity with the more commonly discussed 1920s methods will help you appreciate the differences.

Complex Numbers

Before we start drawing pictures, let me remind you of a couple of facts from pre-university math about complex numbers. The fundamental imaginary number is the square root of minus one,

which we can multiply by any real number to get another imaginary number, such as 4i or -17i. A complex number is the sum of a real number and an imaginary number, such as 6 + 4i or 11 – 17i.

More abstractly, a complex number w always takes the form u + i v, where u and v are real numbers. We call u the “real part” of w and we call v the “imaginary part” of w. And just as we can draw a real number using the real number line, we can draw a complex number using a plane, consisting of the real number line combined with the imaginary number line; in Fig. 1 the complex number w is shown as a red dot, with the real part u and imaginary part v marked along the real and imaginary axes.

Figure 1: Two ways of representing the complex number w, either as u + i v or as |w|eiφ .

Fig. 1 shows another way of representing w. The line from the origin to w has length |w|, the absolute value of w, with |w|2 = u2 + v2 by the Pythagorean theorem. Defining φ as the angle between this line and the real axis, and using the following facts

• u = |w| cos φ
• v = |w| sin φ
• eiφ = cos φ + i sin φ

we may write w = |w|eiφ , which indeed equals u + i v .

Terminology: φ is called the “argument” or “phase” of w, and in math is written φ = arg(w).

One Object in One Dimension

We’ll focus today only on a single object moving around on a one-dimensional line. Let’s put the object in a “Gaussian wave-packet state” of the sort I discussed in this post’s Figs. 3 and 4 and this one’s Figs. 6 and 7. In such a state, neither the object’s position nor its momentum [a measure of its motion] is completely definite, but the uncertainty is minimized in the following sense: the product of the uncertainty in the position and the uncertainty in the momentum is as small as Heisenberg’s uncertainty principle allows.

We’ll start with a state in which the uncertainty on the position is large while the uncertainty on the momentum is small, shown below (and shown also in Fig. 3 of this post and Fig. 6 of this post.) To depict this wave function, I am showing its real part Re[Ψ(x)] in red and its imaginary part Im[Ψ(x)] in blue. In addition, I have drawn in black the square of the wave function:

• |Ψ(x)|2 = (Re[Ψ(x)])2 + (Im[Ψ(x)])2

[Note for advanced readers: I have not normalized the wave function.]

Figure 1: For an object in a simple Gaussian wave packet state with near-definite momentum, a depiction of the wave function for that state, showing its real and imaginary parts in red and blue, and its absolute-value squared in black.

But as wave functions become more complicated, this way of doing things isn’t so convenient. Instead, it is sometimes useful to represent the wave function in a different way, in which we plot |Ψ(x)| as a curve whose color reflects the value of φ = arg[Ψ(x)] , the argument of Ψ(x). In Fig. 2, I show the same wave function as in Fig. 1, depicted in this new way.

Figure 2: The same wave function as in Fig. 1; the curve is the absolute value of the wave function, colored according to its argument.

As φ cycles from 0 to π/4 to π/2 to 3π/4 and back to 2π (the same as φ = 0), the color cycles from red to yellow-green to cyan to blue-purple and back to red.

Compare Figs. 1 and 2; its the same information, depicted differently. That the wave function is actually waving is clear in Fig. 1, where the real and imaginary parts have the shape of waves. But it is also represented in Fig. 2, where the cycling through the colors tells us the same thing. In both cases, the waving tells us that the object’s momentum is non-zero, and the steadiness of that waving tells us that the object’s momentum is nearly definite.

Finally, if I’m willing to give up the information about the real and imaginary parts of the wave function, and just want to show the probabilities that are proportional to its squared absolute value, I can sometimes depict the state in a third way. I pick a few spots where the object might be located, and draw the object there using grayscale shading, so that it is black where the probability is large and becomes progressively lighter gray where the probability is smaller, as in Fig. 3.

Figure 3: The same wave function in Figs. 1 and 2, here showing only the probabilities for the object’s location; the darker the grey, the more likely the object is to be found at that location.

Again, compare Fig. 3 to Figs. 1 and 2; they all represent information about the same wave function, although there’s no way to read off the object’s momentum using Fig. 3, so we know where it might be but not where it is going. (One could add arrows to indicate motion, but that only works when the uncertainty in the momentum is small.)

