The quantum double-slit experiment, in which objects are sent toward and through a pair of slits in a wall,and are recorded on a screen behind the slits, clearly shows an interference pattern. It’s natural to ask, “where does the interference occur?”
The problem is that there is a hidden assumption in this way of framing the question — a very natural assumption, based on our experience with waves in water or in sound. In those cases, we can explicitly see (Fig. 1) how interference builds up between the slits and the screen.
But when we dig deep into quantum physics, this way of thinking runs into trouble. Asking “where” is not as straightforward as it seems. In the next post we’ll see why. Today we’ll lay the groundwork.
Independence and Interference
From my long list of examples with and without interference (we saw last time what distinguishes the two classes), let’s pick a superposition whose pre-quantum version is shown in Fig. 2.
Here we have
- particle 1 going from left to right, with particle 2 stationary at x=+3, OR
- particle 1 going from right to left, with particle 2 again stationary at x=+3.
In Fig. 3 is what the wave function Ψ(x1,x2) [where x1 is the position of particle 1 and x2 is the position of particle 2] looks like when its absolute-value squared is graphed on the space of possibilities. Both peaks have x2=+3, representing the fact that particle 2 is stationary. They move in opposite directions and pass through each other horizontally as particle 1 moves to the right OR to the left.
This looks remarkably similar to what we would have if particle 2 weren’t there at all! The interference fringes run parallel to the x2 axis, meaning the locations of the interference peaks and valleys depend on x1 but not on x2. In fact, if we measure particle 1, ignoring particle 2, we’ll see the same interference pattern that we see when a single particle is in the superposition of Fig. 1 with particle 2 removed (Fig. 4):
We can confirm this in a simple way. If we measure the position of particle 1, ignoring particle 2, the probability of finding that particle at a specific position x1 is given by projecting the wave function, shown above as a function of x1 and x2, onto the x1 axis. [More mathematically, this is done by integrating over x2 to leave a function of x1 only.] Sometimes (not always!) this is essentially equivalent to viewing the graph of the wave function from one side, as in Figs. 5-6.
Because the interference ridges in Fig. 3 are parallel to the x2 axis and thus independent of particle 2’s exact position, we do indeed find, when we project onto the x1 axis as in Fig. 5, that the familiar interference pattern of Fig. 4b reappears.
Meanwhile, if at that same moment we measure particle 2’s position, we will find results centered around x2=+3, with no interference, as seen in Fig. 6 where we project the wave function of Fig. 3 onto the x2 axis.
Why is this case so simple, with the one-particle case in Fig. 4 and the two-particle case in Figs. 3 and 5 so closely resembling each other?
The Cause
It has nothing specifically to do with the fact that particle 2 is stationary. Another example I gave had particle 2 stationary in both parts of the superposition, but located in two different places. In Figs. 7a and 7b, the pre-quantum version of that system is shown both in physical space and in the space of possibilities [where I have, for the first time, put stars for the two possibilities onto the same graph.]
You can see that the two stars’ paths will not intersect, since one remains at x2=+3 and the other remains at x2=-3. Thus there should be no interference — and indeed, none is seen in Fig. 8, where the time evolution of the full quantum wave function is shown. The two peaks miss each other, and so no interference occurs.
If we project the wave function of Fig. 8 onto the x1 axis at the moment when the two peaks are at x1=0, we see (Fig. 9) a single peak (because the two peaks, with different values of x2, are projected onto each other). No interference fringes are seen.
Instead the resemblance between Figs. 3-5 has to do with the fact that particle 2 is doing exactly the same thing in each part of the superposition. For instance, as in Fig. 10, suppose particle 2 is moving to the left in both possibilities.
(In the top possibility, particles 1 and 2 will encounter one another; but we have been assuming for simplicity that they don’t interact, so they can safely pass right through each other.)
The resulting wave function is shown in Fig. 11:
The two peaks cross paths when x1=0 and x2=2. The wave function again shows interference at that location, with fringes that are independent of x2. If we project the wave function onto the x1=0 axis, we’ll get exactly the same thing we saw in Fig. 5, even though the behavior of the wave function in x2 is different.
