What is really going on in the quantum double-slit experiment? The question raised in this post’s title seems to lie at the heart of the matter. In this experiment, which I recently reviewed here, particles of some sort are aimed, one at a time, at a wall with two slits, and their arrival is recorded on a screen behind the wall. As a parade of particles proceeds, one by one, past the wall, an interference pattern somehow appears, emerging gradually like a spectre on the screen.
Interference is a familiar effect, commonly seen in water waves and sound waves. If water waves passed through a pair of slits in a wall, interference would be observed and no one would be surprised. But here we have one particle passing through the wall at a time; it’s not at all the same thing. How can we explain the interference effect in this case?
It’s natural to imagine that somehow either
- each particle acts like a wave, goes through both slits, and interferes with itself, or
- the quantum wave function that describes each particle (or all the particles [?]) goes through both slits and interferes with itself.
So… which is it? Did the particle go through both slits, or did the wave function?
In 1920s quantum physics, there is a very simple answer to this question.
The answer is,…
“No.“
No — neither the particle nor the wave function [not its wavy pattern or its peak(s) or any other part of it] goes through the two slits.
- What!? Then how can there be interference?
That question I will answer in a later post, probably next week or the following. But first, let’s confront the title of this post in a simpler context, so that we can see clearly why — in 1920s quantum physics — the answer to its question is “neither one”.
The Double Door Experiment
Key to understanding the double-slit experiment is to simplify it down to its bare essence. Having a two-dimensional problem where particles are going through slits in a wall is more complicated than necessary. Instead, let’s take a one-dimensional problem that we’ve already looked at, where an object is in a superposition state of going to the left OR going to the right. We already saw that this object is not to be viewed as both going to the left AND going to the right. By setting up measurements on both sides, we saw that it can only be measured to be doing one or the other, and never both. Superposition is an OR, not an AND. And a true particle can only have one position at a time.
The Particle and the Doors: A First Look
In this context, let’s ask the question: can a particle simultaneously go through two doors on opposite sides of a room? This is the same question as the two-slit question, because I can turn one into the other using tubes behind the slits, as in Fig. 1.
By sending a particle toward two slits, we can arrange for its wave function to be in the superposition state we want (Fig. 2)
and then we can ask whether we can observe it going through both doors.
Well, in a recent post we put two balls in the same locations that we now want to place the doors, and we asked if a particle in this very same superposition state can hit both balls simultaneously. The answer was “no”. The same argument applies here; the particle cannot be observed to pass through both doors simultaneously. It can only go through one or the other.
Why? A particle, which has a position and a momentum (even if unknown to us), cannot have two positions. If it starts between the two doors, it can move through one door or the other, but it cannot do both, because then it would have two positions at once.
Maybe you’re not immediately convinced. If not, stay tuned, as I’ll come back to this again later.
The Wave Function and the Doors: A First Look
But for now, let’s turn to the other question arising from my post’s title. Why can’t the particle’s wave function move through both doors, just like a water wave or sound wave does?
Actually, since wave functions don’t move (they just describe particles that do) what we really want to know is slightly different. The initial wave function has a wavy pattern; does this wavy pattern go through both doors?
Certainly water waves and sound waves could go through both doors. They are waves in physical space. So are the doors (or slits) they they can pass through.
But the wave function is a wave in the space of possibilities, and not in physical space. Conversely, the doors do not exist in the space of probabilities; doors are physical objects. Therefore the wave function (and its wavy patterns) cannot pass through the doors at all!
The very idea is nonsensical, the sort of thing that René Magritte would have enjoyed painting. Having the wave function (or its pattern) pass through physical doors would be akin to you entering into Shakespeare’s Romeo and Juliet to save the lovers from their fates, or enjoying the taste of an apple painted by Rembrandt, or walking through a giant hole in a physicist’s argument. Physical space and the space of possibilities are conceptually different; they have distinct meanings. At best one space merely represents what is happening in the other. And so the objects that exist in one don’t exist as objects in the other. (Even when these spaces have the same shape, which sometimes they do, they represent different things, as indicated in the fact that they have different axes.)
To convince you further of these statements, let’s take a look at a simpler example. Consider a system where there is just a single door, but there are two particles. Let’s see why the wave function of these particles (and its wave pattern) can’t even pass through one door, much less two.
The Wave Function of Two Particles and a Single Door
We’ll put the door on the right in physical space. Superposition states aren’t needed here, so instead we’ll send both particles rightward toward the door, in simple wave-packet states. These two particles will be given the same near-definite momentum, but their poorly known positions are shifted apart, so that they are separated in physical space. In pre-quantum language, the set-up is shown in Fig. 3.
What wave function do we need to describe this? It’s simplest to put the two particles in wave packet states with near-definite momentum, somewhat separated in space but with similar motion. You might first think the wave function for such a system would roughly look like Fig. 4:
But no! That’s a trap that’s super-easy to fall into; such a wave function describes one particle in a superposition of two locations, not two particles.
Instead, because the first particle has position x1 and the second has position x2 respectively, their wave function, a function that exists in the two dimensional space of possibilities with axes x1 and x2, takes the form ψ(x1,x2). If we start with x1 near 2 and x2 near 0, as in Fig. 3, then the wave function for these two particles looks like Fig. 5: it has a peak near x1=2 and x2=0. (The dashed black lines are just there to guide your eyes.) The peak indicates that x1 near 2 and x2 near 0 are the most probable values for the two particles’ positions.
Now, where’s the door that we want to try to make the wave function go through? Great question. In physical space it is located at x=+4. Let’s now draw that door in the space of possibilities, whose axes are x1 and x2. How should we do that?