Although this third method is quite intuitive when it works, it often can’t be used (at least, not as I’ve described it here.) It’s often useful when we have just one object to worry about, or if we have multiple objects that are independent of one another. But if they are not independent — if they are correlated, as in a “superposition” [more about that concept soon] — then this technique usually does not work, because you can’t draw where object number 1 is likely to be located without already knowing where object number 2 is located, and vice versa. We’ve already seen examples of such correlations in this post, and we’ll see more in future.

So now we have three representations of the same wave function — or really, two representations of the wave function’s real and imaginary parts, and two representations of its square — which we can potentially mix and match. Each has its merits.

How the Wave Function Changes Over Time

This particular wave function, which has almost definite momentum, does indeed evolve by moving at a nearly constant speed (as one would expect for something with near-definite momentum). It spreads out, but very slowly, because its speed is only slightly uncertain. Here is its evolution using all three representations. (The first was also shown in this post’s Fig. 6.)

I hope that gives your intuition some things to hold onto as we head into more complex situations.

Two More Examples

Below are two simple wave functions for a single object. They differ somewhat from the one we’ve been using in the rest of this post. What do they describe, and how will they evolve with time? Can you guess? I’ll give the full answer tomorrow as an addendum to this post.

Two different wave functions; in each case the curve represents the absolute value |Ψ(x)| and the color represents arg[Ψ(x)], as in Fig. 2. What does each wave function say about the object’s location and momentum, and how will each of them change with time?

Pioneer Works is “an artist and scientist-led cultural center in Red Hook, Brooklyn that fosters innovative thinking through the visual and performing arts, technology, music, and science.” It’s a cool place: if you’re in the New York area, check them out! Among many other activities, they host a series called “Picture This,” in which scientists ruminate over scientific images that they particularly like. My own contribution to this series has just come out, in which I expound upon the importance and meaning of this graph from the CMS experimental collaboration at the Large Hadron Collider [LHC]. (The ATLAS experimental collaboration at the LHC has made essentially identical images.)

The point of the article is to emphasize the relation between the spikes seen in this graph and the images of musical frequencies that one might see in a recording studio (as in this image from this paper). The similarity is not an accident.

Each of the two biggest spikes is a sign of an elementary “particle”; the Z boson is the left-most spike, and the Higgs boson is the central spike. What is spiking is the probability of creating such a particle as a function of the energy of some sort of physical process (specifically, a collision of objects that are found inside protons), plotted along the horizontal axis. But energy E is related to the mass m of the “particle” (via E=mc2) and it is simultaneously related to the frequency f of the vibration of the “particle” (via the Planck-Einstein equation E = hf)… and so this really is a plot of frequencies, with spikes reflecting cosmic resonances analogous to the resonances of musical instruments. [If you find this interesting and would like more details, it was a major topic in my book.]

The title of the article refers to the fact that the Z boson and Higgs boson frequencies are out of tune, in the sense that if you slowed down their frequencies and turned them into sound, they’d be dissonant, and not very nice to listen to. The same goes for all the other frequencies of the elementary “particles”; they’re not at all in tune. We don’t know why, because we really have no idea where any of these frequencies come from. The Higgs field has a major role to play in this story, but so do other important aspects of the universe that remain completely mysterious. And so this image, which shows astonishingly good agreement between theoretical predictions (colored regions) and LHC data (black dots), also reveals how much we still don’t understand about the cosmos.

[An immediate continuation of Part 1, which you should definitely read first; today’s post is not stand-alone.]

The Asymmetry Between Location and Motion

We are in the middle of trying to figure out if the electron (or other similar object) could possibly be of infinitesimal size, to match the naive meaning of the words “elementary particle.” In the last post, I described how 1920’s quantum physics would envision an electron (or other object) in a state |P0> of definite momentum or a state |X0> of definite position (shown in Figs. 1 and 2 from last time.)

If it is meaningful to say that “an electron is really is an object whose diameter is zero”, we would naturally expect to be able to put it into a state in which its position is clearly defined and located at some specific point X0 — namely, we should be able to put it into the state |X0>. But do such states actually exist?

Symmetry and Asymmetry

In Part 1 we saw all sorts of symmetry between momentum and position:

• the symmetry between x and p in the Heisenberg uncertainty principle,
• the symmetry between the states |X0> and |P0>,
• the symmetry seen in their wave functions as functions of x and p shown in Figs. 1 and 2 (and see also 1a and 2a, in the side discussion, for more symmetry.)