This makes the pattern clear: if, in each part of the superposition, particle 2 behaves identically, then particle 1 will be subject to the same pattern of interference as if particle 2 were absent. Said another way, if the behavior of particle 1 is independent of particle 2 (and vice versa), then any interference effects involving one particle will be as though the other particle wasn’t even there.
Said yet another way, the two particles in Figs. 2 and 10 are uncorrelated, meaning that we can understand what either particle is doing without having to know what the other is doing.
Importantly, the examples studied in the previous post did not have this feature. That’s crucial in understanding why the interference seen at the end of that post wasn’t so simple.
Independence and Factoring
What we are seeing in Figs. 2 and 10 has an analogy in algebra. If we have an algebraic expression such as
- (a c + b c),
in which c is common to both terms, then we can factor it into
- (a+b)c.
The same is true of the kinds of physical processes we’ve been looking at. In Fig. 10 the two particles’ behavior is uncorrelated, so we can “factor” the pre-quantum system as follows.
What we see here is that factoring involves an AND, while superposition is an OR: the figure above says that (particle 1 is moving from left to right OR from right to left) AND (particle 2 is moving from right to left, no matter what particle 1 is doing.)
And in the quantum context, if (and only if) two particles’ behaviors are completely uncorrelated, we can literally factor the wave function into a product of two functions, one for each particle:
- Ψ(x1,x2)=Ψ1(x1)Ψ2(x2)
In this specific case of Fig. 12, where the first particle is in a superposition whose parts I’ve labeled A and B, we can write Ψ1(x1) as a sum of two terms:
- Ψ1(x1)=ΨA(x1) + ΨB(x1)
Specifically, ΨA(x1) describes particle 1 moving left to right — giving one peak in Fig. 11 — and ΨB(x1) describes particle 2 moving right to left, giving the other peak.
But this kind of factoring is rare, and not possible in general. None of the examples in the previous post (or of this post, excepting that of its Fig. 5) can be factored. That’s because in these examples, the particles are correlated: the behavior of one depends on the behavior of the other.
Superposition AND Superposition
If the particles are truly uncorrelated, we should be able to put both particles into superpositions of two possibilities. As a pre-quantum system, that would give us (particle 1 in state A OR state B) AND (particle 2 in state C OR state D) in Fig. 13.
The corresponding factored wave function, in which (particle 1 moves left to right OR right to left) AND (particle 2 moves left to right OR right to left), can be written as a product of two superpositions:
- Ψ(x1,x2)=Ψ1(x1)Ψ2(x2) = [ΨA(x1)+ΨB(x1)] [ΨC(x2)+ΨD(x2)]
In algebra, we can expand a similar product
- (a+b)(c+d)=ac+ad+bc+bd
giving us four terms. In the same way we can expand the above wave function into four terms
- Ψ(x1,x2)=ΨA(x1)ΨC(x2)+ΨB(x1)ΨC(x2)+ΨA(x1)ΨD(x2)+ΨB(x1)ΨD(x2)
whose pre-quantum version gives us the four possibilities shown in Fig. 14.
The wave function therefore has four peaks, one for each term. The wave function behaves as shown in Fig. 15.
The four peaks interfere in pairs. The top two and the bottom two interfere when particle 1 reaches x1=0, creating fringes that run parallel to the x2 axis and thus are independent of x2. Notice that even though there are two sets of interference fringes when particle 1 reaches x1=0 in all the superpositions, we do not observe this if we only measure particle 1. When we project the wave function onto the x1 axis, the two sets of interference fringes line up, and we see the same single-particle interference pattern that we’ve seen so many times (Figs. 3-5). That’s all because particles 1 and 2 are uncorrelated.
If at the same moment we measure particle 2 ignoring particle 1, we find (Fig. 17) that particle 2 has equal probability of being near x=2.5 or x=-0.5, with no interference effects.
