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.]
This is Not Two Particles Colliding or Passing
Sure, this looks like two particles approaching each other in one dimension of physical space, which we’ll call the x axis. Naively, the wave function I’ve given you looks like a spread out version of Fig. 2.
But this is a wave function. And a wave function does not exist in physical space. It exists in the space of possibilities. For our two particles, which have position x1 and x2 that tell us where they are along the x axis, there is a two-dimensional space of possibilities. A point in the space of possibilities involves a specific choice of x1 and x2; in short, x1 and x2 are the two coordinates in the space of possibilties. In pre-quantum physics, the system in the space of possibilities is located at a point with two coordinates, as in Fig. 3. By contrast, the particles in physical space are located at two points, each with one coordinate, as in Fig. 2.
What about in quantum physics? Well, the wave function for two particles approaching each other has to be a single function of the two particle positions: Ψ(x1,x2).
- It is not a single function of one variable Ψ(x), because such a wave function exists in a space of possibilities with only one coordinate, and cannot describe two objects moving in one dimension.
- Nor is it two functions Ψ1(x) and Ψ2(x), because this system, isolated from the rest of the world but consisting of objects that are not isolated from each other, has a single wave function, not one per particle.
Two Non-Interacting Particles Passing By
What would a wave function look like if we had two particles approaching each other and passing each other without interacting? Well, if they are not exactly identical [subtleties with identical particles is something I’ll return to another time], then (using the second of the three wave-function representations that I discussed in this post, which you may want to read or review), the wave function looks like what is shown in Fig. 4. Its location in the space of possibilities is not precisely specified as it would be in pre-quantum physics — see Fig. 3 — but the uncertainty is not that bad; and the change of its location with time is quite clear. Initially, the wave function tells us that x1 is most likely positive, x2 is most likely negative, and that x1 is approximately -x2. You see that, as in Figs. 2 and 3, the most likely value of x1 decreases while that of x2 increases (showing the two particles are approaching each other). Then they cross paths when x1=x2=0. Finally they continue onward to where x1<0 and x2>0.
Notice there is no interference! These particles do not interact at all, so they do not even notice each others’ presence.
Two Particles Colliding and Bouncing Off
If the two particles feel each other’s presence so strongly that it is impossible for one to pass the other, then they may bounce off. The precise details depend on exactly what their interaction is, but the wave function for this case will look something like the following:
On the one hand, the wave function for the system starts out the same way as in Fig. 4. There is one difference: the probability that x1<x2 isn’t just small: it’s essentially zero, and thus has no argument (“phase”), which is why it is shown with a constant red color.
But in the middle, where x1 and x2 are nearly zero, the particles are likely to be in the same place, and the interaction between them becomes important. This makes the two particles bounce off each other — and makes the wave function turn around and go back the other way, sending x1 back to larger positive values and x2 back to larger negative values.
And where the wave function turns around, the height of the wave briefly shows up-and-down structure. This is characteristic of interference, similar to what we see in the wave function I gave you to start with, but now in two dimensions rather than just one. What is causing this interference effect? Are these the particles interfering with each other?
No. They’re particles. Particles don’t interfere. The wave function is doing the interfering: as the front part of the wave function begins to reflect (i.e. as the probability that the two particles are in the same place and interact becomes non-zero), it interferes with the part of the wave function that is still incoming (indicating that it is still possible that the particles have not yet met and have not yet interacted.) The possibility that the particles have bounced off is interfering with the possibility that the particles have not yet bounced off.
Is that weird? Yes. But the goal here is to make sure we don’t confuse the facts of the matter, weird as they may be.
Once the probability that the particles have met and bounced is large enough, and the probability that they have not yet met is very small, then the interference between the two possibilities ceases. As the bounced particles return from whence they came in physical space, the system as a whole returns from whence it came in the space of possibilities.
Again, the details depend on exactly how the particles interact. It often happens that there is both a certain probability that the particles bounce and a certain probability that they don’t. That might be fun to look at on another day, but for now, let’s leave these wrong ideas about two particles and return to the right one, which involves just one particle.
What This Wave Function Actually Shows
Since the wave function is plotted as a function of only one variable, the system can have only one coordinate, and that must be the location of one particle in one dimension.
What does it mean that the wave function of one particle initially has two equal humps, moving in opposite directions? That simply means:
There is some probability it is located to the left of x=0 and is moving to the right.
There is an equal probability that it is located to the right of x=0 and is moving to the left.
And why is there interference when the two humps cross near x=0? Is the particle interfering with itself? No! Particles cannot ever interfere, and a single particle even less so. Instead, it is the wave function that shows interference, because the possibility that the particle is near x=0 and moving to the right is interfering with the possibility that it is near x=0 and moving to the left.
It is the interference of possibilities that makes quantum physics different from pre-quantum physics. Even in pre-quantum physics, life can be uncertain; we might not actually know where the particle is located, and might have to assign it some probability — a positive real number — for being one place or another. But then probabilities would just combine simply; there would be no interference between them. The interference comes about because the wave function for each possibility is a complex number.
Is this weird? Yes! But data shows that it is true. There has never been an experiment which showed that this is false, ever.
So in summary, what is this? Somehow [I’ll give you an example in a minute], somebody arranged to give a particle equal probabilities to be (1) heading to the right from the left and (2) heading to the left from the right. [In pre-quantum physics we would draw this as in Fig. 6; note the “OR”, and compare with Fig. 2.] As these two possibilities bring the particle to the same location, the possibilities interfere, to the point of creating a set of locations where the particle has zero probability of being located. Then, as time goes by, one possibility takes the particle away to the left, the other takes it away to the right, and with the particle in different locations in the two possibilities, the interference stops.