This symmetry would seem to imply that if we could put any object, including an elementary particle, in the state |P0>, we ought to be able to put it into a state |X0>, too.

But this logic won’t follow, because in fact there’s an even more important asymmetry. The states |X0> and |P0> differ crucially. The difference lies in their energy.

Who cares about energy?

There are a couple of reasons we should care, closely related. First, just as there is a relationship between position and momentum, there is a relationship between time and energy: energy is deeply related to how wave functions evolve over time. Second, energy has its limits, and we’re going to see them violated.

Energy and How Wave Functions Change Over Time

In 1920s quantum physics, the evolution of our particle’s wave function depends on how much energy it has… or, if its energy is not definite, on the various possible energies that it may have.

Definite Momentum and Energy: Simplicity

This change with time is simple for the state |P0>, because this state, with definite momentum, also has definite energy. It therefore evolves in a very simple way: it keeps its shape, but moves with a constant speed.

Figure 5: In the state |P0>, shown in Fig. 1 of Part 1, the particle has definite momentum and energy and moves steadily at constant speed; the particle’s position is completely unknown at all times.

How much energy does it have? Well, in 1920s quantum physics, just as in pre-1900 physics, the motion-energy E of an isolated particle of definite momentum p is

• E = p2/2m

where m is the particle’s mass. Where does this formula come from? In first-year university physics, we learn that a particle’s momentum is mv and that its motion-energy is mv2/2 = (mv)2/2m = p2/2m; so in fact this is a familiar formula from centuries ago.

Less Definite Momentum and Energy: Greater Complexity

What about the compromise states mentioned in Part 1, the ones that lie somewhere between the extreme states |X0> and |P0>, in which the particle has neither definite position nor definite momentum? These “Gaussian wave packets” appeared in Fig. 3 and 4 of Part 1. The state of Fig. 3 has less definite momentum than the |P0> state, but unlike the latter, it has a rough location, albeit broadly spread out. How does it evolve?

As seen in Fig. 6, the wave still moves to the left, like the |P0> state. But this motion is now seen not only in the red and blue waves which represent the wave function itself but also in the probability for where to find the particle’s position, shown in the black curve. Our knowledge of the position is poor, but we can clearly see that the particle’s most likely position moves steadily to the left.

Figure 6: In a state with less definite momentum than |P0>, as shown in Fig. 3 of Part 1, the particle has less definite momentum and energy, but its position is roughly known, and its most likely position moves fairly steadily at near-constant speed. If we watched the wave function for a long time, it would slowly spread out.

What happens if the particle’s position is better known and the momentum is becoming quite uncertain? We saw what a wave function for such a particle looks like in Fig. 4 of Part 1, where the position is becoming quite well known, but nowhere as precisely as in the |X0> state. How does this wave function evolve over time? This is shown in Fig. 7.

Figure 7: In a state with better known position, shown in Fig. 4 of Part 1, the particle’s position is initially well known but becomes less and less certain over time, as its indefinite momentum and energy causes it to move away from its initial position at a variety of possible speeds.

We see the wave function still indicates the particle is moving to the left. But the wave function spreads out rapidly, meaning that our knowledge of its position is quickly decreasing over time. In fact, if you look at the right edge of the wave function, it has barely moved at all, so the particle might be moving slowly. But the left edge has disappeared out of view, indicating that the particle might be moving very rapidly. Thus the particle’s momentum is indeed very uncertain, and we see this in the evolution of the state.

This uncertainty in the momentum means that we have increased uncertainty in the particle’s motion-energy. If it is moving slowly, its motion-energy is low, while if it is moving rapidly, its motion-energy is much higher. If we measure its motion-energy, we might find it anywhere in between. This is why its evolution is so much more complex than that seen in Fig. 5 and even Fig. 6.

Near-Definite Position: Breakdown

What happens as we make the particle’s position better and better known, approaching the state |X0> that we want to put our electron in to see if it can really be thought of as a true particle within the methods of 1920s quantum physics?

Well, look at Fig. 8, which shows the time-evolution of a state almost as narrow as |X0> .

Figure 8: the time-evolution of a state almost as narrow as |X0>.

