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In my last post and the previous one, I put one or two particles in various sorts of quantum superpositions, and claimed that some cases display quantum interference and some do not. Today we’ll start looking at these examples in detail to see why interference does or does not occur. We’ll also encounter a difficulty asking where the interference occurs — a difficulty which will lead us eventually to deeper understanding.

First, a lightning review of interference for one particle. Take a single particle in a superposition that gives it equal probability of being right of center and moving to the left OR being left of center and moving to the right. Its wave function is given in Fig. 1.

Figure 1: The wave function of a single particle in a superposition of moving left from the right OR moving right from the left. The black curve represents the absolute-value-squared of the wave function, which gives the probability of finding the particle at that location. Red and blue curves show the wave function’s real and imaginary parts.

Then, at the moment and location where the two peaks in the wave function cross, a strong interference effect is observed, the same sort as is seen in the famous double slit-experiment.

Figure 2: A closeup of the interference pattern that occurs at the moment when the two peaks in Fig. 1 perfectly overlap. An animation is shown here.

The simplest way to analyze this is to approach it as a 19th century physicist might have done. In this pre-quantum version of the problem, shown in Fig. 3, the particle has a definite location and speed (and no wave function), with

• a 50 percent chance of being left of center and moving right, and
• a 50 percent chance of being right of center and moving left.

Figure 3: A pre-quantum view of Fig. 1, showing a single particle with equal probability of moving right or moving left. The particle will reach x=0 in both possibilities at the same time, but in pre-quantum physics, nothing special happens then.

Nothing interesting, in either possibility, happens when the particle reaches the center. Either it reaches the center from the left and keeps on going OR it reaches the center from the right and keeps on going. There is certainly no collision, and, in pre-quantum physics, there is also no interference effect.

Still, something abstractly interesting happens there. Before the particle reaches the center, the top and bottom of Fig. 3 are different. But just when the particle is at x=0, the two possibilities in the superposition describe the same object in the same place. In a sense, the two possibilities meet. In the corresponding quantum problem, this is the precise moment where the quantum interference effect is largest. That is a clue.

Two Particles, Two Orderings

So now let’s look in Fig. 4 at the example that I gave as a puzzle, a sort of doubling of the single particle example in Fig. 1.

Figure 4: Two particles in a superposition of moving left or moving right — a sort of doubling of Fig. 3.

Here we have two particles moving from left to right OR from right to left, with 50% probability for each of the two possibilities. I haven’t drawn the corresponding quantum wave function for this yet, but I will in a moment.

We might think something interesting would happen when particle 1 reaches x=0 in both possibilities (Fig. 5a), just as something interesting happens when the particle in Figs. 1-3 reaches x=0 in both of its possibilities. But in fact, there is no interference. Nor does anything interesting happen when the blue particle at the top and the orange particle at the bottom arrive at x=1 (Fig. 5b). Similarly, no interference happens when particle 2 reaches x=0 in both possibilities (Fig. 5c). These “events” are really non-events, as far as quantum physics is concerned. Why is this?

Figure 5a: After the two particles in Fig. 4 have moved slightly, the blue particle is at the same point in both halves of the superposition. Yet in the quantum version of this picture, no interference occurs. Figure 5b: As in Fig. 5a, but slightly later; again no interference occurs. Figure 5c: As in Fig. 5a; yet again there is no quantum interference. The Puzzle’s Puzzling Lack of Interference

To understand why interference never occurs in this case, we have to look at the system’s wave function and how it evolves with time.

Before we start, let’s make sure we avoid a couple of misconceptions:

• First, we don’t have two wave functions (one for each particle);
• Second, the wave function is not defined on physical space (the x axis).

Instead we have a single wave function Ψ(x1,x2), defined on the space of possibilities, which has an x1-axis, (which I will draw horizontal), giving the position of particle 1 (the blue one), and an x2 axis (which I will draw vertical) giving the position of particle 2 (the orange one). The square of the wave function’s absolute value at a specific possibility (x1,x2) tells us the probability of simultaneously finding particle 1 at position x1 and particle 2 at position x2.

In Fig. 6, I have shown the absolute-value-squared of the initial wave function, corresponding to Fig. 4.

Figure 6: Graph of the squared absolute value of the initial wave function, |Ψ(x1,x2)|2, corresponding to Fig. 4. The function is shown dark where it is large and white where it is very small. The two peaks are located at (x1,x2)=(-1,-3) and at (x1,x2)=(+1,+3).

In the first possibility in Fig. 4, we have x1=-1 and x2=-3. One peak of the wave function is located at that position, at lower left in Fig. 6. The other peak of Fig. 6, corresponding to the second possibility in Fig. 4, is located at the position x1=+1 and x2=+3, exactly opposite the first peak.

Fig. 7 now shows the exact solution to the Schrodinger equation, which shows how the wave function of Fig. 6 evolves with time.

Figure 7: How the wave function starting from Fig. 6 evolves over time; there is no interference.

What do we see? The two peaks move generally toward each other, but they miss. They never overlap, so they cannot interfere. This is what makes this case different from Fig. 1; the wave function’s peaks in Fig. 1 do meet, and that is why they interfere.