Meanwhile, the left two and the right two peaks in Fig. 15 subsequently interfere when particle 2 reaches x2=1, creating fringes that run parallel to the x1 axis, and thus are independent of x1; these will show up near x=1 in measurements of particle 2’s position. This is shown (Fig. 18) by projecting the wave function at that moment onto the x2 axis.
Locating the Interference?
So far, in all these examples, it seems that we can say where the interference occurs in physical space. For instance, in this last example, it appears that particle 1 shows interference around x=0, and slightly later particle 2 shows interference around x=1.
But if we look back at the end of the last post, we can see that something is off. In the examples considered there, the particles are correlated and the wave function cannot be factored. And in the last example in Fig. 12 of that post, we saw interference patterns whose ridges are parallel neither to the x1 axis nor to the x2 axis. . .an effect that a factored wave function cannot produce. [Fun exercise: prove this last statement.]
As a result, projecting the wave function of that example onto the x1 axis hides the interference pattern, as shown in Fig. 19. The same is true when projecting onto the x2 axis.
Consequently, neither measurements of particle 1’s position nor measurements of particle 2’s position can reveal the interference effect. (This is shown, for particle 1, in the previous post’s Fig. 13.) This leaves it unclear where the interference is, or even how to measure it.
But in fact it can be measured, and next time we’ll see how. We’ll also see that in a general superposition, where the two particles are correlated, interference effects often cannot be said to have a location in physical space. And that will lead us to a first glimpse of one of the most shocking lessons of quantum physics.
One More Uncorrelated Example, Just for Fun
To close, I’ll leave you with one more uncorrelated example, merely because it looks cool. In pre-quantum language, the setup is shown in Fig. 20.
Now all four peaks interfere simultaneously, near (x1,x2)=(1,-1).
The grid pattern in the interference assures that the usual interference effects can be seen for both particles at the same time, with the interference for particle 1 near x1=1 and that for particle 2 near x2=-1. Here are the projections onto the two axes at the moment of maximal interference.
17 Responses
Hi Matt, tu sum up and to clear my thoughts: If we have a non factorizable wave function, interference pattern exists in possibility space only (compare figures 12 and 13 from your previous post).
If wave functions are factorizable, interference also exists in physical space. Basically if wave function is factorizable interference ‘leaks’ from the abstract possibility space to the physical world. Am I on the right track? Thanks
Matt,
after letting it all sink in, I (think I) understand that the x1 – x2 plots as superpositions (literally: one-atop-the-other) of alternatives show the mechanism best. The interference fringes always perpendicular to the ‘motion’ of the probability peaks.
btw, an other inconsistency in fig. 20 – 21 : if the orange and blue wavicles are to meet at (1,-1) after a unit step, then fig. 20 should have them move
(0,0) -> (1,-1) ; (0,-2) -> (1,-1); (2,0) -> (1,-1) and (2,-2) ->(1,-1) and not
(1,3) -> (0,2) ; (1,0) -> (0,1); (-2,3) -> (-1,2) and (-2,0) -> (-1, 1)
or am I utterly misinterpreting it?
Towards The question where does interference occur and from discussion on wavicles can we infer that the answer is perhaps space time decoupled over antidesitter space where particles preserved from matter anti matter annihilation in cosmogonists via a perhaps thermal misalignment then find in tensions between field the stiffness via the p violation and cp violation. A gesture towards decoupling of space time as in Einsteins idea of gravity as a collapse through multiple fields and forces helps inform towards relating quark color as cohesive to eigenvectors more mathematical object like and electromagnetism with linear eigenvectors closer to idea of a “number”. The physics preserves the space of possibilities in the difference of its eigenvectors as tension between a mathematical object and mathematical space embedded in the course of the linear progress of photons in space time.If we substitute log symmetry for the bilateral symmetry of supersymmetry would this help relate the object to number states which seem the goal of finding the structure of a decoupling of time space in the sense of quarks more time like and electromagnetism more space like… ? via fuzzy number indices skewing the Cayley like expectations of determinants via differentials?..which string theory perhaps finds local structure via knots through vacua in cosmogenesis topological defects with consequences in quantum flux borrowed from space itself in pseudo particle process influencing potential gravity relative guage readings.