Now we can’t even say if the particle is going to the left or to the right! It may be moving extremely rapidly, disappearing off the edges of the image, or it may remain where it was initially, hardly moving at all. Our knowledge of its momentum is basically nil, as the uncertainty principle would lead us to expect. But there’s more. Even though our knowledge of the particle’s position is initially excellent, it rapidly degrades, and we quickly know nothing about it.

We are seeing the profound asymmetry between position and momentum:

• a particle of definite momentum can retain that momentum for a long time,
• a particle of definite position immediately becomes one whose position is completely unknown.

Worse, the particle’s speed is completely unknown, which means it can be extremely high! How high can it go? Well, the closer we make the initial wave function to that of the state |X0>, the faster the particle can potentially move away from its initial position — until it potentially does so in excess of the cosmic speed limit c (often referred to as the “speed of light”)!

That’s definitely bad. Once our particle has the possibility of reaching light speed, we need Einstein’s relativity. But the original quantum methods of Heisenberg-Born-Jordan and Schrödinger do not account for the cosmic speed limit. And so we learn: in the 1920s quantum physics taught in undergraduate university physics classes, a state of definite position simply does not exist.

Isn’t it Relatively Easy to Resolve the Problem?

But can’t we just add relativity to 1920s quantum physics, and then this problem will take care of itself?

You might think so. In 1928, Dirac found a way to combine Einstein’s relativity with Schrödinger’s wave equation for electrons. In this case, instead of the motion-energy of a particle being E = p2/2m, Dirac’s equation focuses on the total energy of the particle. Written in terms of the particle’s rest mass m [which is the type of mass that doesn’t change with speed], that total energy satisfies the equation

For stationary particles, which have p=0, this equation reduces to E=mc2, as it must.

This does indeed take care of the cosmic speed limit; our particle no longer breaks it. But there’s no cosmic momentum limit; even though v has a maximum, p does not. In Einstein’s relativity, the relation between momentum and speed isn’t p=mv anymore. Instead it is

which gives the old formula when v is much less than c, but becomes infinite as v approaches c.

Not that there’s anything wrong with that; momentum can be as large as one wants. The problem is that, as you can see for the formula for energy above, when p goes to infinity, so does E. And while that, too, is allowed, it causes a severe crisis, which I’ll get to in a moment.

Actually, we could have guessed from the start that the energy of a particle in a state of definite position |X0> would be arbitrarily large. The smaller is the position uncertainty Δx, the larger is the momentum uncertainty Δp; and once we have no idea what the particle’s momentum is, we may find that it is huge — which in turn means its energy can be huge too.

Notice the asymmetry. A particle with very small Δp must have very large Δx, but having an unknown location does not affect an isolated particle’s energy. But a particle with very small Δx must have very large Δp, which inevitably means very large energy.

The Particle(s) Crisis

So let’s try to put an isolated electron into a state |X0>, knowing that the total energy of the electron has some probability of being exceedingly high. In particular, it may be much, much larger — tens, or thousands, or trillions of times larger — than mc2 [where again m means the “rest mass” or “invariant mass” of the particle — the version of mass that does not change with speed.]

The problem that cannot be avoided first arises once the energy reaches 3mc2 . We’re trying to make a single electron at a definite location. But how can we be sure that 3mc2 worth of energy won’t be used by nature in another way? Why can’t nature use it to make not only an electron but also a second electron and a positron? [Positrons are the anti-particles of electrons.] If stationary, each of the three particles would require mc2 for its existence.

If electrons (not just the electron we’re working with, but electrons in general) didn’t ever interact with anything, and were just incredibly boring, inert objects, then we could keep this from happening. But not only would this be dull, it simply isn’t true in nature. Electrons do interact with electromagnetic fields, and with other things too. As a result, we can’t stop nature from using those interactions and Einstein’s relativity to turn 3mc2 of energy into three slow particles — two electrons and a positron — instead of one fast particle!

For the state |X0> with Δx = 0 and Δp = infinity, there’s no limit to the energy; it could be 3mc2, 11mc2, 13253mc2, 9336572361mc2. As many electron/positron pairs as we like can join our electron. The |X0> state we have ended up with isn’t at all like the one we were aiming for; it’s certainly not going to be a single particle with a definite location.

