Last time, I showed you that a simple quantum system, consisting of a single particle in a superposition of traveling from the left OR from the right, leads to a striking quantum interference effect. It can then produce the same kind of result as the famous double-slit experiment.
The pre-quantum version of this system, in which (like a 19th century scientist) I draw the particle as though it actually has a definite position and motion in each half of the superposition, looks like Fig. 1. The interference occurs when the particle in both halves of the superposition reaches the point at center, x=0.
Then I posed a puzzle. I put a system of two [distinguishable] particles into a superposition which, in pre-quantum language, looks like Fig. 2.
with all particles traveling at the same speed and passing each other without incident if they meet. And I pointed out three events that would happen in quick succession, shown in Figs. 2a-2c.
And I asked the Big Question: in the quantum version of Fig. 2, when will we see quantum interference?
- Will we see interference during events 1, 2a, 2b, and 3?
- Will we see interference during events 1 and 3 only?
- Will we see interference during events 2a and 2b only?
- Will we see interference from the beginning of event 1 to the end of event 3?
- Will we see interference during event 1 only?
- Will we see no interference?
- Will we see interference at some time other than events 1, 2a, 2b or 3?
- Something else altogether?
So? Well? What’s the correct answer?
The correct answer is … 6. No interference occurs — not in any of the three events in Figs. 2.1-2.3, or at any other time.
- But wait. . . how can that make sense? How can it be that particle 1 interferes with itself in the case of Fig. 1 and does not interfere with itself in the case of Fig. 2?!
How, indeed?
Perhaps thinking of the particle as interfering with itself is . . . problematic.
Perhaps imagining individual particles interfering with themselves might not be sufficient to capture the range of quantum phenomena. Perhaps we will need to focus more on systems of particles, not individual particles — or more generally, to consider physical systems as a whole, and not necessarily in parts.
Intuition From Other Examples
To start to gain some intuition, consider some other examples. Some have interference, some do not. What distinguishes one class from the other?
For example, the case of Fig. 4 looks almost like Fig. 2, except that the two particles in the bottom part of the superposition are switched. Is there interference in this case?
Yes.
How about Fig. 5. In this case, the orange particle is stationary in both parts of the superposition. Is there interference?
Yes, there is.
And Fig. 6? Again the orange particle is stationary in either part of the superposition.
No interference this time.
What about Fig. 7 and Fig. 8?
Yes, interference in both cases. And Figs. 9 and 10?
There is interference in the example of Fig. 10, but not that of Fig. 9.
To understand the twists and turns of the double-slit experiment and its many variants, one must be crystal clear about why the above examples do or do not generate interference. We’ll spend several posts exploring them.
What’s Happening (and Where)?
Let’s focus on the cases where interference does occur: Figs. 1, 4, 5, 7, 8, and 10. First, can you identify what they have in common that the cases without interference (Figs. 2, 6 and 9) lack? And second — bringing back the bonus question from last time, which now comes to the fore — in the cases that show interference, exactly when does it happen, and how can we observe it?
Next time we will start the process of going through the examples in Fig. 2 and Figs. 4-10, to see in each case
- How does the wave function actually behave?
- Why is there (or is there not) interference?
- If there is interference,
- where does it occur?
- how exactly can it be observed?
From what we learn, we will try to extract some deep lessons.
If you are truly motivated to understand our quantum world, I promise you that this tour of basic quantum phenomena will be well worth your time.
16 Responses
Thank you all for your comments; many of you have understood the point now. I have started explaining the answers in more detail in this post: https://profmattstrassler.com/2025/03/26/quantum-interference-3-what-is-interfering/
would a gamma ray, xray through slit have a different take on the uncertainty principle via a different tension or stiffness in its spectra as a readout manifold… perhaps a neutrino as well might map a memory burden to its oscillations motivating this question is perception of photons one by one through slit as also registering to lense: if high energy states unify forces and return in a sense to primordial unverse before Higgs split of electro weak and electro magnetic, does string theory perbran stacks in fermionic and bosonic indices over modular and unity conjugation have potential to transcend the difference between domain walls of weak charge in the parity violation construct and the electric stable strong force condition via higher dimensional construction upon the principle of te dilaton to create a saddlepoint referent bridge where charges are conjugated differentially upon string magnitudes and gravity as possibly a magnification operator become commuator where functions become entangled and in transpose experience the catastrophe inversions creating thereby scale difference and eigenvector influence on reading complex to compact dimensionality?
Thank you Matt, what seems clear to me now is we need to consider the physical systems composed of both 2 particles in superposition (4 amplitudes) as a whole. The interference patten can appear only when and where they overlap. In other words, the interference only occurs when both particles in both halves of the superposition reaches simultaneosly the same point respectively.
First a remark on how I think Dr. Strassler is doing an excellent job with these posts, and I appreciate the effort that goes into assembling the figures and the animations.
As for my comment on this post, I believe that one has to recall the two-particle wave function animations which were provided a few weeks ago on the measurement topic. In particular, for Figure 2, consider a Cartesian plane with the horizontal x1 as one axis representing the possible positions of the blue particle and the vertical x2 as the other axis representing the possible positions of the orange particle. In Figure 2 the two-particle wave function has two blobs of non-zero values. The upper half of Figure 2 has a blob in the third (i.e. lower left) quadrant and lower half of Figure 2 has a blob in the first (upper right) quadrant of the Cartesian plane.
In Figure 2.1 the blob which was in the third quadrant has started to move into the first quadrant by way of the fourth quadrant. The blob which was in the first quadrant has started to move into the third quadrant by way of the second quadrant. In Figure 2.2, the first blob is in the fourth quadrant while the second blob is in the second quadrant. In Figure 2.3 the first blob is almost completely in the first quadrant while the second blob is almost completely in the fourth quadrant.
