So far, in the context of 1920s quantum physics, I’ve given you a sense for what an ultra-microscopic measurement consists of, and how one can make a permanent record of it. [Modern (post-1950s) quantum field theory has a somewhat different picture; please keep that in mind. We’ll get to it later.] Along the way I’ve kept the object being measured very simple: just an incoming projectile with a fairly definite motion and moderately definite position, moving steadily in one direction. But now it’s time to consider objects in more interesting quantum situations, and what it means to measure them.
The question for today is: what is a quantum superposition?
I will show you that a quantum superposition of two possibilities, in which the wave function of a system contains one possibility AND another at the same time, does not mean that both possibilities occur; it means that one OR the other may occur.
Instead of a projectile that has a near definite motion, as we’ve considered in recent posts, let’s consider a projectile that is in a quantum superposition of two possible near-definite motions:
- maybe it is moving to the left at a near-definite speed, or
- maybe it is moving to the right at a similar near-definite speed.
This motion is along the x-axis, the coordinate of a one-dimensional physical space. If the projectile is isolated from the rest of the world, we can write a wave function for it alone, which might initially look like
in which case its evolution over time will look like this:
Again I emphasize this is not the wave function of two particles, despite what you might intuitively guess. This is the wave function of a single particle in a superposition of two possible behaviors. For a similar example that we’ll return to in a few weeks, see this post.
Because the height and speed of the two peaks is the same, there is a left-to-right symmetry between them. We can therefore conclude, before we even start, that there’s a 50-50 chance of the particle going right versus going left. More generally, whatever we observe to the left (x<0) will happen with the same probability as what we observe to the right (x>0).
Today I will show you that even though the wave function has one peak moving to the left AND one peak moving to the right, nevertheless this wave function does not describe a projectile that is moving to the left AND moving to the right. Instead, it means that the projectile is moving to the left OR moving to the right. Superposition is an OR, not an AND. In other words, in pre-quantum language, we have either
We never have both.
But don’t take my word for it. Let’s see how quantum physics actually works.
First Measurement: A Ball to the Left
Our first goal: to detect the projectile if it is moving to the left.
Let’s start by doing almost the same thing we did in this post, which you may want to read first in order to understand the pictures and the strategy that I’ll present below. To do this, we’ll put a measurement ball on the left, which the projectile will strike if it is moving to the left.
Since we now have a system of two objects rather than one, the space of possibilities for the system now has to be two-dimensional, to include both the position x1 of the projectile and the position x2 of the ball. This now requires us to consider a wave function for not just the projectile alone, as we did in Figs. 1 and 2, but for the projectile and the ball together. This wave function will give us probabilities for each possible arrangement of the projectile and ball — for each choice of x1 and x2.
We’ll put the ball at x2 = -1 initially — to the left of the projectile initially — so that the initial wave function looks like Fig. 4, which shows its absolute value squared as a function of x1 and x2.
This wave function has the same shape in x1 as the wave function in Fig. 1, but now centered on the line x2=-1. A collision between projectile and ball will become likely when a peak of the wave function approaches the point x1=x2=-1.
As usual, let’s try to think about this in a pre-quantum language first. If I’m right about wave functions, we have two options:
- The projectile is heading to the left and the measurement ball will react OR
- The projectile is heading to the right and the measurement ball will not react.
Since our wave function is left-to-right symmetric, each option is equally likely, and so if we do this experiment repeatedly, we should see the ball react about half the time.
Here are the two pre-quantum options shown in the usual way, with
- a picture in physical space on the left, and
- a picture of the same event in the space of possibilities on the right.
In the first possibility (Fig. 5a), the projectile moves left, strikes the ball, and the ball recoils to the left. As the ball moves to the left in physical space, the system moves down (toward more negative x2) in the space of possibilities.
OR
In the second possibility (Fig. 5b), the projectile moves right and the ball remains unscathed; in this case, viewed in the space of possibilities, x2 remains at -1 during the entire process while x1 changes steadily toward more positive values.
What about in quantum physics? The wave function should include both options in Figs. 5a and 5b.
Here is an actual solution to the Schrödinger wave equation, showing that this is exactly what happens (and it has more details than the sketches I’ve been doing in my measurement posts, such as this one or this one.) The two peaks spread out more quickly than in my sketches (and I have consequently adjusted the vertical axis as time goes on so that the two bumps remain easily visible.) But the basic prediction is correct: there are indeed two peaks, one moving like the pre-quantum system in Fig 5a, changing direction and moving toward more negative x2, and the other moving like the pre-quantum system in Fig. 5b, moving steadily toward more positive x1.