Think it over…
Unsure? Confused?
That’s fine; there’s no reason not to be confused the first time you think about it. Here’s the answer.
Particle 1 goes through the door when x1=+4, which is the vertical blue dashed line in Fig. 6. Meanwhile, particle 2 goes through it when x2=+4, so that’s the horizontal blue dashed line.
Does that look like a door to you? Certainly it doesn’t look like the door in physical space. And that’s because in the space of possibilities, the dashed lines are not a door, with mass and thickiness and a material make-up. Instead the lines represent a certain set of possibilities, namely that one of the particles is at the location of the physical door.
In fact, there’s a special point, the intersection at x1=x2=4 where the lines cross, that represents the possibility that both particles are simultaneously at the location of the door. No such intersection of lines exists in physical space. This is an intersection of two classes of possibilities, and such a thing can only exist in the space of possibilities.
The lines divide the space into four regions, representing four more general classes of possibilities, shown in Fig. 7. In the lower left region, both particles are to the left of the door. At far right, particle 1 is to the right of the door while particle 2 is to the left; the reverse is true in the upper left region. Finally, at upper right is the region where both particles are to the right of the door.
The wave function’s pattern moves in the two dimensions that are spanned by the x1 and x2. axes. Both particles are moving to the right in physical space, at approximately the same speed. Consequently, the wave function, as it evolves, carries the most probable state of the system across three of the regions:
- initially both particles are to the left of the door
- then particle 1 is to the right of the door while particle 2 remains to the left
- and finally both particles are to the right of the door.
In pre-quantum physics the path traversed by the two-particle system would look like Fig. 8:
The wave function evolves as shown in Fig. 9, very similarly to Fig. 8.
Now, is the wave function going through the door? Again, there is no door here; there are simply lines that tell us when one particle is coincident with the door, as well as a point where both particles are coincident with the door. It’s true that the wavy pattern and the peak of the wave function (which indicates the possibilities where the system is most likely to be found are passing across the lines. But can we say the wave function (or its wavy pattern) passes through the lines?
No: moving through a door involves moving in physical space, along the x-axis, through a gap in a door frame. This is something the wave function does not — cannot — do.
The Wave Function of Two Particles and Two Doors
If you’re still not yet entirely convinced, consider what happens if we have two doors, one on each side. Let’s put our two particles each in a superposition state of moving leftward or moving rightward. Again we’ll set one particle off before the other, so that the first will reach the doors well before the second does.
Our pre-quantum view of such as system is that it now has four possibilities: particle 1 can be going right or left, and particle 2 can be going right or left, as shown in Fig. 10. Only the upper-right option appeared in Fig. 3.
This requires a wave function that initially looks like Fig. 11, with four peaks, one for each general possibility sketched in Fig. 10.
But where are the two doors? They appear in the space of possibilities on four lines; in addition to the blue lines we had in Figs. 6-9, corresponding to particles 1 or 2 being at the righthand door, we now have two more, shown in green in Fig. 12, corresponding to one or the other particle being at the lefthand door.
Notice the two intersections between the blue and green lines! What are these?! No such intersections between the doors can possibly occur in physical space. So this makes it even clearer that these lines cannot be identified with the doors.
What do these two intersections actually represent? The upper left intersection combines the possibility that particle 1 is at the left door and simultaneously particle 2 is at the right door. It’s the other way around at the lower right intersection.
Note there is no intersection, or any point at all, corresponding to particle 1 being at the left door and simultaneously being at the right door. Indeed, such a point would have x1=-4 AND x1=+4, which is impossible; every point in the space of possibilities has a unique value of x1.
How does the wave function behave over time? It behaves as shown in Fig. 13:
What are the four peaks, and what are they doing? They correspond to the four possibilities shown in Fig. 10. Clockwise from the rightmost peak (which appeared in Fig. 9,)
- particle 1 goes through the right door, followed by particle 2.
- particle 1 goes through the right door, after which particle 2 goes through the left door.
- particle 1 goes through the left door, followed by particle 2.
- particle 1 goes through the left door, after which particle 2 goes through the right door.
You can tell which is which by looking at which colored line is crossed first, and which is crossed second, by each peak.
Any one of these four things may happen. Two, or more, may not. And not one of them includes the possibility that either particle goes through both doors simultaneously.
Variation on the Theme
As another instructive variation, suppose we send particles 1 and 2 off at exactly the same moment. Then the wave function, shown in Fig. 13, looks very similar to Fig. 12, except that every peak goes through an intersection of the dashed lines, because the two particles arrive at the doors simultaneously.
Again there are four possibilities,
- both particles can arrive at the right door simultaneously,
- both can arrive at the left door simultaneously,
- particle 1 can arrive at the left door just as the particle 2 arrives at the right door
- or vice versa.
And so it certainly is possible for particle 1 to go through the left door and particle 2 to go through the right door simultaneously. No problem with that.
However, it is simply impossible — illogical, in fact — for particle 1 to go through both doors simultaneously. There’s no spot in the space of possibilities that could even represent that inconsistent scenario.
Lessons for the Double Slit Experiment
The lesson? Many fascinating things happen in quantum physics once we have superpositions and multiple particles and multiple doors. But two things that cannot happen (in 1920’s quantum physics) include the following:
- No particle can go through two doors at once; it can have only one position at a time.
- No wave function, living as it does in the space of possibilities, can pass through any door in physical space.
The same applies for the two slits in the quantum double-slit experiment. One cannot make sense of that experiment without learning this lesson: from the viewpoint of 1920’s quantum physics, the interference effects do not arise from any object, whether particle or wave function or pattern in the wave function, physically moving through the physical slits in physical space.