The crucial point is that at no time is there a substantial overlap of the two blobs (i.e. being in the same quadrant) which would lead to there being a superposition of their real and imaginary amplitudes, resulting in interference.
Figure 4 can be analyzed similarly. The first blob starts again in the third quadrant while the other blob starts in the first quadrant. The first blob is moving to the first quadrant by way of the fourth quadrant (as in Figure 2). However, the second blob starts moving from the first quadrant to the third quadrant by way of the fourth quadrant as well. So over some interval of time the two blobs are both in the fourth quadrant and can interfere with each other by adding their respective amplitudes.
In Figure 5 the orange particle is fixed at a positive x2 (i.e. either in quadrant 1 or quadrant 2). When the blue particle in the top half of the figure gets to x=0 and the blue particle in the bottom half gets to 0, there is a brief interval when both blobs overlap in quadrant 1.
The other figures should follow the same analysis, checking if at any time the two blobs will be in the same quadrant of the (x1, x2) plane.
Hi Matt, now it seems clear to me (I hope!): when there is interference, the two alternatives are distinguishable when there is no interference with respect to the case at hand I cannot distinguish which alternative has occurred. The inability to distinguish the paths is the central point so your comment ‘complex numbers are symptoms not the cause’ should be read in this key. I hope I have understood the point correctly.Thanks
sorry in writing I confused the terms!: when there is interference the paths are indistinguishable, when there is no interference they are distinguishable
Excellent articles about the foundations of QM, the best I have ever seen! Can’t wait for the next one.
Regarding today’s puzzle:
Interference takes place when the two possibilities come to the same result, e.g. the same position for the blue particle.
If now consulting the orange particle can answer the question which of the 2 possibilites “really” happened then the interference is destroyed.
If the “measurement” of the orange particle leaves this question open then the interference survives.
in general we see presented one to one correspondence of unlike terms as are placed to a bilinear expression of Pauli Matrices but with an added distinction of a “color” identification charge and so the presentation subtly draws on the fundamental case of flipping signs in mathematical unlike terms in the context of variety in the mathematical space in relation to bosonic and fermionic space delving into the split electroweak and electromagnetic fields thus mathematics to cosmology in themselves an underlying layer of such comparison structure… one could say likewise the cycle sign relates to multiplication as modular and the phase sign relates to division as intercept with the latter in trace form the object of the Pauli matrices per Clifford identity. The range of comparisons as presented is in the spirit of the Moyla bracket in which fiber bundles in their contrasting topology seek something like a saddleback solution possibly then what one could infer as interference pattern of a sort?
We find a reasonable result when we assume a continuously waving pilot wave field that is constantly correlated with the structures of the entire environment and updated according to its changes. A particle and quantum state entering the field interferes with the finished waving environment – neither with itself nor with its superposition. Symmetries and asymmetries are decisive. When the double slit considered, the interference pattern already exists – energy only makes it visible.
Those are nice words, but this is not very precise. Where precisely is the pilot wave?
Everywhere. I see that the pilot wave set by Bohm is an ad hoc rescue, but when the entire structural environment is understood as an environment of standing waves on average, the interferences are just waiting for energy and momentum pulses.
The field of unification is physically locally tense memory-structured, correlating, not only to the quantum statistics of the immediate environment, but also in the vacuum as a field continuum of matter particles, producing as the sum of its vacuum boost actions a proper accelerating continuum, in relation to which we see gravity as a coordinate system term, a virtual force.
Here described as a slightly more detailed idea. A publication of the whole is in preparation, probably getting out in the next summer.
This is still not clear. What is the “Everwhere” of which you speak. In these examples, everywhere means “everywhere on the x axis”?
Physics can only be done by being precise. Vagueness leads to mistakes.
Everywhere in and between all causal structures is the most precise hint i can give here.
Still, this is all math, Schrödinger’s equation, the “probability space”, but what is really, physically, happening?
Assumption 1: Everything is derived from “energy”.
Assumption 2: Energy is a thing, i.e. it has mass and momentum.
Assumption 3: The variables, time and position are illusionary manifestations of quantum entanglement, i.e. they are real, they are part of the math we humans use to try and characterize reality.
So, the obvious question then is: how can you create particles, fermions, which have mass and momentum from a waves, bosons with zero rest mass? Answer: impossible.
So, it means that EVERYTHING, all energy, has mass and momentum and we are still not capable of observing and measuring the “rest mass” of the fundament field that creates everything.
Could this “aether” be a micro black hole, so tiny the electromagnetic spectrum it produces cannot observe it for the obvious reason, it create it so it is impossible to get the resolution to see it, i.e. a singularity.
Does the interference have something to do with both parts of the superposition reaching a shared state? Like the example from last post, there is never a position where both particles are at the same positions in both halves of the superposition, but for the examples today there are states where the particles on both halves are all in the same positions.
That’s got to be it. It’s consistent with the ones that have interference and which ones don’t; 4, 5, 7, 8 and 10 all eventually reach a state where both parts match, but 6 and 9 don’t.
I think what this means is that, as the Prof said, it’s not that a particle or wavicle is interfering with itself: it’s that the different parts of a superposition will interfere with each other if they match up.
I’m not sure if that applies to cases where the superposition “collapses,” such as when the position is finally measured? You would expect such, naively; if the positions in each part of the superposition match, then you know the position regardless of which part we’re “true.” So, of only for a moment, you have much higher precision, as if the superposition had “collapsed.”
But, as Prof. Strassler has said in previous posts, the notion of superpositions “collapsing” is misleading in some ways. I’m just not sure how that applies here.