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.
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.
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.
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.
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?
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.
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.
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.
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.
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.)
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.
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.
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.
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?
33 Responses
Here’s a question that I think hints at my confusion here: image you have a two-particle system that looks like your first example, but the second particle is always very far away (whatever that should mean..? maybe way outside the experimental apparatus for measuring a particle’s position?). Shouldn’t it then somehow revert to the single-particle case when measuring the first particle, so get interference?
Ah yes, shouldn’t it somehow? But It Doesn’t.
Like it or not, experiment says so. See today’s post. https://profmattstrassler.com/2025/03/31/quantum-interference-5-coming-unglued/
If one particle starts interacting with other objects, then the physics for the other particle starts reverting to the one-particle case. But that happens whether the particles are close or far.
And if the two particles remain isolated from other objects for long enough that they can end up far away, then no, the physics does not revert to the one-particle case. This was the point of Einstein, Padolsky and Rosen in their paper in which they coined the phrase: “Spooky action at a distance.”
I can’t reconcile the absence of interference in figure 5a with what I’ve heard about quantum entanglement experiments. Bell inequality experiments, for instance, need to measure both particles to show entanglement; but you seem to say, by contrasting figure 5a to the one-particle case, that you can detect entanglement by measuring only one particle. What am I missing?
I thought that the answer was that if you measure a single particle, you are not sampling a point in the 2-particle wavefunction but rather sampling the integral over a line. Even though the peaks don’t overlap in 2D, they both contribute to the integral, producing interference in the one-particle measurement for figure 5a.
Your last sentence is half correct. ” Even though the peaks don’t overlap in 2D, they both contribute to the integral, ” that’s right, as long as what you mean is “we integrate the square of the wave function”. But they do not produce interference. I have shown you this in Fig. 7. We do not integrate the wave function itself, just its square, which is always positive, so there’s no way the integral can produce an interference effect if it is absent to start with from both peaks.
The question about detecting entanglement by measuring one particle: Just seeing a bump without fringes does not prove that the particle is entangled. To know that the lack of interference implies entanglement, you would need to know more about the wave function. That additional information would indeed settle the issue, but the point is that you could not settle the issue without it.
What your question is really referring to, though, is a different issue, related to Fig. 12 and Fig. 13. Fig. 12 shows a case where there is entanglement and interference, but Fig. 13 shows that the interference is not visible by measuring a single particle. Only by measuring both particles can you see that there is an interference pattern, and the fact that both particles are required proves they must be entangled, while Fig. 13 is not sufficient for that purpose. (This comes up again briefly at the end of the following post and will be central in the discuss of the one early next week, so please stay tuned.)
Hi Matt. I realize my mistake in thinking of the wavefunction as a hump moving along the x1 axis and a hump moving along the x2 axis – such a wavefunction would not be differentiable with respect to both x1 and x2, so couldn’t be a solution of the Schrodinger equation. We need a smooth function in two dimensions.
No, that’s not the real issue with your suggestion. Much more importantly, your choice of wave function simply does not correspond to the top possibility in Figure 4.
Each part of the superposition in Fig 4 corresponds to ONE and only one hump in the wave function, and the peak corresponding to the top possibility is centered at (x1,x2) = -1, 3, not two peaks at (x1,x2) = (-1,0) and (x1,x2) = (0,-3).
After all, wave functions are about probabilities, and peaks are where things are likely. If you have a hump on the x1 axis centered at x1=-1, that gives a finite probability for particle 1 to be at x=-1 and particle 2 to be at x=0; and the second hump on the x2 axis would give a finite probability for particle 2 to be at x=-3 and particle 1 to be at x=0. In trying to write the top possibility in Figure 4 as two humps, you are writing that single possibility as a superposition of two other possibilities, neither of which is correct.
Since your starting point is wrong, it doesn’t matter what happens *after* that with the Schrodinger equation.