Our relativistic version of 1920s quantum physics simply cannot handle this proliferation. As I’ve emphasized, an isolated physical system has only one wave function, no matter how many particles it has, and that wave function exists in the space of possibilities. How big is that space of possibilities here?

Normally, if we have N particles moving in d dimensions of physical space, then the space of possibilities has N-times-d dimensions. (In examples that I’ve given in this post and this one, I had two particles moving in one dimension, so the space of possibilities was 2×1=2 dimensional.) But here, N isn’t fixed. Our state |X0> might have one particle, three, seventy one, nine thousand and thirteen, and so on. And if these particles are moving in our familiar three dimensions of physical space, then the space of possibilities is 3 dimensional if there is one particle, 9 dimensional if there are three particles, 213 dimensional if there are seventy-one particles — or said better, since all of these values of N are possible, our wave function has to simultaneously exist in all of these dimensional spaces at the same time, and tell us the probability of being in one of these spaces compared to the others.

Still worse, we have neglected the fact that electrons can emit photons — particles of light. Many of them are easily emitted. So on top of everything else, we need to include arbitrary numbers of photons in our |X0> state as well.

Good Heavens. Everything is completely out of control.

How Small Can An Electron Be (In 1920s Quantum Physics?)

How small are we actually able to make an electron’s wave function before the language of the 1920s completely falls apart? Well, for the wave function describing the electron to make sense,

• its motion-energy must be below mc2, which means that
• p has to be small compared to mc , which means that
• Δp has to be small compared to mc , which means that
• by Heisenberg’s uncertainty principle, Δx has to be large compared to h/(4πmc)

This distance (up to the factor of 1/4π) is known as a particle’s Compton wavelength, and it is about 10-13 meters for an electron. That’s about 1/1000 of the distance across an atom, but 100 times the diameter of a small atomic nucleus. Therefore, 1920s quantum physics can describe electrons whose wave functions allow them to range across atoms, but cannot describe an electron restricted to a region the size of an atomic nucleus, or of a proton or neutron, whose size is 10-15 meters. It certainly can’t handle an electron restricted to a point!

Let me reiterate: an electron cannot be restricted to a region the size of a proton and still be described by “quantum mechanics”.

As for neutrinos, it’s much worse; since their masses are much smaller, they can’t even be described in regions whose diameter is that of a thousand atoms!

The Solution: Relativistic Quantum Field Theory

It took scientists two decades (arguably more) to figure out how to get around this problem. But with benefit of hindsight, we can say that it’s time for us to give up on the quantum physics of the 1920s, and its image of an electron as a dot — as an infinitesimally small object. It just doesn’t work.

Instead, we now turn to relativistic quantum field theory, which can indeed handle all this complexity. It does so by no longer viewing particles as fundamental objects with infinitesimal size, position x and momentum p, and instead seeing them as ripples in fields. A quantum field can have any number of ripples — i.e. as many particles and anti-particles as you want, and more generally an indefinite number. Along the way, quantum field theory explains why every electron is exactly the same as every other. There is no longer symmetry between x and p, no reason to worry about why states of definite momentum exist and those of definite position do not, and no reason to imagine that “particles” [which I personally think are more reasonably called “wavicles“, since they behave much more like waves than particles] have a definite, unchanging shape.

The space of possibilities is now the space of possible shapes for the field, which always has an infinite number of dimensions — and indeed the wave function of a field (or of multiple fields) is a function of an infinite number of variables (really a function of a function [or of multiple functions], called a “functional”).

Don’t get me wrong; quantum field theory doesn’t do all this in a simple way. As physicists tried to cope with the difficult math of quantum field theory, they faced many serious challenges, including apparent infinities everywhere and lots of consistency requirements that needed to be understood. Nevertheless, over the past seven decades, they solved the vast majority of these problems. As they did so, field theory turned out to agree so well with data that it has become the universal modern language for describing the bricks and mortar of the universe.

Yet this is not the end of the story. Even within quantum field theory, we can still find ways to define what we mean by the “size” of a particle, though doing so requires a new approach. Armed with this definition, we do now have clear evidence that electrons are much smaller than protons. And so we can ask again: can an elementary “particle” [wavicle] have zero size?

We’ll return to this question in later posts.

This is admittedly a provocative title coming from a particle physicist, and you might think it tongue-in-cheek. But it’s really not.

