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% of the students think this shows two particles. It does not. And that’s why I always give this example.]
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[Note added Friday morning: Okay, I have added the promised explanation. There are a lot of new explanations here that I haven’t written out before. There will surely be passages that aren’t yet as clear as they could be, so please point them out in the comments.]
Two Particles Colliding or Passing? No!
[If you’re already convinced that this wave function doesn’t show two particles, and you don’t feel you need further clarification, you can skip this section and go on to the next one. But you may still find it interesting to return to later.]
Sure, this looks like two particles approaching each other in one dimension of physical space, which we’ll call the x axis. At first glance, 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 possibilities, whereas x is the coordinate of physical space.
In pre-quantum physics, the space of possibilities also has two coordinates, as in Fig. 3. Within that space, the system is at a single, definite spot, shown by the star. (Compare this with Fig. 2; in physical space, the particles lie at two points, each with one coordinate, as in Fig. 2.) The change in the system over time, as particle 1 moves to smaller x1 and particle 2 moves to larger x2, is shown by the red arrow.
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? Let’s assume the two particles are not exactly identical [subtleties with identical particles is something I’ll return to another time]. In that case, 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 the animation shown in Fig. 4. [It runs for a short time, then goes briefly blank before repeating.]
The system’s location within the space of possibilities is no longer precisely specified as it would be in pre-quantum physics (see Fig. 3). But it lies mostly within the visible bump on the plot, whose uncertainty along the x1, x2 axes is not that severe, and its change with time is pretty clear.
- Initially, the wave function tells us that x1 is most likely positive, x2 is most likely negative, and that x1 is approximately –x2.
- Similarly to the pre-quantum case shown in Figs. 2 and 3, the most likely value of x1 decreases over time, while that of x2 increases, showing the two particles are tending to approach one another.
- The particles then cross paths when x1=x2=0.
- Finally they continue onward, with x1 most likely negative and x2 most likely positive.
Notice there is no interference! These particles do not interact at all, so they do not even notice each others’ presence even when they pass each other.
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 will collide and recoil from one another. 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 is significantly less than x2 isn’t just small, it’s essentially zero. The number zero has no argument (“phase”), and so the wave function in this region has zero magnitude and is drawn in red to indicate it has no meaningful phase.
But when x1 and x2 are nearly zero, the particles are likely to be in the same place, and the interaction between them becomes important. The interaction causes the two particles to bounce off each other, as a result of which the wave function turns around and goes back the way it came, sending x1 back to large positive values and x2 back to large negative values.
Most notably, just when the wave function is turning 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 becomes nonzero that the two particles are in the same place and interacting), the reflecting part of the wave function 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.) In short, 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 let’s not get distracted; the goal here is to make sure we get the facts of the matter straight, 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 ideas about two particles aside. The point is that they can’t explain our original wave function, which is a function in only one dimension.
What This Wave Function Actually Shows
Since the wave function can be plotted as a function of only one variable, the system’s space of possibilities can have only one coordinate. That can only be the location of one particle in one dimension of physical space.
So what does it actually mean that the wave function of one particle initially has two equal humps, moving in opposite directions? Simply this:
- There is some probability that the particle 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. Only waves interfere.
Read this carefully. 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 in one place versus another. But in that context, probabilities combine simply; there is no interference between them. The interference seen in quantum physics comes about because the wave function for each possibility is a complex number.
Is this weird? Yes!! But data shows that it is true. Never in history has an experiment shown that this is ever false.
So in summary, what are we looking at in this wave function? Somebody or something has arranged to give a single particle equal probabilities to be
- heading to the right from the left, or
- 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.] When these two possibilities bring this one-and-only particle to the same location in physical space, the possibilities interfere, to the point of creating a set of locations where the wave function is exactly zero, and 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. Once the particle is in different locations within 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 — at least, that’s the way we describe it in 1920s quantum physics.
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 guy’s version of the double-slit experiment”) For a review of the double slit experiment, check out 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 secretly 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. As we move to other locations on the screen, L1 and L2 change.
