Now finally, we come to the heart of the matter of quantum interference, as seen from the perspective of in 1920’s quantum physics. (We’ll deal with quantum field theory later this year.)
Last time I looked at some cases of two particle states in which the particles’ behavior is independent — uncorrelated. In the jargon, the particles are said to be “unentangled”. In this situation, and only in this situation, the wave function of the two particles can be written as a product of two wave functions, one per particle. As a result, any quantum interference can be ascribed to one particle or the other, and is visible in measurements of either one particle or the other. (More precisely, it is observable in repeated experiments, in which we do the same measurement over and over.)
In this situation, because each particle’s position can be studied independent of the other’s, we can be led to think any interference associated with particle 1 happens near where particle 1 is located, and similarly for interference involving the second particle.
But this line of reasoning only works when the two particles are uncorrelated. Once this isn’t true — once the particles are entangled — it can easily break down. We saw indications of this in an example that appeared at the ends of my last two posts (here and here), which I’m about to review. The question for today is: what happens to interference in such a case?
Correlation: When “Where” Breaks Down
Let me now review the example of my recent posts. The pre-quantum system looks like this
Notice the particles are correlated; either both particles are moving to the left OR both particles are moving to the right. (The two particles are said to be “entangled”, because the behavior of one depends upon the behavior of the other.) As a result, the wave function cannot be factored (in contrast to most examples in my last post) and we cannot understand the behavior of particle 1 without simultaneously considering the behavior of particle 2. Compare this to Fig. 2, an example from my last post in which the particles are independent; the behavior of particle 2 is the same in both parts of the superposition, independent of what particle 1 is doing.
Let’s return now to Fig. 1. The wave function for the corresponding quantum system, shown as a graph of its absolute value squared on the space of possibilities, behaves as in Fig. 3.
But as shown last time in Fig. 19, at the moment where the interference in Fig. 3 is at its largest, if we measure particle 1 we see no interference effect. More precisely, if we do the experiment many times and measure particle 1 each time, as depicted in Fig. 4, we see no interference pattern.
We see something analogous if we measure particle 2.
Yet the interference is plain as day in Fig. 3. It’s obvious when we look at the full two-dimensional space of possibilities, even though it is invisible in Fig. 4 for particle 1 and in the analogous experiment for particle 2. So what measurements, if any, can we make that can reveal it?
The clue comes from the fact that the interference fringes lie at a 45 degree angle, perpendicular neither to the x1 axis nor to the x2 axis but instead to the axis for the variable 1/2(x1 + x2), the average of the positions of particle 1 and 2. It’s that average position that we need to measure if we are to observe the interference.
But doing so requires we that we measure both particles’ positions. We have to measure them both every time we repeat the experiment. Only then can we start making a plot of the average of their positions.
When we do this, we will find what is shown in Fig 5.
- The top row shows measurements of particle 1.
- The bottom row shows measurements of particle 2.
- And the middle row shows a quantity that we infer from these measurements: their average.
For each measurement, I’ve drawn a straight orange line between the measurement of x1 and the measurement of x2; the center of this line lies at the average position 1/2(x1+x2). The actual averages are then recorded in a different color, to remind you that we don’t measure them directly; we infer them from the actual measurements of the two particles’ positions.
In short, the interference is not associated with either particle separately — none is seen in either the top or bottom rows. Instead, it is found within the correlation between the two particles’ positions. This is something that neither particle can tell us on its own.
And where is the interference? It certainly lies near 1/2(x1+x2)=0. But this should worry you. Is that really a point in physical space?
You could imagine a more extreme example of this experiment in which Fig. 5 shows particle 1 located in Boston and particle 2 located in New York City. This would put their average position within appropriately-named Middletown, Connecticut. (I kid you not; check for yourself.) Would we really want to say that the interference itself is located in Middletown, even though it’s a quiet bystander, unaware of the existence of two correlated particles that lie in opposite directions 90 miles (150 km) away?
After all, the interference appears in the relationship between the particles’ positions in physical space, not in the positions themselves. Its location in the space of possibilities (Fig. 3) is clear. Its location in physical space (Fig. 5) is anything but.