Importantly, even though the system’s wave function displays both possibilities to us at the same time, there is no sense in which the system itself can be in both possibilities at the same time. The system has a near-50% probability of being observed to be within the first peak, near-50% probability of being observed to be within the second, and exactly 0% probability of being observed within both.
Second Measurement: A Ball to the Right
Now let’s put a ball to the right instead, at x=+1. This is a different ball from the previous (we’ll use both of them in a moment) so I’ll color it differently and call its position x3. The pre-quantum behaviors are the same as before, but with x2 replaced with x3 and with the collision happening at positive values of x1 and x3 instead of negative values of x1 and x2.
OR
The quantum version is just a 180-degree rotation of Fig. 6 with x2 replaced with x3.
Third Measurement: A Ball on Both Sides
But what happens if we put a ball on the left and a ball on the right? Initially the balls are at x2=-1 and x3=+1. What happens later?
Now there are four logical possibilities for what might happen:
- The ball on the left responds while the ball on the right does not
- The ball on the right responds while the ball on the left does not
- Neither ball responds
- Both balls respond
Where in the space of possibilities do these four options lie? The four logical possibilities listed above would put the ball’s positions in these four possible places:
- Option 1: x2 < -1 and x3 = +1 (and x1 negative, as in Fig. 5a)
- Option 2: x2 = -1 and x3 > +1 (and x1 positive, as in Fig. 7a)
- Option 3: x2 = -1 and x3 = +1 (and x1 is ???)
- Option 4: x2 < -1 and x3 > +1 (and x1 is ???)
The fact that it is not obvious where to put x1 in the last two options should already make you suscpicious; but just setting their x1 to zero for now, let’s draw where these four options occur in the space of possibilities. In Fig. 9 I’ve drawn the lines x2=-1 and x3=+1 across the box, with option 3 at their crossing point. Option 1 lies below down and to the left of option 3; option 2 is found to the rigt of option 3; and option 4 is found down and to the right.
What does the wave function actually do? Can the simple two-humped superposition at the start, analogous to Fig. 4, end up four-humped?
Not in this case, anyway. Fig. 10, which depicts the peaks of the absoulte-value-squared of the wave function only, shows the output of the Schrödinger equation. Compare the result to Fig. 9; there are peaks only for options 1 and 2, in which one ball moves and the other does not.
With balls on either side of it, the projectile cannot avoid hitting one of them, whether it goes right or left, which rules out option 3. And the wave function does not put a peak at option 4, showing there’s no way the projectile can cause both balls to move. The two peaks in the wave function move only in the x1 direction as the projectile goes left OR right; then the projectile collides with one ball OR the other; then the ball with which it collided moves, meaning that the system moves to more negative x2 (i.e. down in Fig. 10) OR to more positive x3 (i.e. to the right in Fig. 10), just as expected from Fig. 9.
Actually it’s not difficult to get the third option — but we don’t need quantum physics for that!
We simply change the original wave function to contain three possibilities: the projectile moves left, or it moves right, or it doesn’t move at all. If it doesn’t move at all, then neither ball will react, a third option even in pre-quantum physics:
If the projectile were isolated, we would encode this notion in a wave function which looks like this:
and when we include the two balls we would see the wave function with three peaks, one sitting still at the point marked “Neither Ball Moves” in Fig. 9. But this isn’t particularly exciting or surprising, since it’s intuitively obvious that a stationary projectile won’t hit either ball.
Every Which Way
There simply is no wave function you can choose — no initial superposition for the single projectile — which can cause the projectile to collide with both balls. The equations will never let this happen, no matter what initial wave function you feed into them. It’s impossible… because a superposition is an OR, not an AND. There is no way to make the projectile go left AND right — not if it’s a particle in 1920s quantum physics, anyway.
Yes, the wave function itself can have peaks that appear at to be in several places at the same time within the space of possibilities, as in Figs. 6, 8, and 10. But the wave function is not the physical system. The wave function tells us about the probabilities for the system’s possibilities; its peaks are just indicating what the most likely possibilities are.
The system itself can only realize one of the many possibilities — it can only be found (through a later measurement) in one place within the space of possibilities. This is always true, even though the wave function for the system highlights all the most probable possibilities simultaneously.