So where do the interference effects come from? The wave function’s pattern can travel across regions of possibility space that are associated with the slits. We need to work out the consequences of this observation, and interpret it properly. Stay tuned to this channel; the answer is near at hand.
73 Responses
Apologies for asking a question which seems to have a very obvious answer.
In the section ”The Wave Function of Two Particles and Two Doors”: there are 4 scenarios presented. And at the outset you assume that each particle is in a superposition of moving either to the left or to the right.
To make my question clear, let ”+” denotes particle moving to the right and ”-” denotes particle moving to the left.
Therefore, I can write a superposition of these particles as (2+, 0+) OR (2+, 0-) OR (2-, 0+) OR (2-, 0-). But in Fig. 10, we have particles towards the left of the origin as well (negative axis). I don’t see why that is needed, taking cue from Fig. 3.
What am I doing wrong here if you could please help?
The wave packets have approximate positions and approximate momenta, so the correct superposition can better be written as
(2 +, 0.3 +) OR (2 +, -0.3 -) OR (-2 -, 0.3 +) OR (-2 -, -0.3 -)
where the first number gives the purple particle’s approximate position, the following sign gives us the direction of its motion, the second number gives us the blue particle’s approximate position, and the final sign gives us the direction of its motion.
If you are asking: why didn’t I choose
(2 +, 0 +) OR (2 +, 0 -) OR (-2 -, 0 +) OR (-2 -, 0 -)
or
(2 +, 0 +) OR (2 +, 0 -) OR (2 -, 0 +) OR (2 -, 0 -)
the answer is that I was most definitely making a choice. These two other superpositions would have been perfectly consistent; there are many other possible choices, all legitimate. My motivation in this choice is that I wanted Figure 12 to be easy to interpret, and this choice made that easiest: it gave the wave function four independent and symmetrically located peaks, moving in a symmetric way that was relatively easy to understand.
Does that help?
sure makes sense. Thank you Professor for your help.
Also, can I have a superposition state: (2 +, 0 +) OR (2 +, 0 -) OR (2-, 0 +) OR (2 -, 0 -) OR (-2+, 0+) OR (-2+, 0-) OR (-2-, 0+) OR (-2-, 0-) ?
Is this a valid state?
Yes, though as things become more complicated we really need to deal with the fact that here we are summing up products of wave packets without saying precisely what they are. I’ve been cavalier about explaining this because it hasn’t mattered much in my examples (although I’ve been precise in my calculations.) A sufficiently complicated sum of wave packets will stop acting like a simple superposition of independent wave packets and will start displaying other features of interference and other phenomena.
thank you Professor for your reply.
Thank you for this wonderful article. I have a query: why is that x1 and x2 in the space of possibilities orthogonal? At least that is how I see from the figures.
May I suggest the following two articles?
https://profmattstrassler.com/articles-and-posts/quantum-basics/physical-space-and-the-space-of-possibilities-a-crucial-distinction/
https://profmattstrassler.com/articles-and-posts/quantum-basics/physical-space-and-the-space-of-possibilities-a-crucial-distinction/understanding-the-space-of-possibilities-an-example/
If, after reading those articles, you are still confused on this point, please come back and ask your question again.
Thank you. However, I am sorry but I am still confused. I understand from the articles that the space of possibilities is an abstract space. And because it is an abstract space we draw x1 and x2 axes in the space of possibilities mutually perpendicular?
But in my mind, I can very well draw these axes in any orientation. Say x1 makes an angle 30 degrees with x2 in the space of possibilities. Can you please suggest where I am getting it wrong?
Okay, I *think* I understand your question. This is a bit tricky to explain quickly.
There are two different issues. How do we draw the axes, a matter of convenience, and what is the actual geometry of the space — the more physical question of how “close” are different possibilities to one another?
Here’s an animation showing how we might take the original x1, x2 grid where the axes are perpendicular, and tilt it. There are two specific possibilities picked out, marked with blue dots. https://profmattstrassler.com/wp-content/uploads/2025/03/LeaningGrid.gif
In one sense, nothing is happening as the axes are tilted. All I’ve done is draw the axes in a funny way. The locations of the two possibilities, in terms of their coordinates (x1, x2), have not changed. This is no different from taking a graph and looking at it through a lens that distorts it; it may look different, but it’s actually the same. From this point of view, taking the x1 and x2 axes to be perpendicular is a matter of convenience.
However, there’s a more subtle question we can ask. It appears that the distance between the two blue points is shrinking as the axes tilt. The question is: is it actually changing or not?
That is not a question of how we draw the axes, and what the angle between them might be on the page. That is the question: how do we measure distances?
For instance, do we measure the distance L between the two points using the Pythagorean formula L^2=(x1-x1′)^2+(x2-x2′)^2? (Here x1 is the horizontal coordinate of the first dot and x1′ is the coordinate of the second dot.) Even though the points change position on the screen, the values of x1,x2 and of x1′,x2′ do not change. The axes tilt and the points move in their location on the screen, but their coordinates relative to the tilted axes do not change, as the grid makes clear: each point remains within its grid box. So if this is how we measure distance, then the points are the same distance apart no matter what tilt we apply to the axes. And so, tilting the axes really is just a redrawing of the original picture.
If, however, we measure the distance between the points by taking a ruler and applying it to the screen, then we are not using the tilted axes to measure distance but instead are using up-down and right-left axes to measure distance, even after the axes on the screen are tilted. In this case the distance between the points really is changing.
This question of how to measure distances, and what we are holding fixed as we tilt the axes, involves defining a “metric” on a space. Until we define a metric, we have no way to measure distances.