Well, if L1 and L2 are equal, as at the exact center of the screen, the ripples from the two slits are identical. As a result, the two ripples crest at the same time and dip at the same time, so that their effects add, making the waving even more dramatic than for either ripple separately. The same holds 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 with curved paths.
Ordinary Waves 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 for the two-particle system becomes much larger, just as we saw in Figs. 2-5.
The end result? When a quantum particle with rather definite momentum but quite uncertain position moves toward the two slits, with a wave function that initially looks like Fig. 9a, 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, with a similar shape as shown both in Fig. 9b and in the pictures at the very start of this post. These possibilities will interfere when they both describe the particle as being nearly in the middle of the bottom tube.
This is 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 where the particle is located at precisely the moment where the interference effect is most dramatic. If we repeat this many times, repeatedly sending one particle at a time toward the slits, 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 so there is no longer any interference between two possibilities since one of them is gone.
- If we measure the particle’s location as it goes through the slits, we are doing something a little more subtle than closing a slit; but still, the interference will be lost. [The precise details depend on exactly how we do the measurement.] We will soon have to examine how exactly this can happen. But what’s nice about this disguised double-slit experiment is that doing so will be much easier than in the standard double-slit experiment.
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 isn’t as puzzling as it first seems, in quantum physics.
35 Responses
Lacking the bravery of Paul Butcher, I did not post yesterday. I did think of the 2-particle trap, but resisted. By the time I decided to consider the analogy with the double slit experiment as a 1-particle case, others had suggested the same. I was still hesitant, however, because I had also considered an electron in a p-orbital from inorganic chemistry and realized that the puzzle solution had to be 1-dimensional.
So, no guess. Thank you for the puzzle and its explanation.
“If we measure the particle’s location as it goes through the slits, we are doing something a little more subtle than closing a slit; but still, the interference will be lost.”
Of course. By measuring the particle location we changed the space of possibilities. If we measure that the particle goes through one slit, it is impossible it goes through the other.
Not really wrong, but not entirely complete either; I think you’re being a bit too quick to rush to judgment. Understanding how measurement plays out in the quantum context also requires a discussion, which we may get to next week.
Hi Matt, terrific post thanks. To recap: probability space is the place where wave functions live (with the simplifications due to the fact that there is only one wave function). Every point in the probability space is represented by a complex number, complex numbers can be considered vectors whereby vectors having the same module but opposite direction cancel each other, hence constructive or destructive interference. Is this correct?
Two minor corrections.
(A) The space is called “the space of possibilities.” It’s not the space of probabilities, because a probability can only be a positive real number between 0 and 1, and so the space of probabilities is just a segment of the real number line.
The space of possibilties involves all the different ways you can arrange the elements of the system — literally every possibility you can think of that is consistent with physics. In some wave functions, most of the possibilities may have zero probability, and in other wave functions, it may even happen that one and only one possibility has probability exactly one, while all other possibilities have probability exactly zero. We’ll look at this space in more detail in other examples, but the choice of terminology is important to avoid confusion later.
B) Constructive/destructive interference does occur with complex numbers as you say. Keep in mind, though, that the fact that a complex number is like a two-dimensional vector is sufficient to allow interference, but it is not necessary. Interference can also occur with real numbers as long as the real numbers can be positive and negative.
For instance, in sound waves, the air becomes slightly overdense in some places and slightly underdense in others; i.e. the density fluctuates to the positive side and to the negative side. In water ripples, the height of the water goes up and down. That is already enough to give us interference.
Complex numbers just give us the same thing, plus additional phenomena that come from having even more freedom in how they can vary.
Thanks. Btw, the space of possibilities in quantum terms is the Hilbert space?
Yes, in some chosen basis. (The basis dependence of the space of possibilities is an issue I haven’t attempted to delve into yet… hopefully that won’t turn out to be a pedagogical nightmare.)
the composite mapping quality of a wave in weak force array at a mapping level of strong force inverse quasi horizon exhibits as a plasma?