A particle, in the strict sense of the term, is an object with a position and a momentum, even though we cannot know both perfectly at any moment, thanks to Heisenberg’s uncertainty principle. It can only be measured to be in one place, or can only be measured to be traveling in one direction, at a time. In 1920s quantum physics, these statements apply to an electron, which is viewed as a strict particle, and so it cannot go in two directions at once, nor can it be in two places at once. The fact that we are always somewhat ignorant of where an electron is and/or where it is going, and the fact that quantum physics puts ultimate limitations on our ability to know both simultaneously, do not change these basic conceptual lessons… the lessons of (and for) the 1920s.
30 Responses
Ah, but what if both the right and left wavefunction parts collide with a single ball, a different species of ball? That is, while they move apart as you show, from behind another ball comes in a bit faster,
and already spread out bigger than the spread between the two parts? It collides with both parts at the same time. Try again with both balls the same species.
Now THAT’s a quantum test!
Your picture is ambiguous: do you mean that the “ball that comes in faster” has a single peak spreading out so fast that it catches up to the other ball, or do you mean it has two peaks which each move fast enough outward to catch up to the other ball’s peaks from behind?
Either way, we can run that test easily, but I can tell immediately that your problem is that you are thinking in terms of physical space, the classic mistake. You need to think in terms of the 2-dimensional space of possibilities.
Why don’t you try sketching the wave function in that 2-dimensional space, and see if you can figure out what happens? Then write back, clarify your suggestion, and — if you were able to figure it out — tell us what the answer is. If you can’t figure it out, then I’ll address your clarified suggestion and tell you what happens.
Actually — let me ask you this, as a test of your understanding: in eiither interpretation of your scenario, what do you think is the probability that the second ball entirely misses the first ball and doesn’t hit anything?
Ok, try this, in one dimension. One ball acts as you describe to start. The other starts out in two parts, both farther out from the center and heading IN. If you are allowed a thought experiment with
the two parts going out, I’m allowed two parts coming in, in a similar superposition.
There is of course a choice of amplitudes of the two parts of each ball (L+R versus L-R, for each ball). [I suppose as a thought experiment, the 2nd ball could be in two unmoving parts with two separated lobes, slowly spreading per the uncertainty principle.]
In real 2 or 3D, this can be arranged with an electron, a double slit, and static electric fields.
What happens depends on the forces between the balls … repulsive or attractive, and if attractive, if they are over the whole scenario stuck together or not
(in a well).
I hope this is well stated. I have a computer program somewhere which I wrote that actually shows this, but its so old that it only runs on an old 8086 PC with a VGA card.
Yes, you can certainly have them coming in (as I did here: https://profmattstrassler.com/2025/02/20/can-you-interpret-this-quantum-wave-function/ )
However, I will ask you again: what, in your opinion, is the probability that the two balls will collide in your scenario? And if you think it is 100%, when will the collisions occur relative to one another?
In my 1D scenario, of course 100%. If the expectaton value of the position of both balls is zero and the two linear combinations are symmetric, the parts meet at the same time.
If not symmetic, we are talking “spooky action at a distance”. In the symmetric case, of course both balls are at, and remain, at the same spot
(expectation value of both balls and their CM at zero, and their relative velocity is actually zero!)
That’s why I propose it! Students seeing a simulation were stunned.
If the potential is right, you can get mirror image “molecules” vibrating if the balls
are atoms, or bottomonium if they are bottom quarks!
Well, the answer is 50%. You’re simply incorrect. As I said, you are thinking in physical space and not in the space of possibilities, and you are making the classic mistake.
Superposition is an OR, not an AND. There are four peaks to the wave function at all times.
(1) Ball 1 moves L and ball 2 moves R ;
(2) Ball 1 moves R and ball 2 moves L;
(3) Ball 1 moves R and ball 2 moves R.
(4) Ball 1 moves L and ball 2 moves L.
In peaks (1) and (2), the balls collide. In peaks (3) and (4), they do not; instead, the incoming ball trails the outcoming ball at all times. Hence the probability of collision is 50%.
[If the speed of the incoming ball is faster than that of the outgoing ball, then collisions happen in each case, but in peaks 3 and 4 the collisions occur much later than the collisions in peaks 1 and 2, since in 3 and 4 the incoming ball and outgoing ball are going in the same direction and so the former has to catch up to the latter.]
If you can’t convince yourself of this, maybe I’ll explain it in a brief blog post so I can put up the picture of the wave function in the space of possibilities.
I don’t know if your students seeing a simulation were stunned; sounds like you showed them a wrong one.
Oh. I get it. You are talking three particles, two of which you call balls.
I was talking two elementary particles. If your balls are macroscopic,
and in different quantum states if chemically identical, I see what you mean.