I have not told you how to define that metric. However, in all the problems we have been working on so far, the metric really is the simple, Pythagorean one when the axes are perpendicular, and distances defined through that metric with perpendicular axes do not change when we tilt the axes. In other words, we can draw the axes however we like, but it is most convenient to draw them perpendicular, because then the metric is simple and distances can be read off intuitively by eye.
The reason the metric is so simple is that the Schrodinger equation says that rate of change of the wave function is proportional to (d^2/dx1^2 + d^2/dx2^2) of the wave function [let me know if you don’t know calculus and I’ll try to explain that expression more clearly.] The fact that there is no d^2/(dx1 dx2) term in that expression tells us that we have a very simple metric when we use perpendicular axes.
So that fact is implicit in what I’ve been doing here. If I leave that metric the same (in terms of the coordinates) as I tilt the axes, then all the distances remain the same as the axes are tilted and we can draw things however we like. If we changed the metric as we tilt the axes, then distances between possibilities really would change; but then I’d be changing the Schrodinger equation as I tilt the axes, which would mean I’d be changing the physics problem that we are studying.
I hope this helps clarify the issues somewhat. Feel free to follow up.
Thank you so much Professor for your patience in explaining all the basic stuff. It helped me.
The Schroedinger equation part, I will not say that I understood. My knowledge in calculus is minimal.
especially the absence of the term (d2/dx1 dx2) and its connection with perpendicular axes.
Ok, without calculus, the easiest way to say it is that the kinetic energy KE (which determines how possibilities evolve with time) can be written in terms of the momentum of particle 1, p1, and the momentum of particle 2, p2, as
The absence of any term proportional to
in this expression (and the absence of potentially more complex forms that depend on the positions of the particles too) assures that the metric is simple: it gives us Pythagorean distances when we use simple perpendicular (“Cartesian”) coordinates.
Thank you for your help. Much appreciated.
The Holo in the Phycists Non Holomorphic Magritte:
“Figure 5: Graph of the initial wave function of our system, with particle 1 located near x1=2 and particle 2 located near x2=0. The dashed lines are there to guide the eye. The vertical axis is the absolute value of the wave function, whose square at a certain point gives the probability for the corresponding possibility. The colors represent the wave function’s complex argument [or “phase”].
Thus the gnomon plus a ratio is the probability argument and pertains as well to the way acceleration corresponds to an inverse product meaning the diagonal placed at end or side of square as a ratio conform or projections.
The diagram gives the complex expression…. In two ways , the block mass as excursion or diversion mappable in end to equations of motion s to be sure but also the distance between quanta as also the complex argument like spaces between marks in a drawing as in Durer for example…quanta here a term purposely being used rather loosely to mean quasi smooth intervals of shall we say berms and wales…
Speaking of travail and travel through the holes of an argument :Magritte’s” This Is not an apple” is part rebus e.g. (this is not an appellation) and part noema or hyper complex riddle in the sense that the suspended apple as an elevation representing it’s norm (hanging from tree in quasi suspension but is in a Galilean enclosure most likely an Einstein elevator…thus it is possibly suspended in free fall which term brings the riddle of the hierarchy puzzle… given the manifold of vacua as a topological descent or ascent relative dimensional tunnelings posed to symmetry collapses or parity violation and vacuum nucleation which one are we in?
This in turn raises the question as to how such things as quasi particles as define a physical world comport to something of wave function analog where the particle , like the apple has no idea of its borrowing or burrowing into another dimension..not knowing what floor it is or is not on and is suspended in an indirect function which is quasi physical or as Magritte would have it “surreal” Surreal and Hyper Real mathematics are in fact formulations… Surreal deals with a unity model whereas Hyper real turns to relating logic to mathematics…These realizations enlivening much of the input of Duchamp for example who had considerable influence on Penrose to say the least.
So; to summarize or build a coast: the pattern of recursive form which mirrors between the Higgs construct and the vaccua can be taken a step further into analyzing the language forms which direct in mathematics and physics the concept if invariance and dependence over fields and their fields of thought.
At figure six the gnomon structure door is a position where a centrifugal spiral formed in unity 1 smaller square)has diagonal on which unity two (the larger square) would be centripetal…therefore building a complex term on that diagonal could state that like a Hamiltonian the door is only half visible and symmetries in parity violation are the projection of mass where then the door is not the opening but the dimensional aspect complete to the three dimensional apparition. This can only be distracted to a door by framing the door entirely to a particle state which can only enter with its mass intact thus its definition migrates from complex to real.. the reality of the particle then flattens the door to let it in and the particle itself is the door of its perception in which physical dimensionality of a door becomes the mass of the particle and so one could give an experimental hypothesis that the mass of the particle could frame the door itself to a ripple quanta reflective of the concept of wave function if tme itself is experimentally given a kind of added dimension… the idea is not generally supported but it remains suggestive within the overall all argument which confines wave fx and wavicle to Pauli non Pauli versions…
My adhoc progression is not so different from your argument up to figure 13… where it gets really interesting to essentially exploit 4 pi space as far as one can follow it in graph form which is precisely three dimensional space then modified in more symbolic forms to higher dimensions. The active centripetal and centrifugal clock over dimension seems underlying status of spin in phase state translations.
Hey, Matt. I’m afraid I don’t like your 1-D ‘door’ explanation because there isn’t a door there, even in physical space/representation. Where is it? There are no ‘sides’ in a 1-D system to mark the ‘door’ — you can just have a wall at some x position, or not.
