The quantum double-slit experiment, in which objects are sent toward a wall with two slits and then recorded on a screen behind the wall, creates an interference pattern that builds up gradually, object by object. And yet, it’s crucial that the path of each object on its way to the screen remain unknown. If one measures which of the slits each object passes through, the interference pattern never appears.
Strange things are said about this. There are vague, weird slogans: “measurement causes the wave function to collapse“; “the particle interferes with itself“; “electrons are both particles and waves“; etc. One reads that the objects are particles when they reach the screen, but they are waves when they go through the slits, causing the interference — unless their passage through the slits is measured, in which case they remain particles.
But in fact the equations of 1920s quantum physics say something different and not vague in the slightest — though perhaps equally weird. As we’ll see today, the elimination of interference by measurement is no mystery at all, once you understand both measurement and interference. Those of you who’ve followed my recent posts on these two topics will find this surprisingly straightforward; I guarantee you’ll say, “Oh, is that all?” Other readers will probably want to read
- this post on measurement
- (and perhaps this one too about how to make a measurement permanent)
- these posts on interference for one particle and for two particles
The Interference Criterion
When do we expect quantum interference? As I’ll review in a moment, there’s a simple criterion:
- a system of objects (not the objects themselves!) will exhibit quantum interference if the system, initially in a superposition of possibilities, reaches a single possibility via two or more pathways.
To remind you what that means, let’s compare two contrasting cases (covered carefully in this post.) Figs. 1a and 1b show pre-quantum animations of different quantum systems, in which two balls (drawn blue and orange) are in a superposition of moving left OR moving right. I’ve chosen to stop each animation right at the moment when the blue ball in the top half of the superposition is at the same location as the blue ball in the bottom half, because if the orange ball weren’t there, this is when we’d expect it to see quantum interference.
But for interference to occur, the orange ball, too, must at that same moment be in the same place in both parts of the superposition. That does happen for the system in Fig. 1a — the top and bottom parts of the figure line up exactly, and so interference will occur. But the system in Fig. 1b, whose top and bottom parts never look the same, will not show quantum interference.
In other words, quantum interference requires that the two possibilities in the superposition become identical at some moment in time. Partial resemblance is not enough.
The Measurement
A measurement always involves an interaction of some sort between the object we want to measure and the device doing the measurement. We will typically
- use a small object to carry out the basic initial measurement, and then
- amplify the result so that it can be made permanent.
For today’s purposes, the details of the second step won’t matter, so I’ll focus on the first step.
Setting Up
We’ll call the object going through the slits a “particle”, and we’ll call the measurement device a “measuring ball” (or just “ball” for short.) The setup is depicted in Fig. 2, where the particle is approaching the slits and the measuring ball lies in wait.
If No Measurement is Made at the Slits
Suppose we allow the particle to proceed and we make no measurement of its location as it passes through the slits. Then we can leave the ball where it is, at the position I’ve marked M in Fig. 3. If the particle makes it through the wall, it must pass through one slit or the other, leaving the system in a superposition of the form
- the particle is near the left slit [and the ball is at position M]
OR - the particle is near the right slit [and the ball is at position M]
as shown at the top of Fig. 3. (Note: because the ball and particle are independent [unentangled] in this superposition, it can be written in factored form as in Fig. 12 of this post.)
From here, the particle (whose motion is now quite uncertain as a result of passing through a narrow slit) can proceed unencumbered to the screen. Let’s say it arrives at the point marked P, as at the bottom of Fig. 3.
Crucially, both halves of the superposition now describe the same situation: particle at P, ball at M. The system has arrived here via two paths:
- The particle went through the left slit and arrived at the point P (with the ball always at M),
OR - The particle went through the right slit and arrived at the point P (with the ball always at M).
Therefore, since the system has reached a single possibility via two different routes, quantum interference may be observed.
Specifically, the system’s wave function, which gives the probability for the particle to arrive at any point on the screen, will display an interference pattern. We saw numerous similar examples in this post, this post and this post.
If the Measurement is Made at the Slits
But now let’s make the measurement. We’ll do it by throwing the ball rapidly toward the particle, timed carefully so that, as shown in Fig. 4, either
- the particle is at the left slit, in which case the ball passes behind it and travels onward,
OR - the particle is at the right slit, in which case the ball hits it and bounces back.
(Recall that I assumed the measuring ball is lightweight, so the collision doesn’t much affect the particle; for instance, the particle might be an heavy atom, while the measuring ball is a light atom.)
The ball’s late-time behavior reveals — and thus measures — the particle’s behavior as it passed through the wall:
- the ball moving to the left means the particle went through the left slit;
- the ball moving to the right means the particle went through the right slit.
[Said another way, the ball and particle, which were originally independent before the measurement, have been entangled by the measurement process. Because of the entanglement, knowledge concerning the ball tells us something about the particle too.]
To make this measurement complete and permanent requires a longer story with more details; for instance, we might choose to amplify the result with a Geiger counter. But the details don’t matter, and besides, that takes place later. Let’s keep our focus on what happens next.
The Effect of the Measurement
What happens next is that the particle reaches the point P on the screen. It can do this whether it traveled via the left slit or via the right slit, just as before, and so you might think there should still be an interference pattern. However, remembering Figs. 1a and 1b and the criterion for interference, take a look at Fig. 5.
Even though the particle by itself could have taken two paths to the point P, the system as a whole is still in a superposition of two different possibilities, not one — more like Fig. 1b than like Fig. 1a. Specifically,
- the particle is at position P and the ball is at location ML (which happens if, in Fig. 4, the particle was near the left slit and the ball continued to the left);
OR - the particle is at position P and the ball is at location MR (which happens if, in Fig. 4, the particle was near the right slit and the ball bounced back to the right).
The measurement process — by the very definition of “measurement” as a procedure that segregates left-slit cases from right-slit cases — has resulted in the two parts of the superposition being different even when they both have the particle reaching the same point P. Therefore, in contrast to Fig. 3, quantum interference between the two parts of the superposition cannot occur.
And that’s it. That’s all there is to it.
Looking Ahead.
The double-slit experiment is hard to understand if one relies on vague slogans. But if one relies on the math, one sees that many of the seemingly mysterious features of the experiment are in fact straightforward.
I’ll say more about this in future posts. In particular, to convince you today’s argument is really correct, I’ll look more closely at the quantum wave function corresponding to Figs. 3-5, and will reproduce the same phenomenon in simpler examples. Then we’ll apply the resulting insights to other cases, including
- measurements that do not destroy interference,
- measurements that only partly destroy interference,
- destruction of interference without measurement, and
- double-slit experiments whose interference can’t be located in physical space,
- etc.
