Did the Particle Go Through the Two Slits, or Did the Wave Function?

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.

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

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.

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:

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.

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.

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.

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:

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

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

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.

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:

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.

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.