Still, I can imagine you pondering whether it might somehow make sense to assign the interference to poor, unsuspecting Middletown. For that reason, I’m going to make things even worse, and take Middletown out of the middle.
A Second System with No Where
Here’s another system with interference, whose pre-quantum version is shown in Figs. 6a and 6b:
The corresponding wave function is shown in Fig. 7. Now the interference fringes are oriented diagonally the other way compared to Fig. 3. How are we to measure them this time?
The average position 1/2(x1+x2) won’t do; we’ll see nothing interesting there. Instead the fringes are near (x1-x2)=4 — that is, they occur when the particles, no matter where they are in physical space, are at a distance of four units. We therefore expect interference near 1/2(x1-x2)=2. Is it there?
In Fig. 8 I’ve shown the analogue of Figs. 4 and 5, depicting
- the measurements of the two particle positions x1 and x2, along with
- their average 1/2(x1+x2) plotted between them (in yellow)
- (half) their difference 1/2(x1-x2) plotted below them (in green).
That quantity 1/2(x1-x2) is half the horizontal length of the orange line. Hidden in its behavior over many measurements is an interference pattern, seen in the bottom row, where the 1/2(x1-x2) measurements are plotted. [Note also that there is no interference pattern in the measurements of 1/2(x1+x2), in contrast to Fig. 4.]
Now the question of the hour: where is the interference in this case? It is found near 1/2(x1-x2)=2 — but that certainly is not to be identified with a legitimate position in physical space, such as the point x=2.
First of all, making such an identification in Fig. 8 would be like saying that one particle is in New York and the other is in Boston, while the interference is 150 kilometers offshore in the ocean. But second and much worse, I could change Fig. 8 by moving both particles 10 units to the left and repeating the experiment. This would cause x1, x2, and 1/2(x1+x2) in Fig. 8 to all shift left by 10 units, moving them off your computer screen, while leaving 1/2(x1-x2) unchanged at 2. In short, all the orange and blue and yellow points would move out of your view, while the green points would remain exactly where they are. The difference of positions — a distance — is not a position.
If 10 units isn’t enough to convince you, let’s move the two particles to the other side of the Sun, or to the other side of the galaxy. The interference pattern stubbornly remains at 1/2(x1-x2)=2. The interference pattern is in a difference of positions, so it doesn’t care whether the two particles are in France, Antarctica, or Mars.
We can move the particles anywhere in the universe, as long as we take them together with their average distance remaining the same, and the interference pattern remains exactly the same. So there’s no way we can identify the interference as being located at a particular value of x, the coordinate of physical space. Trying to do so creates nonsense.
This is totally unlike interference in water waves and sound waves. That kind of interference happens in a someplace; we can say where the waves are, how big they are at a particular location, and where their peaks and valleys are in physical space. Quantum interference is not at all like this. It’s something more general, more subtle, and more troubling to our intuition.
[By the way, there’s nothing special about the two combinations 1/2(x1+x2) and 1/2(x1-x2), the average or the difference. It’s easy to find systems where the intereference arises in the combination x1+2x2, or 3x1-x2, or any other one you like. In none of these is there a natural way to say “where” the interference is located.]
The Profound Lesson
From these examples, we can begin to learn a central lesson of modern physics, one that a century of experimental and theoretical physics have been teaching us repeatedly, with ever greater subtlety. Imagining reality as many of us are inclined to do, as made of localized objects positioned in and moving through physical space — the one-dimensional x-axis in my simple examples, and the three-dimensional physical space that we take for granted when we specify our latitude, longitude and altitude — is simply not going to work in a quantum universe. The correlations among objects have observable consequences, and those correlations cannot simply be ascribed locations in physical space. To make sense of them, it seems we need to expand our conception of reality.
In the process of recognizing this challenge, we have had to confront the giant, unwieldy space of possibilities, which we can only visualize for a single particle moving in up to three dimensions, or for two or three particles moving in just one dimension. In realistic circumstances, especially those of quantum field theory, the space of possibilities has a huge number of dimensions, rendering it horrendously unimaginable. Whether this gargantuan space should be understood as real — perhaps even more real than physical space — continues to be debated.
