A very curious thing about quantum physics, 1920’s style, is that it can create observable interference patterns that are characteristic of overlapping waves. It’s especially curious because 1920’s quantum physics (“quantum mechanics”) is not a quantum theory of waves. Instead it is a quantum theory of particles — of objects with position and motion (even though one can’t precisely know the position and the motion simultaneously.)
(This is in contrast to quantum field theory of the 1950s, which [in its simplest forms] really is a quantum theory of waves. This distinction is one I’ve touched on, and we’ll go into more depth soon — but not today.)
In 1920s quantum physics, the only wave in sight is the wave function, which is useful in one method for describing the quantum physics of these particles. But the wave function exists outside of physical space, and instead exists in the abstract space of possibilities. So how do we get interference effects that are observable in physical space from waves in a weird, abstract space?
However it works, the apparent similarity between interference in 1920s quantum physics and the interference observed in water waves is misleading. Conceptually speaking, they are quite different. And appreciating this point is essential for comprehending quantum physics, including the famous double slit experiment (which I reviewed here.)
But I don’t want to address the double-slit experiment yet, because it is far more complicated than necessary. The complications obscure what it is really going on. We can make things much easier with a simpler experimental design, one that allows us to visualize all the details, and to explore why and how and where interference occurs and what its impacts are in the real world.
Once we’ve understood this simpler experiment fully, we’ll be able to discard all sorts of misleading and wrong statements about the double-slit experiment, and return to it with much clearer heads. A puzzle will still remain, but its true nature will be far more transparent without the distracting cloud of misguided clutter.
The Incoming Superposition Experiment
We’ve already discussed what can happen to a particle in a superposition of moving to the left or to the right, using a wave function like that in Fig. 1. The particle is outgoing from the center, with equal probability of going in one direction or the other. At each location, the square of the wave function’s absolute value (shown in black) tells us the probability of finding the particle at that location… so we are most likely to find it under one of the two peaks.
But now let’s turn this around; let’s look at a superposition in which the particle is incoming, with a wave function shown in Fig. 2. This is just the time-reversal of the wave function in Fig. 1. (We could create this superposition in a number of ways. I have described one of them previously — but let’s not worry today about how we got here, and keep our attention on what will happen when the two peaks in the wave function meet.)
Important Caution! Despite what you may intuitively guess, the two peaks will not collide and interrupt each others’ motion. Objects that meet in physical space might collide, with significant impact on their motion — or they might pass by each other unscathed. But the peaks in Fig. 2 aren’t objects; the figure is a graph of a probability wave — a wave function — describing a single object. There’s no other object for our single object to collide with, and so it will move steadily and unencumbered at all times.
This is also clear when we use my standard technique of first viewing the system from a pre-quantum point of view, in which case the superposition translates into the two possibilities shown in Fig. 3: either the particle is moving to the right OR it is moving to the left. In neither possibility is there a second object to collide with, so no collision can take place.
The wave function for the particle, Ψ(x1), is a function of the particle’s possible position x1. It changes over time, and to find out how it behaves, we need to solve the famous Schrödinger equation. When we do so, we find Ψ(x1) evolves as depicted in Figs. 4a-4c, in which I’ve shown a close-up of the two peaks in Fig. 2 as they cross paths, using three different visualizations. These are the same three approaches to visualization shown in this post, each of which has its pros and cons; take your pick. [Note that there are no approximations in Fig. 4; it shows an exact solution to the Schrödinger equation.]
The wave function’s most remarkable features are seen at the “moment of crossing,” which is when our pre-quantum system has the particle reaching x=0 in both parts of the superposition (Fig. 5.)
At the exact moment of crossing, the wave function takes the form shown in Figs. 6a-c.
Figure 6c: The absolute-value-squared of the wave function at the crossing moment, indicated in gray scale; larger values of |Ψ(x1)|2 are shown darker, with |Ψ(x1)|2=0 shown in white.
The wiggles in the wave function are a sign of interference. Something is interfering with something else. The pattern superficially resembles that of overlapping ripples in a pond, as in Fig. 7.