We live in a cosmos with quantum physics, relativity, gravity, and a bunch of elementary fields, whose ripples we call elementary particles. These elementary “particles” include objects like electrons, photons, quarks, Higgs bosons, etc. Now if, in ordinary conversation in English, we heard the words “elementary” and “particle” used together, we would probably first imagine that elementary particles are tiny balls, shrunk down to infinitesimal size, making them indivisible and thus elementary — i.e., they’re not made from anything smaller because they’re as small as could be. As mere points, they would be objects whose diameter is zero.

But that’s not what they are. They can’t be.

I’ll tell this story in stages. In my last post, I emphasized that after the Newtonian view of the world was overthrown in the early 1900s, there emerged the quantum physics of the 1920s, which did a very good job of explaining atomic physics and a variety of other phenomena. In atomic physics, the electron is indeed viewed as a particle, though with behavior that is quite unfamiliar. The particle no longer travels on a path through physical space, and instead its behavior — where it is, and where it is going — is described probabilistically, using a wave function that exists in the space of possibilities.

But as soon became clear, 1920s quantum physics forbids the very existence of elementary particles.

In 1920s Quantum Physics, True Particles Do Not Exist

To claim that particles do not exist in 1920s quantum physics might seem, at first, absurd, especially to people who took a class on the subject. Indeed, in my own blog post from last week, I said, without any disclaimers, that “1920s quantum physics treats an electron as a particle with position x and momentum p that are never simultaneously definite.” (Recall that momentum is about motion; in pre-quantum physics, the momentum of an object is its mass m times its speed v.) Unless I was lying to you, my statement would seem to imply that the electron is allowed to have definite position x if its momentum p is indefinite, and vice versa. And indeed, that’s what 1920s quantum physics would imply.

To see why this is only half true, we’re going to examine two different perspectives on how 1920s quantum physics views location and motion — position x and momentum p.

1. There is a perfect symmetry between position and momentum (today’s post)
2. There is a profound asymmetry between position and momentum (next post)

Despite all the symmetry, the asymmetry turns out to be essential, and we’ll see (in the next post) that it implies particles of definite momentum can exist, but particles of definite position cannot… not in 1920s quantum physics, anyway.

The Symmetry Between Location and Motion

The idea of a symmetry between location and motion may seem pretty weird at first. After all, isn’t motion the change in something’s location? Obviously the reverse is not generally true: location is not the change in something’s motion! Instead, the change in an object’s motion is called its “acceleration” (a physics word that includes what in English we’d call acceleration, deceleration and turning.) In what sense are location and motion partners?

The Uncertainty Principle of Werner Heisenberg

In a 19th century reformulation of Newton’s laws of motion that was introduced by William Rowan Hamilton — keeping the same predictions, but rewriting the math in a new way — there is a fundamental symmetry between position x and momentum p. This way of looking at things is carried on into quantum physics, where we find it expressed most succinctly through Heisenberg’s uncertainty principle, which specifically tells us that we cannot know a object’s position and momentum simultaneously.

This might sound vague, but Heisenberg made his principle very precise. Let’s express our uncertainty in the object’s position as Δx. (Heisenberg defined this as the average value of x2 minus the squared average value of x. Less technically, it means that if we think the particle is probably at a position x0, an uncertainty of Δx means that the particle has a 95% chance of being found anywhere between x0-2Δx and x0+2Δx.) Let’s similarly express our uncertainty about the object’s momentum (which, again, is naively its speed times its mass) as Δp. Then in 1920s quantum physics, it is always true that

• Δp Δx > h / (4π)

where h is Planck’s constant, the mascot of all things quantum. In other words, if we know our uncertainty on an object’s position Δx, then the uncertainty on its momentum cannot be smaller than a minimum amount:

• Δp > h / (4π Δx) .

Thus, the better we know an object’s position, implying a smaller Δx, the less we can know about the object’s momentum — and vice versa.

This can be taken to extremes:,

• if we knew an object’s motion perfectly — if Δp is zero — then Δx = h / (4π Δp) = infinity, in which case we have no idea where the particle might be
• if we knew an object’s location perfectly — if Δx is zero — then Δp = h / (4π Δx) = infinity, in which case we have no idea where or how fast the particle might be going.

You see everything is perfectly symmetric: the more I know about the object’s location, the less I can know about its motion, and vice versa.