Indeed, the lessons of quantum interference are ones that physicists and philosophers have been coping with for a hundred years, and their efforts to make sense of them continue to this day. I hope this series of posts has helped you understand these issues, and to appreciate their depth and difficulty.
Looking ahead, we’ll soon take these lessons, and other lessons from recent posts, back to the double-slit experiment. With fresher, better-informed eyes, we’ll examine its puzzles again.
36 Responses
That was a series well done! I haven’t seen a clearer presentation at that level.
“Whether this gargantuan space should be understood as real — perhaps even more real than physical space — continues to be debated.”
Maybe it is a “me” problem, but I have general problems with concepts like probability being real when they can be formulated in various, non-identical, but useful ways. E.g. frequentist probability (which assigns probabilities to data such as transitions and scattering) and bayesian probability (which assigns probabilities to hypotheses such as wavefunction models from data, say “A size-consistent wave-function ansatz built from statistical analysis of orbital occupations”, V Chuiko, PW Ayers arXiv:2304.10484).
Also, being real is often taking to mean that you get a reaction on an action (“kick a stone and it kicks back”). But as you say elsewhere, wavefunctions can’t hurt you (and you can’t hurt them, you act on state configurations).
I guess my question is: what do they mean with “real” when they say likelihoods (or wavefunctions) should be understood as real? Are there definitions other than the action-reaction one?
Hei Professor,
after reading your posts on quantum interference some doubts have surfaced. When we say a beam of light with a wavelength of 500 nm, what does this imply for the photons?
From your earlier posts I have understood that E = hf and therefore, each photon helping build up that beam of light will have an energy E = h*c/500 nm. Does this mean that a single photon has a feature called wavelength? I am trying to understand if these photons have a size.
If not, when does this distinction from a fixed quanta of energy (photon), to a beam of light with a non-zero wavelength arise? I hope my question is clear.
A photon in a light beam of definite wavelength has the same frequency and wavelength as the beam, but (unlike the beam) has a quantized amplitude and therefore a definite energy, E = h f. So in the case you started with, the photons have a wavelegth of 500 nm. (This is part of why I prefer to call photons “wavicles” rather than “particles”.)
That said, as quantum objects, photons can be in superposition states with many different wavelengths. That’s true of photons emitted from a glowing wire, for instance. By contrast, in a radiative atomic decay, where an atom spits out one photon, the wavelength is quite sharply defined. In annihilation of positronium (an electron/positron atom), it can go either way: if the atom is in a spin-0 state, the two photons from the anihilation have definite wavelength, whereas if it is a spin-1 state the three photons are in a broad superposition of possible wavelengths. (This distinction between decays to two objects versus decays to three or more objects holds more generally for all other kinds of elementary “particles”.)
Thank you Professor.
So lets for brevity assume that we have a beam of light of a specific wavelength. Then the photon has a wavelength of 500 nm. And now for the stupid question, if the size (wavelength) of the photon is 500 nm, it seems a bit weird that it comes out of an excited atom whose size is way smaller. If you can please give some pointers as to how to think of these concepts, it will be helpful. I am learning how light interacts with matter and therefore, have been thinking about this.
**I read that your blogs will touch upon quantum field theory in some time and maybe I will get my answer there.
I will indeed cover this precise issue at some point. But the essential fact is that emitting the photon takes time… of order 500 nm divided by the speed of light, which is the inverse of the light’s frequency. Low frequency radio waves, say at 150 MHz, are emitted by a wire whose current goes back and forth at 150 million times per second. Clearly it can’t form a wave faster than 1/150 millionths of a second. Similarly, an atom that is emitting light at 500 nm has a beat frequency between its initial and final state of order 600 trillion Hz, which means that it takes at leastt 1/600 trillionths of a second to emit any light, consistent with its wavelength of 500 nm.
It’s true that it is very confusing if you imagine the emission is of a tiny dot and that the emission takes place instantaneously. So… don’t. That’s why photons of definite frequency should not be imagined as tiny dots. In this sense, they are wavicles, not particles.
Okay, so now the linear harmonic oscillator model of an atom makes sense, it is oscillating. Thank you Professor.