If this pattern reminds you of the one seen in the double-slit experiment, that’s for a very good reason. What we have here is a simpler version of exactly the same effect (as briefly discussed here; we’ll return to this soon.)
These wiggles have a consequence. The quantity |Ψ(x1)|2, the absolute-value-squared of the wave function, tells us the probability of finding this one particle at this particular location x1 in the space of possibilities. (|Ψ(x1)|2 is represented as the black curve in Fig. 6a, as the square of the curve in Fig. 6b, and as the gray-scale value shown in Fig. 6c.) If |Ψ(x1)|2 is large at a particular value of x1, there is a substantial probability of measuring the particle to have position x1. Conversely, If |Ψ(x1)|2=0 at a particular value of x1, then we will not find the particle there.
[Note: I have repeated asserted this relationship between the wave function and the probable results of measurements, but we haven’t actually checked that it is true. Stay tuned; we will check it some weeks from now.]
So if we measure the particle’s position x1 at precisely the moment when the wave function looks like Fig. 5, we will never find it at the grid of points where the wave function is zero.
More generally, suppose we repeat this experiment many times in exactly the same way, setting up particle after particle in the initial superposition state of Fig. 2, measuring its position at the moment of crossing, and recording the result of the measurement. Then, since the particles are most probably found where |Ψ(x1)|2 is large and not where it is small, we will find the distribution of measured locations follows the interference pattern in Figs. 6a-6c, but only appearing one particle at a time, as in Fig. 8.
This gradual particle-by-particle appearance of an interference pattern is similar to what is seen in the double-slit experiment; it follows the same rules and has the same conceptual origin. But here everything is so simple that we can address basic questions. Most importantly, in this 1920’s quantum physics context, what is interfering with what, and where, and how?
- Is each particle interfering with itself?
- Is it sometimes acting like a particle and sometimes acting like a wave?
- Is it simultaneously a wave and a particle?
- Is it something in between wave and particle?
- Is each particle interfering with other particles that came before it, and/or with others that will come after it?
- Is the wave function doing the interfering, as a result of the two parts of the superposition for particle 1 meeting in physical space?
- Or is it something else that’s going on?
Well, to approach these questions, let’s use our by now familiar trick of considering two particles rather than one. I’ll set up a scenario and pose a question for you to think about, and in a future post I’ll answer it and start addressing this set of questions.
Checking How Quantum Interference Works
Let’s put a system of two [distinguishable] particles into a superposition state that is roughly a doubling of the one we had before. The superposition again includes two parts. Rather than draw the wave function, I’ll draw the pre-quantum version (see Fig. 3 and compare to Fig. 2.) The pre-quantum version of the quantum system of interest looks like Fig. 9.
Roughly speaking, this is just a doubling of Fig. 3. In one part of the superposition, particles 1 and 2 are traveling to the right, while in the other they travel to the left. To keep things as simple as possible, let’s say
- all particles in all situations travel at the same speed; and
- if particles meet, they just pass through each other (much as photons or neutrinos would), so we don’t have to worry about collisions or any other interactions.
In this scenario, several interesting events happen in quick succession as the top particles move right and the bottom particles move left.
Event 1 (whose pre-quantum version is shown in Fig. 10a): at x=0, particle 1 arrives from the left in the top option and from the right in the bottom option.
Events 2a and 2b: (whose pre-quantum versions is shown in Fig. 10b):
- at x=+1, particle 1 arrives from the left in the top option while particle 2 arrives from the right in the bottom option
- at x=-1, particle 2 arrives from the left in the top option while particle 1 arrives from the right in the bottom option
Event 3 (whose pre-quantum version is shown in Fig. 10c): at x=0, particle 2 arrives from the left in the top option and from the right in the bottom option.
So now, here is The Big Question. In this full quantum version of this set-up, with the full quantum wave function in action, when will we see 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?
And a bonus question: in any events where we see interference, where will the interference occur, and what roughly will it look like? (I.e. will it look like Fig. 6, where we had a simple interference pattern centered around x=0, or will it look somewhat different?)