Hi Matt. Still unclear on one point. For the case shown in figure 4, I understand that there is a single wavefunction for the two particles, not a separate wavefunction for each particle. If we take the top possibility in figure 4, why can’t this single wavefunction initially consist of a hump on the x1 axis centred at x1 = -1 and a hump on the x2 axis centred at x2 = -3, with the hump on the x1 axis moving along the x1 axis to higher x1 at a certain speed, and the hump on the x2 axis moving along the x2 axis to higher x2 at a certain speed. And for the bottom possibility in figure 4 why can’t this single wavefunction initially consist of a hump on the x1 axis centred at x1 = +1 and a hump on the x2 axis centred at x2 = +3, with the hump on the x1 axis moving along the x1 axis to lower x1 at a certain speed, and the hump on the x2 axis moving along the x2 axis to lower x2 at a certain speed. Then we will have overlap of the wavefunctions for the top and bottom possibilities when x1 for the two possibilities is the same and when x2 for the two possibilities is the same, and overlap of the wavefunctions implies interference. Is this because the Schr(o)dinger equation does not have a solution giving the wavefunctions I have described with the time development I have described, but does have a solution representing the wavefunctions you have described with the time development you have described in the above article?
Matt, write down for us, in the Schrodinger picture, the complete complex wave function, time dependant, in position space (not momentum space), for each case.
Unentangled, I presume correctly?
Without that, I’m confused.
I’m sorry you’re confused, but that was clear from your earlier posts. I’m certainly not going to write down all of the wave functions for all of my examples in previous posts, but for Figs 7 and 12, they can be written (though this can be further simplified; I don’t have time to do that now.) Writing x1=x and x2=y to keep formulas simpler to read, these are
At t=0, these simplify. For instance the first one becomes
which is a sum of products of Gaussian wave packets moving with equal speed. The second is a similar product except that x and y are exchanged in the second term.
Certainly the two particles are entangled; if you integrate out y, you get a mixed state of x, not a pure state. After all the wave function cannot be written as a product of a function of x and a function of y, nor can this be done for any linear combination of x and y.
I was confused by not knowing the wave function. Seeing the ones you cooked up makes it clear. I was thinking simple unentangled ones.
Matt, when you write ‘These “events” are actually non-events, as far as quantum physics is concerned’ (fig.5a-c), do you mean to say that only ‘classical’ events happen in this context when I make a measurement? Thanks
No, that’s not what I mean. These are non-events classically, too. Nothing happens physically. The only thing that happens is that the possibilities at those moments are more limited; for instance, at the beginning particle 1 can be at position x1=1 or -1, but at the first “event” particle 1 can only be at position x1=0. It’s not a physical event that is reflected in the behavior of particle 1, but an abstract fact about the possibilities that I might find if I make a measurement.
The point is that in quantum physics it’s the same thing. Essentially nothing interesting happens that doesn’t happen in classical physics.
I’m thoroughly enjoying your posts and eagerly anticipating the upcoming discussion on quantum fields. As you delve into that topic, I’m particularly looking forward to understanding how wavefunction interference manifests itself (if indeed that is even possible) within the vast space of possible field configurations. I need to be patient.
So do I. This is a complicated web of ideas, interwoven with misleading vines that have to be cut, and it can only be disentangled slowly.
Are you getting to the point that in these examples there is no “interference” in the real world? Each “particle” goes where it goes and no where else, but in an individually unpredictable manner. What we interpret as physical interference is the overall pattern of distribution of a multitude of these paths. This pattern reflects the “interference” distribution in the space of possibilities. If this is your point, then the next explanation must be how the effect is accomplished. Am I following your reasoning?
Well, you might be reading in too much. All I’m trying to do is lay out clearly what happens and how the Schroedinger equation correctly describes it. Then we can see which statements about quantum physics are inconsistent, allowing us focus on the ones that aren’t.
Your statements aren’t inconsistent, but they’re not entirely clear, and not entirely necessary either. So let’s go a little further, so that we can see exactly how things happen (for instance, why measurement in the double-slit experiment makes interference go away) and then step back and take stock.
Also, remember that everything I’m doing here is for 1920’s quantum physics. When we get to quantum field theory — the real world — we’ll have to almost completely redo the whole thing.