You don’t have to like this or feel comfortable with it. But this is how the world actually works — in 1920s quantum physics.
And just wait til we get to quantum field theory and have to redo this almost from scratch.
The Double-Slit Experiment in Disguise
The reason this example is so interesting, aside from the fact that it’s a good test of whether you’ve really understood that a wave function lives in a space of possibilities, is that it is a disguised version of the double-slit experiment (or you can call it “this guys version of the double-slit experiment”) That’s the one I outlined in my first post on this subject of quantum basics.
Most importantly this disguised version is a simpler version, simple enough that we’re going to be able to study all its features in much greater detail than we’d be able to in the standard double-slit experiment. Whereas the standard double-slit experiment uses two dimensions of physical space, making all sorts of things harder to depict and to visualize, we’ll take full advantage of the fact that here we only need one-dimension.
Now, why is this the double-slit experiment? Let’s quickly review the double-slit experiment and why it shows an interference pattern.
The Double-Slit Revised
In Figure 7a, an ordinary wave — perhaps a sound wave or an ocean wave — approaches a wall with two slits cut into it where the wave can pass through. In Figure 7b, the wave reaches the wall, and the parts that are passing through the slits begin to spread out in a circular pattern from both slits. The two circular patterns soon intersect and interfere, as shown in Figure 7c for a wave that is continuously moving through.
How do we understand why the interference pattern for this ordinary wave consists of a series of approximately equally spaced active and inactive patches? This is illustrated in Fig. 8, where we focus on one part of the screen and look at the ripples arriving there from the two slits. One set of ripples has to travel a distance L1 from slit to screen, while the other set travels a different distance L2. At different points on the screen, L1 and L2 are different.
Well, if L1 and L2 are equal, as at the exact center of the screen, the ripples from the two slits are identical, and so they crest and the same time and dip at the same time, so their effects add, making even more dramatic waving.. The same is true at other points where the difference between L1 and L2 is equal to an integer number of wavelengths (the wavelength being the spacing between one wave crest, shown in blue, and the next); in this case, the ripples crest and trough at the same time, and the effects add.
But if L1 and L2 differ by half a wavelength, or by 3/2, 5/2, 7/2, etc. of a wavelength, then the crests of one ripple arrive at the same time as the troughs of the other, causing the waves to cancel out. These are the dark patches on the screen, where no waving is taking place. And so the effect of interference — of the overlapping waves — is an alternating pattern, in which patches where the ripples add together are separated by dark patches where they cancel each other out.
What does this have to do with our example above? Well, imagine that instead of an open area behind the slits into which the waves can spread out, we put curved tubes behind the slits, as in Fig. 9a, that force the ripples from the slits to travel along paths that leave them aimed at each other, as in Fig. 9b.
And why do we get an interference pattern? Pick a point near the center of the bottom tube. If the distance from the left slit to that point, as measured along the green arrow at the left, differs from the corresponding distance along the green arrow at the right by an integer number of wavelengths, then the ripples will add; if the difference in the distance is a half-integer number of wavelengths, they will cancel. It is exactly the same effect as in Fig. 8, just along a curved path
Ordinary Wave vs. a Wave Function for a Single Particle
So far, this has been for ordinary waves, like those of sound or water, traveling in physical space inside the slits and tubes. But for a single quantum particle, something very special happens:
- the space of possibilities and physical space have the same shape — they appear the same;
- the quantum wave function for a single particle satisfies almost the same type of equation as a sound wave or water wave;
and so the behavior of the wave function in the space of possibilities for a single particle traveling amid these slits and tubes is quite similar, and shows the same interference effects, as an actual, physical sound wave or water wave moving through the slits and tubes.
Caution! The wave function itself is not moving through the slits and tubes. Those are in physical space, and only physical particles and waves can move through them. Instead, the wave function is describing the possibilities for a single particle moving through the slits and tubes, and that space of possibilities looks the same — but is not the same — as the physical space in which this single particle exists. This becomes obvious if we try to consider two particles rather than one; the physical space remains the same, but the space of possibilities becomes much larger.
The end result? When a quantum particle with rather definite momentum but quite uncertain position moves toward the two slits, it ends up with two possibilities: that it moved through the left slit and ended up moving to the right along the bottom tube, or that it moved through the right slit and ended up moving to the left along the bottom tube. These two possibilities are described by a quantum wave function, and will interfere when they both describe the particle as being nearly in the middle of the bottom tube. Just as for the double-slit experiment, and for the same reason.
All the effects seen in the double-slit experiment are seen here too. For instance:
The interference pattern is measurable: all we have to do is measure, at precisely the moment where the interference effect is most dramatic, where the particle is. If we repeat this many times, with one particle at a time over and over, we will never find a particle at the locations where the wave function is zero — where the interference between the possibilities perfectly cancels.
If we close one slit, only one possibility remains, and there is no longer any interference.
If we measure the particle’s location as it goes through the slits, the interference will be lost. The precise details depend on exactly how we do the measurement, but what’s nice about this disguised two-slit experiment is that we will be able to look at how this happens in great detail.
So this is good news! We have preserved all of the features of the double slit experiment that seem so strange, and done so with simpler math and pictures than in the standard version of the experiment. Looking ahead, we will use this system (and variants of it) over and over again as we come to grips with what is so puzzling (as well as what seems puzzling, but actually isn’t) in quantum physics.