What’s your vote? Make your educated guesses, silently or in the comments as you prefer. I’ll give you some time to think about it.
25 Responses
I’m on the option 6 bandwagon due to the two “bumps” in the wave function in the 2d configuration space never getting close enough to interfere with each other.
If that’s right, then I think I see how this simple setting demonstrates the surprising features of the double slit experiment. With one particle, the wave function can interfere with itself and produce a fringe pattern. But when that particle is entangled with something else (the second particle in the toy example, a detector at one slit in the double slit experiment), the fringe pattern disappears.
If 6 isn’t right, then I guess I don’t understand after all.
I’m going with option 6, as long as the packet widths are narrow. When I drew the trajectories in pre-quantum 2-d possibility space the two OR trajectories never intersected. (Unless i made a mistake.) Narrow enough 1920’s OR packets should not come close enough to interfere.
If you swap the orange and blue particles for one of the OR paths then I think the paths cross. It would be 10b with the colors lining up. I think I would expect interference at that event.
I’m guessing option 2 because in events 1 and 3 you have particles right on top of each other. This could be bogus because the wave function for the system might be distributed differently than the particles and the particles might be a weird combo (neutrino and a photon) but this is my best guess.
Yeah, don’t worry about neutrino vs photon vs something else. It won’t matter. The only important thing is that the two particles are not of the same exact type. (If they were, you’d have to worry about symmetry between them, and my pictures would be misleading… a point we’ll get to in a few weeks.)
I’m guessing that it would be either 1 or 2. In the single particle case, the wave function had interference at the equivalent to events 1 and 3, so if we’re ignoring collisions those events should show the same pattern.
The only remaining question is whether the pre-quantum position of the particles overlapping *between possibility options* would cause interference in the wave function. I’m leaning towards no, if only because yes seems like a too obvious answer? But also it feels like that interference would only happen if the particles *did* collide, which we’re ignoring the possibility of in this case. But I’m still not sure.
There are four particles in total, so it seems like we might need a 4D configuration space? I’m gonna have to do some doodling to try to figure it out. (As a WAG, I’ll guess all crossing events generate interference since, other than color, the particles seem to be identical.)
The particles *are* distinguishable, so they are not identical; I haven’t even told you they have the same mass, and for my question, they don’t have to. Or they could have the same mass and different charges; for example, one could be an electron and one could be a positron.
Remember, also, that superposition is an OR, not an AND, as I’ve been emphasizing in many of my recent posts.
I’m thinking we will only see interference during 2a and 2b in the pre-quantum system. Maybe this gets more complicated with proper quantum systems, but I’m going with my intuition for now (a very dangerous thing, but oh well)
Unlike photons, neutrinos are fermions. So how is it that they can pass through each other and act like photons?
ray.stefanski@gmail.com
It’s simply that the interactions among photons are tiny, as are the interactions among neutrinos, as are the interactions between neutrinos and photons. They are extraordinarily unlikely to collide (though strictly speaking the probability isn’t exactly zero.) The distinction between fermions versus bosons doesn’t play a role.
If neutrinos only interact via the weak force and gravity, how can they collide with a photon? Why isn’t the cross-section zero?
I would be interested in this as well. I thought particles that did not interact with any common forces cannot interact.
Excellent question. There are numerous cases of particles that do not interact directly but do interact *indirectly* due to quantum effects.
The point is that neutrinos interact with W bosons and electrons via the weak force, and W bosons and electrons interact with photons via the electromagnetic force. This allows neutrinos to scatter off of photons, though at a very tiny rate… but that rate grows rapidly, and when the energy of the two colliding particles each becomes larger than 100 GeV or so, it is no longer small. The diagram shown here
https://i.sstatic.net/4T5uy.png
(as well as one other where the electron and W boson trade places) can allow a photon to scatter off a neutrino, or for a neutrino + anti-neutrino to turn into two photons, or for two photons to turn into a neutrino and anti-neutrino. But I don’t know of a circumstance where these effects are yet measurable.