A door(way)/slit to be at all meaningful has to put some restriction on the possible paths/configurations/… that are allowed. A physical-space slit (in two or higher dimensions) has corresponding allowed & disallowed points in the space of possibilites, so in that sense, the slit *does* exist there.
Bill, the point here is that the physical door plays no actual role. Nor do the physical slits. The sole role of the slits is to create a superposition of a particle going through the left slit and the particle going through the right slit. We could get exactly the same result if we had no slits and instead had tubes or simply some magnetic fields that guide the particles. Focusing on the slits, or on the doors, is a distraction, and it makes the problem much harder to understand than it actually is.
Now, you are welcome to put in the full two-dimensional door and work things out in the four-dimensional space of possibilities for the two particles moving in four dimensions. You will get exactly the same answers, except a lot more complicated to work out and impossible to draw. You will learn nothing new from doing so, but I advise you to try.
Oh, and by the way, we will see all the effects seen in the double-slit case — including the fact that if you measure the particle’s position, the interference disappears (unless you do exactly the right experiment, and then it doesn’t) — using this one-dimensional example. So complain again at the end of this series of posts, if you still don’t see why I’m taking this route.
yet to do more than skim it but i would argue that of course a particle can be in two places at once, e.g. it could be 1m away or 2m away. of course it depends on your frame of reference. with the gallilean transformation ofc we can unify these into a single transformation. but (ignoring einstein) c ofc is different and a different frame of reference that will become separated from your material reference the more it moves. and ofc relative motion is necessary to create waves of light, so naturally you would expect different particles with different motion to have a different idea of c relative to matter, hence it is in 2 places at once wrt 2 particle motion paths.
In physics, statements are always implicitly made in one frame of reference at a time, using one set of coordinates at a time.
If I was allowed to consider all possible clocks, then I could say that all events take place at all possible times. This is not what we are talking about here.
So no, a particle cannot be two places at once. The coordinates you assign to the particle depend on your reference frame — on your coordinate system for space and time — but in any chosen coordinate system, the particle is in one position only.
And no, particles do not have a different idea of “c”; Einstein’s point is that space and time are such as to assure that “c” is reference-frame-independent.
It’s rare to read such a lucid text about how quantum mechanics is actually a statistical science! Great understanding!
Only a very recent study “dared” to also consider that entangled states do not have to arise in a common event…
thanks for the reply. i believe c=time/distance was already known to be reference independent before einstein and the received wisdom is that his theory implies that two different times can occur at once, e.g. 1 second and 2 seconds from some point. personally i have found the view that two different distances can occur at once to be sometimes helpful in some very elementary level physics. i doubt either is right but rather like newton rough approximations
if unlike me, you are familiar with quantum mechanics, and want to reconcile a particle being in two places at once with a common interpretation of probability waves, it might help to think of a particle-motion-path as an “observation,” thus electromagnetic phenomena is dependent on the particle doing the “observation,” and being composed of multiple particles with different motions due to heat, our view of such phenomena necessarily takes a statistical character
as einstein wrote, “before the advent of the theory of relativity it had always been assumed in physics that the statement of time had and absolute significance, i.e. that it is independent of the state of motion of the body of reference.” of course he formed his theory before the advent of quantum mechanics, in light of which it is reasonable to ask if his assumption that position has and absolute significance with respect to the state of motion of the body of reference was perhaps mistaken
oops, his belief was that position was independent of the motion of the reference. i would argue this is incorrect, just as einstein argues the same but for time
oops, einstein recognized that distance was dependent on relative motion but perhaps not past relative motion which is my point
anyway my point is if you assume the position of light in space is dependent on the motion-path of the “observing” particle, einsteins motivation for the lorentz transform disappears whilst measurement of anything travelling at c takes on a statistical nature due to heat etc, and it is very easy to calculate parallax within observational error. not saying its right, but worth thinking about
if you were to insist on an einsteinian analogy, you could say each particle, having its own motion and being its own reference, observes distances differently, and hence observations of distance depend on the particles and their motion which like temperature is a statistical measure
so take einsteins example of two lightning flashes at A and B. one point M observes them to happen at the same time and that they are at the same distance. another point N on a train is observed by M to be at M when AB flash, moving toward B. einstein predicts that N will observe B to flash before A, and so says that N observes B to flash before A. you can thus begin to see that objective truth regarding c depends on motion, and so given observers like us composed of particles jiggling and spinning physical truth becomes statistical. einsteins prediction flouts the principle of relativity but nonetheless some statistical nature necessarily arises because of the contradictory principles of relativity and c, the accuracy of einsteins prediction only affecting the exact rules it obeys
Not expecting an expert answer here, and I am a philosophy enthusiast not a physicist.
I have just always had a problem with the inserting of consciousness into this problem, I know you have not done that here, but I just thought maybe you might have an opinion.
I have seen some descriptions of the double split experiment that say that when the particle is measured before passing through the slits it does not display the wave pattern, and when it isn’t measured before entering the split it displays that wave pattern.
I have always thought it was just intuitive that the “particle” collapses into a particle when measured, so when it is measured before the slits it has a definite placement in space and so only goes through one slit and so the wave pattern is not what we see, we see the random dispersement.
But when it is measured only after the slits it can collapse into any space described by the self-interfering wave function, so the wave pattern is what we see in this scenario.
So I would think this collapse happens because of some physical causing of the wave function to collapse by the method of measurement.
Do you have any insight into this? If not feel free to ignore.
Explaining how this works is indeed one of the important goals of this set of articles. Using the set of tools that I have been building up for readers, we will see this play out in explicit examples, within two or three weeks; and we’ll even see hints of it within a week. We’ll see the interference pattern appear and disappear in many different cases, and we’ll look at why this happens. If my explanations don’t entirely answer your question within a month, please re-ask it.