We’ll find the interference in the correlation of the particles position. If we measure the position of both particles simultaneously and graph the difference x2-x1, the interference pattern emerged. This is clean from the picture 12; to see the hill and valleys, you must look at the interference pattern diagonally.
Or, more precisely, by measuring the positions of both particles and graphing them to two-dimensional graph, we will partially reconstruct the wave function in the “space of possibilities”, including the interference pattern. This is evident – by repeating the experiment many times, we will get (almost) any possible result. However, we can never measure the phase of the wave function, thus it is only partial reconstruction.
Definitely on the right track here.
I think it’s high time we stop using the term “particle” and start calling it for what it is, a “wavicle”. Your term, I think? 🙂
In Fig. 12, when the interference occurs, is there a whole lot of refraction going on? In general, could you say one way to explain the Pauli exclusion principle is that the fermion wave function refract not just with other fermions but with themselves, too? So, an isolated election does have a wave function because of refraction, hence one can think of “confinement” as a shell where the refraction is zero and becomes all reflection, trapping the energy within that tiny space, orbital.
Maybe, I should not have used the term, orbital, because I don’t want to infer to an election within the atom, but one which is “free” and still maintains a wave function, i.e. remains confined outside the atom.
I didn’t invent the term. Arthur Eddington invented it 100 years ago.
No, there’s no refraction going on in Fig. 12; just interference.
But, how can two fermions interference? Is this why physicist use the “imaginary” part?
Isn’t refraction more logical? There could be a threshold of energy density in a given space that you will get refraction since the medium is “different”.
Refraction could also explain confinement within a black hole. The event horizon is a shield with such a high energy density that ALMOST all energy within will be reflected back in. There maybe a threshold, i.e. some fluctuations to allow some refraction, Hawking’s radiation?
This imaginary stuff just doesn’t make any sense to me, seems like one of those pseudomathematics.
My attempt to answer the last questions:
To see (measure) effects from interference in the space of possibilities of Fig. 12, both particles need to be measured at the same time when there is maximum interference in the space of possibilities at (x1,x2)=(+1,-1). Repeat until you see the effects.
(Matt, I really appreciate your book and the series you started here in 2025. Even though I only took basic physics in college, I’ve always liked quantum physics and I’m learning so much more here vs random internet articles. Thanks!)
I don’t like it but what a great demonstration that possibility space is not physical space. I thought we had a nice little 1D physics problem going on so my intuition is not to be trusted for QM 😖
I just want to say, I love this series of posts, thank you very much!
To get from Fig 12 (3D wavefunction), to Fig 13 (1D observation), I am visualising the camera coming down, until the X2 plane is back and forth, and we have X1 as a base line. All those peaks and valleys are now hidden.
To view those peaks and troughs, I want to visualise the camera coming down at a 45° angle. So the baseline now becomes X1-X2. I think the correlation between the two particles is what shows observable interference.
Figure 7 on Do Quantum Wave Functions Collapse? shows what I’m referring to.
No wait. I’m 90° off. X1 minus X2 is constant. That’s how it’s defined.
So it’s repeated measurements of X1 plus X2 that shows the the interference building up over time. What is that physically? Hmm. Unclear. Damn thought I was getting somewhere
You are getting somewhere, and we’ll see how close you are early next week.
Hi Matt, my try: interference is another way of calculating the combined behaviour of different alternatives whereby interference itself occurs in an abstract space, abstract in the sense that it is abstracted from the real physical world (fig. 10a,10b).the key point is: when there is interference the paths are indistinguishable, when there is no interference they are distinguishable. Thanks
Well, this is pretty good — though it’s not that the paths are completely indistinguishable, but that they cross. Two (or more) ways to get to the same result.
And the question of whether the physical world is the real one, or whether the space of possibilities is the real one, is something we’re going to have to ask when we have explored a bit more.
thank you. Does it mean that when the alternative paths do not intersect, the wave function describes results analogous to the classical ones, and only when they intersect do we obtain the results peculiar to quantum physics (e.g. when only one detector detects particles whereas classically they should both)? thanks