[These types of Feynman diagrams, with a loop of “virtual particles” (which aren’t particles, and are more general disturbances in a quantum field) are at the core of what makes quantum field theory so different from pre-quantum field theory. I’ll discuss this more later this year, when I explain more about how a real interacting quantum field theory actually works.]
Similar issues allow the Higgs boson to decay to two photons. Photons do not have a mass, unlike W and Z bosons, because they do not directly interact with the Higgs field. However, the photon interacts with top quarks and with W bosons (both of which carry electric charge), and both of them in turn interact with the Higgs field, allowing a Higgs boson (the particle of the Higgs field) to decay to two photons, as in these diagrams: https://www.researchgate.net/profile/Oezer-Oezdal/publication/330116851/figure/fig4/AS:711042095124482@1546537054866/Higgs-boson-decay-to-two-photon-via-top-quark-or-W-boson-loop.ppm
[See also https://profmattstrassler.com/articles-and-posts/the-higgs-particle/the-standard-model-higgs/decays-of-the-standard-model-higgs/ , notably Fig. 3, which shows how the top quark both allows gluon collisions to create Higgs bosons and allows Higgs bosons to decay to gluons or photons.]
These types of indirect effects are very important and are measured in many experiments. [They are one of the easiest ways to spot a mistake in an amateur’s theory of everything, because inevitably they aren’t accounted for.] However, in the case of photons and neutrinos, they are tiny.
The largest effect in one photon scattering off another photon is the direct electromagnetic process photon+photon –> electron + positron, which is observed (and the reverse process is observed all the time.) The largest effect in neutrinos scattering off neutrinos, which is really tiny, is the direct weak-nuclear process neutrino + neutrino –> neutrino + neutrino, but that’s really hard to observe! We do indirectly observe the weak-nuclear process electron + positron –> neutrino + anti-neutrino, so we know the reverse process also occurs, but we don’t directly observe it.
This (and your whole blog and book) was very helpful and interesting. Thank you for your time.
Right, I remember an older post where you explained how, during its travel, a “particle” or wavicle will *mainly* be a ripple in the corresponding field, but some of the energy (and therefore oscillation) will dip in and out of *other* fields; not enough to constitute an actual wavicle, but enough to be of note in a scientific sense. I think this was what you said was the more accurate interpretation of Feynman diagrams of photons “separating” into “virtual” electron-positron pairs before annihilating. Was that what it was?
If so, this could possibly explain how particles that normally do not interact have a chance to do so anyway, if enough of their energy had dipped into a field that *did* normally interact.
Yes, that’s a perfectly good way of saying it. I’ll try to be more precise later in the year, if I can figure out how; we can at least look at a simple example or two.
Because I’m a glutton for punishment, I’m going to do this in public:
I believe the answer is option 6. Because there are two particles, we’re talking about a 2-dimensional phase space which will initially have two peaks, one centred around (-1, -3) and another at (1, 3). These will move through the space such that they never (assuming they’re well localised) really overlap. Therefore no interference.
Is it a phase space?
Ah! No, I guess not. Configuration space would be more correct?
Right — no matter what the answer, we need to be in the configuration space, which is what I called the space of possibilities here.
Hi Matt, my guess: we do not see any interference because you have depicted particles in a physical space where there are no waves that can interfere with each other but only particles.
I would say that interference occurs in possibility space as a ‘dance’ between complex numbers, but in a previous answer you told me that complex numbers are the symptom not the cause, so I am at an impasse. Thanks
Ok, but in Figure 3 I did the same thing — rememeber that’s a pre-quantum version of Fig. 2 — and there was interference. The pictures that describe the example in this final section are pre-quantum pictures of a quantum situation. In that quantum superposition, there is a wave function; I haven’t drawn it, but it does play a central role in the discussion.
If the measurement is of the simultaneous location of both particles then I would have said option 6. But then the bonus question doesn’t seem necessary. So I must have this wrong.
If the measurement is just of one or the other particle then I might guess option 2.
I might be getting confused if whether the two particles are entangled or not, or far more likely I am very confused and need to think more about it.
Well I’ll just guess option 6.