I’m not a physicist, but I did take a few years of physics classes in college. I can report that the only absolute certainty is that your current teacher will tell you that all of your previous assumptions and models regarding the Two Slit Experiment are wrong. It doesn’t matter if your last professor won the Nobel Prize. Their explanation simply doesn’t hold water, and this new one is the correct one! Repeat ad infinitum.
Maybe so. But am am not interpreting the double-slit experiment, unlike most people. I am showing you explicitly how the Schroedinger equation actually works. You will see that many previous assumptions/claims are just plain inconsistent with the Schroedinger equation. How you interpret the Schroedinger equation is up to you, but it is important not to make claims that are in contradiction to the mathematics.
And yes, even Nobel Prize winners have been wrong. That’s necessarily true, since some of them have contradicted each other — meaning that they cannot all have been right.
But there will be no assumptions or models or wordy explanations here. There will simply be illustrative and well-chosen solutions to the Schroedinger equation… and the rest is up to you.
I’ve been enjoying these posts very much. One thing I’m struggling with is this. What exists in real space is a quantum field. To calculate what the field does – and hence what we observe – requires construction of a wavefunction from the field distribution which only exists in configuration space. The wavefunction is therefore not “real” but a computational device that provides only statistical results. Is the statistical nature of the predicted outcomes purely a result of the quantum field distribution being indefinite as a result of Heisenberg uncertainty? Or does uncertainty come in from somewhere else?
No, the logic goes a different way. And it’s complicated and controversial.
First, not everyone agrees the wave function is not real. For instance, the many-worlds interpretation asserts that it is real. I don’t have a firm point of view on this, though I’ll have to discuss this question quite soon.
As for uncertainty — I would say that Heisenberg’s uncertainty is really a consequence, not a cause. The uncertainty arises from the fact that the wave function (in 1920s quantum physics) can be written as a function of positions OR of momenta, not both; and therefore the function can only specify one or the other precisely. And in field theory the wave function can be written as a function of field shapes OR of changes in field shapes, but not both, with a similar effect. A limit on uncertainty follows from this fact. In some sense I touched on this in these two posts: https://profmattstrassler.com/2025/02/10/elementary-particles-do-not-exist-part-1/ and https://profmattstrassler.com/2025/02/11/elementary-particles-do-not-exist-part-2/ .
But ultimately, I think people do not agree about the right way or best way to organize the logic. Why do we need a probabilistic theory, governed by a wave function that is a function of only half the variables needed in pre-quantum physics? We know that this works by comparing with experiment, but there’s no principled explanation about why our world’s nature is quantum. And so it can be hard to say what is principle, what is cause, and what is consequence. (This can be hard to do in classical physics too; it can depend on what formulation you use — i.e., which laws are principles and which ones are derived from those principles may depend on your assumptions and starting point.)
So I’m afraid I don’t have a great answer for you — at least not yet. Maybe I’ll recognize one as I put these posts together.
Veritasium had a recent video going over the relationship between the principle of Action, the Planck constant and the double slit experiment. Would love to read your opinion on that!
Way too early. Maybe at some point.
Thank you for this explanation. I got lost at Figure 5. I’m confused because a wave function is described as existing in a *two-dimensional* space but Figure 5 is a 3-dimenional diagram. Can you elaborate on the diagram more? I apologize if I’m missing something obvious.
Yes: Figure 3 shows the wave function’s absolute value squared |ψ|^2 as a function of the two coordinates x1 and x2, which are the coordinates of the space of possibilities.
This is analogous to the usual graph of a function that we might do in pre-college math: where a function y(x) depends on one coordinate, but is graphed using a curve in two dimensions. Here we have a function |ψ(x1, x2)|^2 that depends on two coordinates, but we graph it using a surface in three dimensions.
The labels on the axes are supposedly there to help you interpret what I’m doing, but I realize that sometimes I leave out steps (especially if I covered those steps in earlier posts in this series.) It’s quite tricky to try to keep things sufficiently self-contained in a subject that has so many layers. I’m sure you’re not the only one who had this question, so thanks for asking it.
What makes me skeptic about this “possibility” vs “physical” space distinction is the fact that everything is quantum. When you say “doors exist in the physical space” it’s almost like you’re ignoring the fact that they’re made of particles, which are nothing more than perturbations in their own fields or wave function. It’s not like doors are static, macro, physical things.
But I’m probably wrong so just forgive my ignorance 🙂
You’re right to be keeping track of this issue carefully. But of course I am well aware of the issue, and would not have started explaining all of these things in this way if I did not know that including the doors in the wave function — which would make the space of possibilities far too large to draw — would not change the conceptual result. In fact, in this case, it would not change the result at all.
If you notice, I didn’t really use the doors at all; I didn’t tell you what they were made of, how wide they are, etc. That’s because the only thing that actually matters in this example is the overall shape of the physical space in which each particle can move. Each door in this case is just a placeholder for a point along the x axis; it doesn’t actually interact with the object passing through it. And that is why I can leave it out of the wave function. [This is not actually true of the slits in the double-slit experiment, or rather, of the wall with the slits, whose atoms literally do obstruct the incoming particles; that’s part of why that experiment is more complicated than necessary, in an way that’s irrelevant to the quantum interference effect of interest.]