This is they very heart of classicalization. But if the “balls” were truly identical,
in the ground state, and arranged symmetrically, you get a superposition, by symmetry, both colliding. That experiment can’t be done. If you don’t agree with that, I really don’t understand what you are saying … there is no arguing with symmetry. Its spooky!
Nope. Wrong again. I’m referring to two balls, or two elementary particles, but in any case, two objects. Your analysis is simply wrong, and you should go back and retake your quantum class. Symmetry doesn’t cut it; there’s perfect symmetry, but 50% probability of collision.
In re: March 11, 2025 at 9:14 AM
If you mean a “collision” that will be “measured” in the future, we do agree.
Yes, in your case its 50%. I had to get your exact terminology right.
how do topologically unstable structures relate mixed degrees beyond the description of activated gasses? Instability in wave fx and material then likewise cross referenced give morphic structure the equivelant independent fx of uncertainty-probability-density thus such differences as cosmological constant, vacuum permitivity to color charge, permitivity to electric…
Hi Matt. I think your very clear analysis shows that the Schrodinger’s Cat paradox, where it is claimed that QM says that a cat can be simultaneously alive and dead, is a misunderstanding of QM, but apparently one that Schrodinger himself had. It is simply shown not to be a paradox by understanding that QM is saying that the cat can be either alive or dead, with equal probabilities, but the probability of it being both alive and dead is zero. Do you have any idea why Schrodinger had a problem with that? Did the Copenhagen interpretation say that it being both alive and dead was a possibility?
I don’t think Schrodinger was confused so much as bemused. His theory tells you that macroscopic objects can in principle be in superposition states, and he understood this correctly. However, the specifics of the experiment are not realistic, and that’s because it’s extremely difficult to keep a macroscopic object in a well-defined superposition — to “maintain coherence” — because macroscopic objects can’t be kept isolated from their environments. That’s why a key part of the thought experiment is keeping the box perfectly isolated from the outside world, which is impossible practically and, arguably, in principle too [long discussion to follow about that.]
Fundamentally the conceptual problem arises from being able to cleanly separate the world into two noninteracting pieces, one small and one large, each with its own wave function, and then at a certain moment cause them to interact, joining their two wave functions into one. Only then do you have the question “What was the cat doing before I (and the rest of my part of the world) knew whether the cat had lived or died?”
But the question of how to resolve the problem is not known. That is, it is not clear how to interpret consistently what the evolving wave function is actually telling you. I do think trying to understand it as “Cat alive AND dead” does not make sense, but “Cat alive OR dead” is still a state where the situation is unknown. That state merely tells you that when you open the box, you will find Alive OR Dead. But it still claims you can’t predict the outcome… and that trying to make continuous, coherent sense of what happened at every stage between when you put the cat in the box and when you opened the box is not possible. The wave function is telling you how probability amplitudes evolve and what the probabilities are when you make the measurement at the end; it is not necessarily telling you what the system actually did over time, unless you decide you want to try to interpret it that way, which then forces you into paradoxes. It’s wanting to have a complete history that forces you to say “the cat was at the time in a state of Alive AND Dead” (which makes no sense since only the entire box has a state, not the cat alone) or better, “the box was at the time in a state in which [the atom had not decayed and the cat was alive] AND [the atom had decayed and the cat was dead]” (which could perhaps make sense), instead of saying that “At the time we did not have any information as to whether the cat was Alive OR Dead, and could only predict the probability of which outcome we would find when we opened the box”, which is correct but feels to many people, Einstein among them, like a cop-out.
Obviously we’ll have to discuss this more. But being precise is key… and difficult.
Hi Matt. You did an exact solution of the Schrodinger equation for the situation shown in figure 6. I just want to make sure I understand this correctly. As I understand it, you invented a wavefunction which at time t = 0 had a magnitude squared equal to the initial situation shown in figure 6. You then plugged this wavefunction into the time dependent Schrodinger equation
i hbar dpsit/dt = H psi = – (hbarsquared/2m) d2psi/dxsquared
where I hope you understand my clumsy symbols and derivatives are really partial derivatives, and solved this equation exactly. Have I got that right?
Sort of, except that the Hamiltonian H includes a derivative term for x1 and a derivative term for x2, with different masses, plus a potential energy that depends on |x1-x2| that prevents the projectile from passing through the ball.
Yes of course – I was forgetting about the ball and the need for interaction between the ball and the projectile.
Would it be possible to post the mathematics on your blog for those who are interested?
At some point, yes.
Hmm. Send a cold neutronn with a well-defined wavelength through a double slit with left and right apertures, then reflect the resulting two-lobed beam from a mirror just long enough to reflect both lobes.