This is very good! I now see where you are going and am eager to read the next post. I have one question, which might be premature, but to clear up my thinking: You say “If the two particles are uncorrelated their behavior can be factored.” Is this equivalent to saying ” If the two particles are uncorrelated their wave function equation can be separated, (mathematically)”?
Matt, this series of articles is great, but I’m afraid that I’m now confused about something I thought I previously understood.
You were kind enough to answer a question I asked here: https://profmattstrassler.com/waves-in-an-impossible-sea/waves-in-an-impossible-sea-commentary-and-discussion/going-beyond-the-book/#comment-459899
In your response you said “First, your intuition for an interferometer using a light beam made of a huge number of photons, such as we can describe by physics of non-quantum fields in early years of undergraduate physics, is correct.” I thought I understood why that was the case, but now I’m not so sure.
To recap: we’re talking about a Michelson interferometer with vastly different beam lengths (one a light second long, one a light hour long). I don’t see how photons emitted 1 second ago can be correlated in any way to photons emitted 1 hour ago, and therefore I don’t understand how there would be any interference?
What am I missing?
Matt, you will be coming to this, I’m sure, but to what extent is what you’re calling correlation above related to the phenomenon described as entanglement? One part of me thinks that they’re likely to be the same thing, but people talk about entanglement being “monogamous” (which I’ve never really understood) and it seems to me that there’s no real limit on how many particles could be correlated?
One reason I’m treading lightly here is that I am an expert on the equations of atomic and particle physicists but not an expert in the language of quantum foundations and quantum information, so I am in danger of using a word incorrectly. Yes, correlation involves entanglement, but I’m not quite ready to be precise about the latter.
(I hope this isn’t too technical of a comment.)
Matt, are you using the (unitary) Cayley form (Crank-Nicolson form) of the discretized Schrödinger equation,
$(1 + 1/2 i H dt) \psi^(n+1) = (1 – 1/2 i H dt) \psi^n$ ?
Also, are you solving the full 2d S. eqns, Or using operator splitting?
In my recent posts, I am solving the full 2d Schrödinger equation, using analytic expressions, not numerics. In these contexts, I find it is both much faster and more reliable than the numerical methods I have easy access to.
Cool! Any good references for these analytic methods?
Any book on wave equations, but there must be lots of tools on-line. At some point I’ll put mine up too, but it takes time.
Cohen-Tannoudji to the rescue!
That will do just fine. At this point we’re just doing two free particles in one dimension, so the equations are very simple. It gets slightly more complicated when we include collisions, which we will do soon.
Matt,
below Fig. 11 you write:
“The two peaks cross paths when x1=0 and x2=2. The wave function again shows interference at that location, with fringes that are independent of x2. If we project the wave function onto the x1=0 axis, we’ll get exactly the same thing we saw in FIG, even though the behavior of the wave function in x2 is different.”
I guess that should be ” … we’ll get exactly the same thing we saw in Fig. 9 , even though …”
Oops – thanks. No, Fig. 5 is what I meant.
Matt,
For these examples, are particles 1 and 2 assumed to be distinguishable e.g have different spin values?
I graduated 40 years ago with a degree in Physics so am a little rusty…
Thanks
Paul
They are distinguishable (hence the different colors). [[Also, if they were not distinguishable, then the wave function would have to satisfy Psi(x1,x2)=Psi(x2,x1), which my examples do not satisfy.]]
Spin isn’t the issue. I didn’t say these were electrons [in which case they would indeed have to have different and fixed spin]. They could be two different atoms. Or one could be an electron and the other a positron (i.e. an anti-electron.) Or an electrically charged Kaon and an electrically neutral Kaon. Anything other than identical particles will do.