If we did include the doors, the result, given the way I’ve set up the experiment, is that the full wave function, including 2 particles on the x axis and the doors, would take a simple form of a product of three sub-wave-functions:
Psi_{full} = Psi_{particles}(x1,x2) Psi_door1(y1,y2,….yN)Psi_door2(z1,z2,….zN)
where x1 and x2 are the positions of the particles on the line, y_i is the position (in three dimensions) of particle i in one door, z_i the position of particles in the other door, and N is the total number of particles in each door. N is something larger than 10^{29}. This product form will remain unchanged by the Schroedinger equation because the doors and the two particles on the line do not interact — meaning that the doors and the two particles are always uncorrelated. The result is that the probability for the two particles to be in some state X AND door2 to be in some quantum state Y AND door1 to be in some quantum state Z is just the product of the possibilities, p_full = p_X p_Y p_Z. Therefore, even though the space of possibilities now has 6N + 2 dimensions, which is spectacularly gigantic, it can be treated as three separate spaces of possibilities, two for the particles in the two doors, which have 10^29 dimensions or more and cannot be drawn — but in any case do not have any influence on the two particles on the line — and a 2-dimensional space of possibilities, which is the relevant one for the two particles, and is the one I drew. The motion of the system through time across the full space of possibilities is given by three independent motions: (1) the part of the Schroedinger equation that acts on Psi_particles, which I kept track of, and (2) a much more complicated but completely irrelevant motion of the system of door 1; (3) a similar complicated motion for door 2.
So hidden behind *all* of my examples is a thought process such as this one, where I carefully justify the choices I make.
Now, what would happen if we really treated the wall and its openings properly? I have already given you examples of how measurements are made — i.e., of how to include the measurement device in the wave function rather than having it sit outside it. Here it really *should* be included, because every measurement device interacts with the object it measures. See https://profmattstrassler.com/2025/02/27/what-is-a-measurement/ and https://profmattstrassler.com/2025/03/03/making-a-measurement-permanent/ .
Of course a real measurement device has, again, 10^{many} particles, and so I cannot draw it. But I have employed measurement devices with one or two particles in my examples in order to illustrate the conceptual points about how to string these concepts together in possibility space, so that you can understand how a full, realistic measurement device would have to work in the undepictable giant spaces of possibilities of ordinary life. You will need to do some thinking about how to use my examples to infer the larger situation; it goes beyond what can be drawn. At some point we can discuss more complex situations, but right now I haven’t given you the conceptual tools to do it, mainly because if I did so now, you wouldn’t be convinced that the tools always work.
So let’s work through some more examples, see why possibility space is absolutely central and essential in understanding quantum interference, and then revisit your questions.
One of the most important parts of a physicist’s training is the art of choosing the right example. This involves learning to be very careful about what is relevant and what is irrelevant to the question at hand, so that the relevant issues can be addressed in mathematics and concepts using the simplest possible example that captures the key issues. Once one understands the issues fully, they can then be generalized to more difficult cases. I am using that methodology in every one of these posts.
If it was a single slit there would be no interference so why the anomaly in two or more slits
We’ll start exploring that over the coming two weeks.
Actually, energy always exists in waves form either in the form of particles or energy. The waves length visible to the physical eyes appears as matter or particles and which is beyond the limit of our physical eyes creates differentiation between matter and particles. There must be some animals in the universe who vision limit might be more than the humans. In that case, they can visualise the universe in the form of waves not as matter or particles what we visualize.
Uh huh.
Can you cover the “Bell Test” sometime? What is your opinion on it? It seems a bit preposterous to think that it is incorrect, yet I’ve never been able to convince myself that it is solid science.
Yes, at some point I will. Maybe late in spring… there’s a lot to do first. It is solid science, but it’s easy to draw the wrong conclusion from it.
Again very well done, Matt, I nearly get the feeling that I nearly understand it.
A (silly) question on fig. 10 : of the four possibilities LL, LR, RL and RR, the pair LR & RL are like artifacts of the axis labels, no? Aren’t particles 1 and 2 indistinguishable? There must be a symmetry at work in the (single) 2-particle wave function. Does it make physical sense to distinguish x1 as position of particle 1 and x2 as that of particle 2? Can this artifact be easily avoided?
If the particles were indeed indistinguishable, then indeed LR and RL would be the same. But that is why I colored them differently — to assure you that they are indeed distinguishable. I didn’t tell you they were of the same type. For example, perhaps one is an electron and the other is a positron.
We’ll have to look more at indistinguishable particles later, but right now the contraints that indistinguishability puts in the space of possibilities and the wave function would just be a confusing distraction.
One framing I’ve heard is that the photon only “exists” at the time of detection. It doesn’t travel from A to B or anything in between. There’s just an electromagnetic field at all points in the apparatus which is constantly varying in time in some complicated way. I’m… not sure that helps 😉
One gripe I had as a physics undergraduate, which I think you’re touching on in these series of articles, is that we were taught physics in more-or-less chronological (or historical) order. I can’t help wondering is many misconceptions or false intuitions could have been avoided if we had just been taught QFT first, and avoided the baggage of “1920’s quantum physics” entirely.
One problem is that most physics faculty don’t know QFT that well if at all. But actually there is a way to avoid this problem. If we simply did quantum mechanics of a long chain of balls and springs, then everything would be much clearer… both for particle physics and the physics of crystals.
As for “the particle doesn’t exist except at the time of detection” — I don’t think that’s correctly formulated. First, the question is whether the *system* of interacting particles — of which we are a part! — exists between measurements. It doesn’t make sense to focus on individual particles if they interact; after all, particles don’t have wave functions, systems do. Second, “measurement” is not a sharply defined term; when exactly is an interaction or sequence of interactions a measurement? I think it is clear that there is no sharp divide between measurement and not-measurement. What happens if the measurement is not completed, or is inaccurate — does the system partially exist at that moment? What happens if a small explosion causes a billion billion photons to rush out into the universe, but no one measurement device is there to collect a significant number of those photons? Did the explosion partially exist? I really think this idea leads nowhere.