Regarding Figures 5&8 and their orange dots/lines, how are the orange dot position measurements of Particle 1 and Particle 2 associated to make the individual orange lines? In order to create the interference pattern (Fig 5 yellow dots, F. 8 green dots), do these P1 & P2 position measurements need to be made simultaneously (a relativity problem), sequentially, or randomly. I think I am asking how ½(X1+X2) & ½(X1-X2) are actually calculated?
Let me first answer a bit simplistically, with a caveat at the end.
Crudely speaking, the positions of particles 1 and 2 must be measured at nearly the same time, because the interference pattern only lasts a finite time. The interference pattern will form and disappear only while the two peaks in the wave function are overlapping, and that time period is finite. The two position measurements must be made during that brief time period. The sequence doesn’t matter, and exact simultaneity isn’t necessary either.
As you say, relativity could make this tricky, but the original 1920s quantum physics doesn’t know about relativity, and can’t address this question precisely. I haven’t thought through what we would have to say in quantum field theory. Relativistic 1920s quantum physics (which isn’t entirely consistent) probably has an answer, because there will be a relativistic notion of where the interference is in the space of possibilities; however, I haven’t gone through a calculation of this yet, so I will have to get back to you on this question.
Now, when do we need to calculate 1/2 (x1+x2) or 1/2 (x1-x2)? If we are measuring x1 and x2 at the time of the interference, then any calculation of their combinations can be carried out whenever we like. If we actually have a device that measures the average position (or position difference) without measuring x1 and x2 separately, then that measurement must be carried out during the time period where the interference is occurring.
So the answer to your final question depends on the experimental apparatus used.
But here’s the caveat: what do you mean by “measurement”? Do you mean the moment that the position was recorded by a device, the moment that the device sent a message to a computer so that the result was stored in memory, or the moment that I turned on the computer screen to look at the result? Only the very earliest stage of the measurement must occur when the interference is underway. The other parts of the process can be delayed arbitrarily long… as when we measure properties of supernovas in the early universe whose positions were “measured” by the photons they emitted. (There’s some hint of this in my post on making a measurement permanent.)
I’ll keep thinking and try to improve my answer to your questions; it is important to have clearer answers.
Good answer! Just what I needed.
I am looking forward to the next episode of this tutorial.
Many thanks!
Lost again here: ”And where is the interference? It certainly lies near 1/2(x1+x2)=0. ” Can you please explain how this mathematical relationship of average was arrived at?
This comes from Figure 3 (and is seen experimentally in Figure 5.) In Fig 3, you can see that the fringes lie at a 45 degree angle, and that’s the first clue.
Fringes which can be seen in particle 1’s position, x1, run perpendicular to the x1 axis (see this post, figures 3-11.) Similarly, fringes that can be seen in particle 2’s position x2 run perpendicular to the x2 axis.
But the fringes in Fig 3 run perpendicular to the line which lies at a 45 degree angle to the x and y axes, from lower left to upper right and passing through the origin. This line is the “x1+x2” axis, in the sense that as you move along this axis, x1+x2 changes but x1-x2 remains zero (similarly to how, on the x axis, moving along this axis changes x but leaves y unchanged at zero.) So these fringes will be observable in a measurement of x1+x2. (The 1/2 in front is mainly for convenience; multiplying by a constant doesn’t change the shape of the interference pattern, just how we measure it.)
That helped, thank you!
Hi Matt. I understand the magnitude squared of the wavefunction shown in figure 3 is giving the probability of particle one being at x1 AND particle two simultaneously being at x2. So it is giving the probability of various combinations of positions of particle 1 and particle 2 occurring simultaneously. The interference fringes we see in figure 3 are telling us that there is no chance of finding particle 1 and particle 2 simultaneously at the values of x1 and x2 corresponding to the white areas but there is a high chance of finding particle 1 and particle 2 simultaneously at the values of x1 and x2 corresponding to the dark areas. Have I got that right?
Yes, that’s correct.
Hi Matt. The interference pattern is always occurring in the space of possibilities, not in physical space. When we only have one particle, then physical space is the same as the space of possibilities, so it makes sense to say that interference is occurring in physical space, but not when there is more than one particle.