(Note: My knowledge can always be worse. If I’ve done a sloppy measurement, I could be very uncertain about the object’s location and very uncertain about its location. The uncertainty principle contains a greater-than sign (>), not an equals sign. But I can never be very certain about both at the same time.)

An Object with Known Motion

What does it mean for an object to have zero uncertainty in its position or its motion? Quantum physics of the 1920s asserts that any system is described by a wave function that tells us the probability for where we might find it and what it might be doing. So let’s ask: what form must a wave function take to describe a single particle with perfectly known momentum p?

The physical state corresponding to a single particle with perfectly known momentum P0 , which is often denoted |P0>, has a wave function

times an overall constant which we don’t have to care about. Notice the ; this is a complex number at each position x. I’ve plotted the real and imaginary parts of this function in Fig. 1 below. As you see, both the real (red) and imaginary (blue) parts look like a simple wave, of infinite extent and of constant wavelength and height.

Figure 1: In red and blue, the real and imaginary parts of the wave function describing a particle of known momentum (up to an overall constant). In black is the square of the wave function, showing that the particle has equal probability to be at each possible location.

Now, what do we learn from the wave function about where this object is located? The probability for finding the object at a particular position X is given by the absolute value of the wave function squared. Recall that if I have any complex number z = x + i y, then its absolute value squared |z2| equals |x2|+|y2|. Therefore the probability to be at X is proportional to

(again multiplied by an overall constant.) Notice, as shown by the black line in Fig. 1, this is the same no matter what X is, which means the object has an equal probability to be at any location we choose. And so, we have absolutely no idea of where it is; as far as we’re concerned, its position is completely random.

An Object with Known Location

As symmetry requires, we can do the same for a single object with perfectly known position X0. The corresponding physical state, denoted |X0>, has a wave function

again times an overall constant. Physicists call this a “delta function”, but it’s just an infinitely narrow spike of some sort. I’ve plotted something like it in Figure 2, but you should imagine it being infinitely thin and infinitely high, which obviously I can’t actually draw.

This wave function tells us that the probability that the object is at any point other than X0 is equal to zero. You might think the probability of it being at X0 is infinity squared, but the math is clever and the probability that it is at X0 is exactly 1. So if the particle is in the physical state |X0>, we know exactly where it is: it’s at position X0.

Figure 2: The wave function describing a particle of known position (up to an overall constant). The square of the wave function is in black, showing that the particle has zero probability to be anywhere except at the spike. The real and imaginary parts (in red and blue) are mostly covered by the black line.

What do we know about its motion? Well, we saw in Fig. 1 that to know an object’s momentum perfectly, its wave function should be a spread-out, simple wave with a constant wavelength. This giant spike, however, is as different from nice simple waves as it could possibly be. So |X0> is a state in which the momentum of the particle, and thus its motion, is completely unknown. [To prove this vague argument using math, we would use a Fourier transform; we’ll get more insight into this in a later post.]

So we have two functions, as different from each other as they could possibly be,

• Fig. 1 describing an object with a definite momentum and completely unknown position, and
• Fig. 2 describing an object with definite position and completely unknown momentum.

CAUTION: We might be tempted to think: “oh, Fig. 1 is the wave, and Fig. 2 is the particle”. Indeed the pictures make this very tempting! But no. In both cases, we are looking at the shape of a wave function that describes where an object, perhaps a particle, is located. When people talk about an electron being both wave and particle, they’re not simply referring to the relation between momentum states and position states; there’s more to it than that.

CAUTION 2: Do not identify the wave function with the particle it describes!!! It is not true that each particle has its own wave function. Instead, if there were two particles, there would still be only one wave function, describing the pair of particles. See this post and this one for more discussion of this crucial point.

Objects with More or Less Uncertainty

We can gain some additional intuition for this by stepping back from our extreme |P0> and |X0> states, and looking instead at compromise states that lie somewhere between the extremes. In these states, neither p nor x is precisely known, but the uncertainty of one is as small as it can be given the uncertainty of the other. These special states are often called “Gaussian wave packets”, and they are ideal for showing us how Heisenberg’s uncertainty principle plays out.

In Fig. 3 I’ve shown a wave function for a particle whose position is poorly known but whose momentum is better known. This wave function looks like a trimmed version of the |P0> state of Fig. 1, and indeed the momentum of the particle won’t be too far from P0. The position is clearly centered to the right of the vertical axis, but it has a large probability to be on the left side, too. So in this state, Δp is small and Δx is large.