The result of a particle (e.g. photon) being split in a beam splitter sometime before the scene. Then being sent towards each other (via mirrors) near the “end” of its journey into the “interferometer” area.
I meant to say “towards itself”, not “towards each other”.
Very good… now you’re getting the hang of it… Except that it can’t be a photon, because quantum physics of the 1920s can’t handle massless particles or relativistic velocities, so this must be a particle traveling far below the speed of light. To describe photons, we need quantum field theory, and the wave function for this process will look completely different. That’s something we’ll come back to in coming weeks.
Duh! only one particle, so it must be getting ‘focussed’ and then ‘de-focussed’?
Something like that… keep it simple…
Two bosons passing through each other, or 2 fermions colliding and rebounding. How to see the difference?
I certainly first thought particles colliding, but since that’s wrong, I’m going to guess — a particle whose velocity is uncertain such that it could be moving either left or right?
That is definitely part of the answer. You can try to be a little more precise than that.
an interferometer or even just the double slit experiment ?
Can you be more explicit? Classic interferometers https://en.wikipedia.org/wiki/File:Interferometer.svg and the classic double slit experiment https://en.wikipedia.org/wiki/File:Doubleslit.svg take advantage of two physical dimensions.
I was worried that the number of dimensions was wrong.
If the dimension was radial dimension then could it perhaps be a quark trapped in a hadron.
Too complicated…
With one dimension it seems like one needs something to reflect the two parts of the wave function, and something to separate them in the first place. A contrived setup would be a half silvered mirror half way between two other mirrors. But that doesn’t seem physically very interesting except maybe that the probability drops to zero in the middle
Any controlled experiment performed by human beings to isolate a phenomeon is a contrivance. Contrivance alone is not a reason to reject an interpretation; far from it.
You’re not entirely far off at this point. But indeed you haven’t yet seen the connection with important experiments… it’s not entirely obvious.
Overnight a navigational device like an accelerometer popped into my head.
As in something like this? https://pubs.aip.org/aip/rsi/article-abstract/38/8/1090/444749/Interference-Accelerometer?redirectedFrom=fulltext That’s not quantum though, just ordinary light wave interference; but you could do something similar with wave function interference and probably people have done so.
Perhaps there’s a resulting superposition state of bounce-back and pass-through?
There is certainly superposition here, in a sense. But if you think there are two particles passing through each other or bouncing off each other, that’s not so.
This looks like two wavicles colliding.
The wavicles travel until they are close to each other. Then there is some interference between the wavicles’ oscilations. There is also what seems to be some rotations, because there is an oscilation between the real and imaginary parts’ amplitudes of the wave function’s complex number.
It also seems to happen some energy lost somehow, because the amplitudes of the resulting wavicles are lower, and more spread out.
That’s what it looks like! But it’s a trap, and you’ve fallen in.
Commenting in the interests of keeping myself honest (so I don’t read the “reveal” tomorrow and fool myself into thinking “yes, of course, that’s exactly what I thought”):
A system that can initially be thought of as two particles approaching each other. The particles collide (interact?) and then scatter off each other, the result being a system of two particles travelling away from each other. I’m not sure whether there’s any useful distinction between thinking of it as the particles bouncing off each other or passing through each other (I suspect either interpretation is equally valid?).
That’s what it looks like! But it’s a trap, and you’ve fallen in.
Cool. I look forward to the reveal tomorrow (with enhanced humility).
Ah! Perhaps this is closer?
A single particle which starts in a superposition of “particle moving left” and “particle moving right” where those two possibilities start far apart, but evolve to a point where they interfere (generating an interference pattern) and then eventually become a superposition of particle moving left and particle moving right, getting further apart.
Now we’re talking…
Looks like what Atlas sees at the LHC. Unlike cars driving on a road, in the quantum realm collisions are practically inevitable. Most collisions don’t quite reach the same velocity and energy as those in a collider but any two particles in any space are as likely to collide as not. In the classical world, there is the persistent urban legend that the first two automobiles in Ohio managed to collide with one another in 1891.
Thanks for the interesting and yet to be revealed Wave patterns
Kind regards Joseph