Matt,
Great job helping me to make picturable (anschaulich) what Heisenberg called unpicturable, or at least a useful analogy!
1. But, going further, I am struck by the apparent coordinance between your “space of possibilities” and Kastner’s “pre-spacetime” in her Possibilist Transactional Interpretation of Quantum Mechanics (The Transactional Interpretation of Quantum Mechanics: A Relativistic Treatment: Kastner, Ruth E.: 9781108830447: Amazon.com: Books). (Not trying to veer into metaphysical interpretation, just observing as you push me to understanding.)
2. More importantly, please point me to where you explain how the imaginary unit “i” comes into the particle wave function. I missed that and cannot readily find it.
1. I don’t know Kastner’s work well enough yet, but the wave function and Schroedinger equation of 1920s quantum mechanics are based on the space of possibilities (or “configuration space”, in the jargon), so one has to work with it no matter what one’s approach might be to interpretation.
I do not know how Kastner addresses quantum field theory, but honestly I would not be terribly interested in her interpretations unless she has explained how her ideas generalize to quantum field theory. (Most ideas about quantum mechanics don’t generalize successfully. At least she’s incorporating relativity… which many authors don’t. But interacting relativistic quantum field theory is much, much more complicated than quantum mechanics, and it organizes “particles” and space very differently from quantum mechanics. As a hint of what is to come, see https://profmattstrassler.com/2025/02/10/elementary-particles-do-not-exist-part-1/ , https://profmattstrassler.com/2025/02/11/elementary-particles-do-not-exist-part-2/)
2. I didn’t explain anything about the “i”. I haven’t been trying to explain where quantum mechanics comes from. Right now I’m only trying to explain how it works.
But in any case, the “i” is something we don’t explain; we observe that to match experiments, we need it. After all, the Schroedinger equation is successful because it allows correct predictions for experiment; if I need an i to make the predictions, then I’ll put it in. (Doing so occurred to Schoedinger because we already had a 19th century wave equation, that of Hamilton and Jacobi, and Schroedinger recognized that the latter could be interpreted as the phase of a complex function — hence the “i”.) There’s nothing weird about this; the wave function is a description of things happening in the physical world, and the rules for using it assures that all the predictions are real quantities even though the function is complex.
So I’m not sure what kind of “explanation” you’re looking for. I can’t explain why the electric field is a vector at each point in space, either; I can only explain how experiments teach me that I need to write math in which it is a vector with three components.
Matt,
Your surmise appears to be correct: Kastner’s Possibilist Transactional Interpretation does advance from QM to relativistic QM but not to quantum field theory. Her four books, and multiple emails, have laid a good foundation for me to move on to your presentation of quantum field theory. BTW: Thanks for clarifying the equivalency of “space of possibilities”, “configuration space”, and “pre-spacetime”.
Your “explanation” of “i” is what I needed. The numbers of quantum field waves are complex. Perchance, is the unlabeled axis of your many graphs imaginary? I think particularly of your 3-dimensional graphs with only X1 & X2 labeled.
No, the “unlabeled” axis is almost always labeled
the square of the absolute value of the wave function [except in a few cases where I plotted the absolute value but not its square]. It certainly is in most of those figures… please let me know if that label is missing somewhere.
Thanks. That makes sense. I missed that.
Matt: As I pursue an understanding of quantum mechanics with your help, I wonder if the necessity of “i” / complex-numbers in quantum mechanics accounts for the probabilistic determination of quantum mechanics? Counter-factually, if quantum mechanics were formulated without “i” / complex-numbers would it simply be discretely determined classical Newtonian mechanics?
No, that’s backwards. We see the probabilistic nature of quantum mechanics, and most importantly, quantum interference affecting those probabilities, in experiments. Then we try to come up with math to describe it. We find the use of complex numbers useful. However, instead of complex numbers in a complex wave function, we could use two real numbers in two real wave functions, and write the Schrodinger equation as two equations. That would match the data too.
Experiment and its data always comes first, and its results are unambiguous. Theory comes later, and since it can always be written in multiple forms, it’s precise details are somewhat arbitrary. Consider the various formulations of Newton’s laws that were invented over the centuries; some don’t even use the word “force”, even though that is central to Newton’s approach.