Does the neutron reflection on the mirror:
(1) Add left- or right-handed angular momentum to the mirror due to only one lobe containing the neutron; or,
(2) Introduce linear momentum to the mirror due to both lobes imparting momentum as they reflect coherently?
We haven’t gotten to interference yet — and we will. Coherence is the key question.
Another key question — complicated for neutrons — is whether we are using 1920s quantum physics or 1950s quantum field theory to describe what happens. Clearly the answers to your questions don’t depend on how we describe the neutron and its passage through a double slit, but we might view the process differently depending on how we describe it.
Using 1920s quantum physics to describe the double slit experiment would you say the particle goes through one slit OR the other slit, or would you say the particle goes through one slit AND the other slit ?
Do you think there is anything to quantum logic where I think words like OR and AND ( or perhaps I should say meet and join ) follow the rules of a non-standard logic ?
Excellent questions, to be in part answered next week.
figure 10 seems to show an inverse mirroring at half plane slice that is prototype of Wilson Loop where a complex sinusoidal covers curved space time? These slicing s then, translating quadrant calculus of pi to something more like a Japanese folding book where you could elicit page combinations of three fold four.. through Pascals triangle and yet borrow from the Schrodinger you worked out the four dimensional illusion by relating of Pascals simplex at root the 3,6,,9 and 11 as essentially complex (i) poles. The Schrödinger drawing you worked out was particularly help-full as indeed also figure ten relativehttps://en.wikipedia.org/wiki/Wilson_loop
Sure, agree. The quibble is about “there is NO sense” (emphasis mine). I suggest there is a sense: but not a classical sense.
Should we say instead, “the system combines information from both possibilities…”?
This avoids the verb “to be” which has such a classical, concrete connotation. However, I suggest that is connotation, not denotation. There are levels of existence. Superposition is a very subtle level that challenges our language.
Well, this is going to come up a lot in a week or two. I agree that language can be our downfall, and it is certainly possible that the language I’m using, much of it for the first time, is going to need adjustment; certainly when I look at my blog posts from 2012 on quantum field theory, there are things I say differently nowadays. Let’s see how it plays out and continue the conversation.
you write, “there is no sense in which the system itself can be in both possibilities at the same time”
Consider interference in a one-electron-at-a-time two slit experiment… how can nature sum both ways if both ways do not exist? Isn’t that like asking, “What is A+B given that A and B do not exist?”
We’re will soon study interference in precisely this case. But look at your words very carefully: you notice that you asked about “both ways”. I wasn’t talking about “ways”; I was talking about “now”. There is a very big difference between saying that “a system now, at this moment, exists in two states” and saying that “a system has two different ways to move from an initial state to a final state.” Interference comes from the latter. But the latter does not imply the former.
Great post , as usual, thanks. You wrote: ‘a system has two different ways of passing from an initial state to a final state’. i.e.: each different way of passing from one state to another is associated with a complex number. These complex numbers somewhat interfere with each other like vectors and sometimes cancel out as happens in the dark areas of the screen in the double-slit experiment. This ‘complex’ dance takes place in a space that does not belong to space-time but in a certain sense determines it because in any case the outcome of the measurement cannot go outside the ‘enclosure’ of amplitudes that live in this abstract space. is this correct or am I off the mark? thanks
You’re not far off, but not exactly right as stated. And the problems with your statements are interesting, as they help me see some issues that I’ll have to clarify as we go forward.
The complex number in the wave function psi(x1,x2,t) at a particular time t tells us the probability of the system being found in a particular possible state (x1,x2) at that same time t.
It does not tell us the probability that the system passed from one state psi(x1,x2,t0) at an earlier time t_0 to a certain final state psi(x1,x2,t) at time t. The Schroedinger equation, which makes the wave function change over time, is what tells us about evolution from time t_0 to time t. It does not tell us about probabilities, but it can cause interference which affects the shape of the wave function at time t, and thus the probabilities for states at time t.
We’ll see examples soon, one or two weeks depending on how many universities are canceled by the rulers in Washington DC.
And yes, this is all taking place in the space of possibilities… which is rather abstract. Up to this point, the question of whether this space is itself real, or just a conceptual device for doing calculations, remains open; it’s something we’ll have to discuss at some point.
‘The complex number in the wave function psi(x1,x2,t) at a particular time t tells us the probability that the system is in a particular possible state (x1,x2) at that same time t.’ OK.
But isn’t it that the evolution of the wave function is just this set of complex numbers almost like frames in a film whereby movement (evolution) arises from static elements? Many thanks.