Figure 3: A wave function similar to that of Fig. 1, describing a particle that has an almost definite momentum and a rather uncertain position.

In Fig. 4 I’ve shown a wave function of a wave packet that has the situation reversed: its position is well known and its momentum is not. It looks like a smeared out version of the |X0> state in Fig. 2, and so the particle is most likely located quite close to X0. We can see the wave function shows some wavelike behavior, however, indicating the particle’s momentum isn’t completely unknown; nevertheless, it differs greatly from the simple wave in Fig. 1, so the momentum is pretty uncertain. So here, Δx is small and Δp is large.

Figure 4: A wave function similar to that of Fig. 2, describing a particle that has an almost definite position and a highly uncertain momentum.

In this way we can interpolate however we like between Figs. 1 and 2, getting whatever uncertainty we want on momentum and position as long as they are consistent with Heisenberg’s uncertainty relation.

Wave functions in the space of possible momenta There’s even another more profound, but somewhat more technical, way to see the symmetry in action; click here if you are interested.

As I’ve emphasized recently (and less recently), the wave function of a system exists in the space of possibilities for that system. So far I’ve been expressing this particle’s wave function as a space of possibilities for the particle’s location — in other words, I’ve been writing it, and depicting it in Figs. 1 and 2, as Ψ(x). Doing so makes it more obvious what the probabilities are for where the particle might be located, but to understand what this function means for what the particle’s motion takes some reasoning.

But I could instead (thanks to the symmetry between position and momentum) write the wave function in the space of possibilities for the particle’s motion! In other words, I can take the state |P0>, in which the particle has definite momentum, and write it either as Ψ(x), shown in Fig. 1, or as Ψ(p), shown in Fig. 1a.

Figure 1a: The wave function of Fig. 1, written in the space of possibilities of momenta instead of the space of possibilities of position; i.e., the horizontal axis show the particle’s momentum p, not its position x as is the case in Figs. 1 and 2. This shows the particle’s momentum is definitely known. Compare this with Fig. 2, showing a different wave function in which the particle’s position is definitely known.

Remarkably, Fig. 1a looks just like Fig. 2 — except for one crucial thing. In Fig. 2, the horizontal axis is the particle’s position. In Fig. 1a, however, the horizontal axis is the particle’s momentum — and so while Fig. 2 shows a wave function for a particle with definite position, Fig. 1a shows a wave function for a particle with definite momentum, the same wave function as in Fig. 1.

We can similarly write the wave function of Fig. 2 in the space of possibilities for the particle’s position, and not surprisingly, the resulting Fig. 2a looks just like Fig. 1, except that its horizontal axis represents p, and so in this case we have no idea what the particle’s momentum is — neither the particle’s speed nor its direction.

Fig. 2a: As in Fig. 1a, the wave function in Fig. 2 written in terms of the particle’s momentum p.

The relationship between Fig. 1 and Fig. 1a is that each is the Fourier transform of the other [where the momentum is related to the inverse wavelength of the wave obtained in the transform.] Similarly, Figs. 2 and 2a are each other’s Fourier transforms.

In short, the wave function for the state |P0> (as a function of position) in Fig. 1 looks just like the wave function for the state |X0> (as a function of momentum) in Fig. 2a, and a similar relation holds for Figs. 2 and 1a. Everything is symmetric!

The Symmetry and the Particle…

So, what’s this all got to do with electrons and other elementary particles? Well, if a “particle” is really and truly a particle, an object of infinitesimal size, then we certainly ought to be able to put it, or at least imagine it, in a position state like |X0>, in which its position is clearly X0 with no uncertainty. Otherwise how could we ever even tell if its size is infinitesimal? (This is admittedly a bit glib, but the rough edges to this argument won’t matter in the end.)

That’s where this symmetry inherent in 1920s quantum physics comes in. We do in fact see states of near-definite momentum — of near-definite motion. We can create them pretty easily, for instance in narrow electron beams, where the electrons have been steered by electric and magnetic fields so they have very precisely defined momentum. Making position states is trickier, but it would seem they must exist, thanks to the symmetry of momentum and position.

But they don’t. And that’s thanks to a crucial asymmetry between location and motion that we’ll explore next time.