Hi Matt, first of all I agree with Phil, this blog is impressively well done! Then, my try: no particle and no wave function passes through the two gates at the same time, there is only the sum of complex numbers relating to alternative paths.There is no movement, understood as a trajectory, of particles only change in the corresponding complex numbers. What we call interference is another way of visualising the above: nothing passes between the two doors at once merely particles appearing here or there without movement or trajectory.
Too far from the right answer? Thank you
In a week you’ll see the answer. Complex numbers are the symptom, not the cause.
I like how this thing is turning into a quantum spy story… 🙂
Maybe alter how it’s observed. Dig deeper and you’ll notice that it’s information that goes way beyond linear thought and into our conscious thought and soul. Right in our belly somewhere. 😇
Maybe, but to my ear it sounds as though you’re not actually that interested in science, which is done with physical experiments and logical inference.
Dark matter is a supersolid that fills the vacuum of space. Particles move through and displace the supersolid dark matter causing it to wave, analogous to a boat and its bow wave. In a double-slit experiment the particle travels through a single slit and the associated wave in the dark matter passes through both.
Thanks for your expert opinion!
Dark matter is clumpy, not a “solid”.
[“Hubble Detects Smallest Known Dark Matter Clumps”. NASA.]
This blog is fantastic! This posting has kind of detonated what I thought I knew from popular science which is I guess a heavily simplified version of the Copenhagen interpretation. This level of thinking serves as a goad to try to go from science consumer to becoming a struggling science amateur by taking like 20 math classes and maybe a half dozen physics classes to no longer be quite this clueless!
🙂 No, you shouldn’t need anything like that. The goal here is to lower the barrier; this series of posts represents my first attempt to figure out how to do this, but probably not my last.
Here is my issue with that. I will go from accepting what let’s say Sean Carroll says to what you say still without really understanding. Maybe this is wrong but Physics without math seems to be a super difficult proposition to teach. There seems to be a middle way developing with Sean Carroll and Anthony Zee of going ahead and including the math but I’m not sure that’s the answer either. If you write down a PDE I can see the terms but lacking that background, I don’t really know what you are saying.
That’s a problem. I think it is important to at least try to think critically about physics. So I hear a talk on QFT and the description is beautiful. Fermion fields and Higgs fields acted upon or acting through a Boson field to yield the behavior you can observe. It is a captivating picture but today, if I ask myself how do we know that, I’m not sure I can get there.
Maybe somebody will figure out how to convert the math of QM and maybe QFT to Algebra so most folks can have a quantitative understanding but till then the animations and pictorial representations here are a huge help. I’m a former engineer so maybe that’s a me problem of wanting to see under the hood
Well, this is my job. I did convert a good bit of the math of basic QFT to first-year university math, and you might find that useful: https://profmattstrassler.com/articles-and-posts/particle-physics-basics/fields-and-their-particles-with-math/ That series is ten years old, and I could supplement it with a bunch of other posts that I’ve written since then, but the basic core is pretty solid.
I apologize for my ignorance of this work. Yes those equations require no math beyond high school math. Thank you!
Oh, please don’t apologize! … just have some fun exploring the website! Use the search function; you’ll find a lot here. I think you’ll enjoy the “Higgs Field (with math)” series, the “Celebrating the Standard Model” series and the “Standard Model More Deeply” series.
“Lost in Superposition”: great title for quantum physics, hey? 🙂
“A particle, which has a position and a momentum (even if unknown to us), cannot have two positions.”
Unless, it is a composite, a “particle” comprised of millions maybe infinite quantum states can be not only in two positions but in infinite positions. Again, Prof. Strassler, I seem to be stuck in this theory that particles, excitations in fields, can be further quantized to lower confined spaces, like a lattice of micro black holes.
If you can assume that the fabric of spacetime is composed a lattice of tiny black holes, which theory (theories) would that violate? I don’t know what experiment to do to prove or disprove this theory and I certainly do have the math to calculate it, but it makes sense, intuitively.
I don’t believe the wave function collapses in the double slit experiment, it just changes with the addition of the observer. The observer is also part of the system, i.e. there is only one reference frame in the universe that all parameters or invariant.
“the interference effects arise do not arise” -> “the interference effects do not arise”
Thanks, fixed.
Is it possible to understand quantum theory by just saying probability is treated differently. So instead of the intuitive way of combining possibilities one uses what seems to be a little crazy “pythagorean” way of combining possibilities ?
So one can just think classically, including superpositions and entanglement, but by using “pythagorean” probabilty and one gets quantum superpositions and quantum entanglement.
By pythagorean I just mean adding amplitudes by squaring them ( like the Pythagoras theorem ). But I guess there is much more to quantum theory because we have the maths behind eigenstates and eigenvalues for measurements as well.
Remember that quantum wavefunction amplitudes are complex valued. Wavefunction “energy” (power, really) is amplitude squared, and that you can sum.
Hello! Thanks for the article — the idea of OR and AND reminds me of one of my professors when I was in physics grad school (Jack Cohn). He worked in quantum foundations back when it certainly wasn’t cool and ultimately wrote a short book “Construction of Quantum Theory” dealing with this OR/AND business. Anyway, just didn’t know if you were familiar with that work at all or if it’s just part of the language orthodoxy in the field. Thanks again!
No, I’m not familiar with Cohn’s book or his work. Thanks for mentioning him.