Hi Matt, great post as always. It has cleared up a lot of the fog that had built up in my understanding. I know we’ve just started taking a look at God’s cards (to quote a well-known book by G. Ghirardi, the G of the GRW theory), but already now I would like, if possible, to know your opinion on the metaphysics of what we’ve learned so far: what is the relationship between the space of possibilities and the physical world? Is it just an instrumental relationship in the sense that it only serves to calculate the probability of what happens in the physical world? I cannot believe that the solution is just ‘shut up and calculate’…Thanks.
I really want to avoid any interpretation and metaphysics until I’ve completed the process of establishing clear understanding. Otherwise any conversation about the meaning of quantum physics will be drowned in misconceptions. We have to do quantum field theory (which, unlike quantum mechanics of the 1920s, agrees with all data) before it makes any sense to attempt interpretation. I cannot tell you how much pointless chatter I have read that centers on the metaphysics of 1920s quantum physics, all while ignoring the fact that the whole thing was superseded 75 years ago by a theory with a difference conception of what elementary “particles” are (specifically, that they are quantized waves, not quantized particles.)
ok, I got it 🙂
In the second paragraph under Figure 8, should “x1, x2, and 1/2(x1-x2) ” be “x1, x2, and 1/2(x1+x2)”? Thanks as always for these posts.
Yes, thanks! Fixed.
“Space of probability.”?
Albert Einstein must be rolling in his grave.
Lost in the math, maybe?
I believe reality is staring at us in the face we are just too blind to see it, yet. I believe gravity (and therefore spacetime) can be quantized and because it’s non-renormalizable doesn’t mean it cannot be done, it means that we don’t have the math, yet, or we cannot use math to characterize it.
Explaining things in one dimension is straight forward, but the real world is multi-dimensional, one that we cannot even visualize the eigenspace let alone the eigenvalues of each “quantum state”.
I believe that the two edges of spacetime have the same mechanism, a black hole. The huge black holes we see in the center of galaxies on one edge and tiny black holes that make up the nucleus of every atom another other edge. from one singularity to another.
Matt,
again extremely well done. Thanks.
First thing that comes to my mind is that ‘location’ in space doesn’t (necessarily) seem to play an essential role in interference phenomena.
Purely mathematically I could imagine two observables (one randomly varying in shape from square to round and another from red to blue, the first being small and growing, the latter being large and shrinking). Some mathematical description of them could show ‘interference’ only when they are approaching the same size : a square shape of one would then never coincide with a blue color of the other. No location, no linear time coordinate, only some correlated variables.
But the physical reality seems to be restricted to :
1./ cases where the 2 observables are ‘of the same type’ like positions on an x-axis
2./ cases where a ‘configuration’ can be realized via more than one alternative route using ‘OR’.
Are both conditions sufficient and necessary for interference to occur?
Good question. Since it’s very precise as a question, I’d better think it through before answering. “Same type” may need better definition; there are a lot of cases I haven’t covered yet that add complexity, such as angular variables where things get mixed up in ways that positions do not. And the word “configuration” needs to be broadened; the space of possibilities doesn’t just refer to positions, of course, which is often what “configuration” means, though it’s open to definition.
Hi Matt Strassler,
I’ve not followed your series closely, but when I took a quick scan of this one, I was puzzled by these extremely cohesive wave… packets?… you show crossing the screen. For the fairly long wavelengths you show when they interfere, shouldn’t those packets look more like spherical waves emanating from the sides of the container?
Sorry, Terry, the posts explain this very carefully. This is not a container; first of all, it is a portion of the space of possibilities, not a physical object, and second, it has no walls (as you can see from the packets that move away).
If the particles were identical would the space of possibilities still have an X1 axis and an X2 axis, or would the geometry of the space of possibilities have to change ?
There would still be an x1 and an x2 axis. The statement that the particles are identical is not that they have the same position but that if I were to exchange them, you couldn’t tell that I had done so. So we still need two positions, and therefore two axes in the space of possibilities. However, the wave function needs to change so that if I switch one particle with the other [i.e. if I switch the x1 axis with the x2 axis] the wave function remains the same.
We’ll do identical particles very carefully soon.