Why, conceptually speaking, do the two peaks miss? We can understand this using the pre-quantum method, drawing the system not in physical space, as in Fig. 4, but in the space of possibilities. The top possibility in Fig. 4 first puts the system at the star in Fig. 8a, moving up and to the right over time. Because the two particles have equal speeds, every change in x1 is matched with an equal change in x2, which means the star moves on a line whose slope is 1 (i.e. it makes a 45 degree angle to the horizontal.) Similarly, the bottom possibility puts the system at the star in Fig. 8b, moving down and to the left.

Figure 8a: In the space of possibilities, the pre-quantum system in the top possibility of Fig. 4 is initially located at the star, and changes with time by moving along the arrow. Figure 8b: Same as Fig. 8a, but for the bottom possibility in Fig. 4.

If the two stars ever did find themselves at the same point, then what is happening in the first possibility would be exactly the same as what is happening in the second possibility. In other words, the two possibilities would cross paths. But this does not happen here; the paths of the stars do not intersect, reflecting the fact that the top possibility and bottom possibility in Fig. 4 never look the same at any time.

Quantum physics combines these two pre-quantum possibilities into the single wave function of Fig. 7. The two peaks follow the arrows of Figs. 8a and 8b, and so they never overlap.

The three (non-)events shown in Figs. 5a-5c above correspond to the following:

• At the time of Fig. 5a, the two peaks in Fig. 7 are on the same vertical line (they have the same x1)
• At the time of Fig. 5b, the two peaks are aligned along the diagonal from lower right to upper left.
• At the time of Fig. 5c, the two peaks are on the same horizontal line (they have the same x2).

The Flipped Order

Let’s now compare this with the next example I gave you in my previous post. It is much like Fig. 4, except that in the second possibility we switch the two particles.

Figure 9: As in Fig. 4, except with the two particles switching places in the bottom part of the superposition.

This case does have interference. How can we see this?

The top possibility is unaltered, and so Fig. 10a is the same as Fig. 8a. But in Fig. 10b, things have changed; the star that was at x1=+1 and x2=+3 in Fig. 8b is now moved to the point x1=+3 and x2=+1. The corresponding arrow, however, still points in the same direction, since the particles’ motions are the same as before (toward more negative x1 and x2.)

Figure 10a: Same as Fig. 8a, except with the point (x1,x2)=(+1,-1) circled. Figure 10b: A new version of Fig. 8b with particles 1 and 2 having switched places. The system now reaches the circled point (x1,x2)=(+1,-1) at the same moment that it does in Fig.10a.

Now the two arrows do cross paths, and the stars meet at the circled location. At that moment, the pre-quantum system appears in physical space as shown in Fig. 11.

Fig. 11: Quantum interference occurs when, in the pre-quantum analogue, the two possibilities put all their particles in the same place.

In both possibilities, the two particles are in the same locations. And so, in the quantum wave function, the two peaks will cross paths and overlap one another, causing interference. The exact wave function is shown in Fig. 12, and its peaks move just like the stars in Fig. 10a-10b, resulting in a striking interference pattern.

Figure 12: The wave function corresponding to Figs. 9-11, showing interference when the peaks overlap. Profound Lessons

What are the lessons that we can draw from this pair of examples?

First, quantum interference occurs in the space of possibilities, not in physical space. It has effects that can be observed in physical space, but we will not be able to visualize or comprehend the interference effect completely using only physical space, whose coordinate in this case is simply x. If we try, we will lose some of its essence. The full effect is only understandable using the space of possibilities, here two-dimensional and spanned by x1 and x2. (In somewhat the same way, we cannot learn the full three-dimensional layout of a room having only a photograph; some information about the room can be inferred, but some part is inevitably lost.)

Second, starting from a pre-quantum point of view, we see that quantum interference is expected when the pre-quantum paths of two or more possibilities intersect. As an exercise, go back to the last post where I gave you multiple examples. In every case with interference, this intersection happens: there is a moment where the top possibility looks exactly like the bottom possibility, as in Fig. 11.

Third, quantum interference is generally not about a particle interfering with itself — or at least, if we try to use that language, we can’t explain when interference does and doesn’t happen. At best, we might say that the system of two particles is interfering with itself — or fails to interfere with itself — based on its peaks, their motions and their potential intersections in the space of possibilities. When the system consists of only one particle, it’s easy to confuse these two notions, because the system interfering looks the same as the particle interfering. More generally, it is very easy to be misled when the space of possibilities has the same number of dimensions as the relevant physical space. But with two or more particles, this confusion is eliminated. For significant interference to occur, at least two possibilities in a superposition must align perfectly, with each and every particle in matching locations. Whether this is possible or not depends on the superposition’s details.

How Do We Observe the Interference?

But now let’s raise the following question. When there is interference, “where” is it? We can see where it is in the space of possibilities; it’s clear as day in Fig. 12. But you and I live in physical space. If quantum interference is really about interfering waves, just like those of water or sound, then the interference pattern should be located somewhere, shouldn’t it? Where is it?

Well, here’s something to think about. The double-slit-like interference pattern in Fig. 2, for one particle in a superposition, produces a real, observable effect just like that of the double-slit experiment. In Fig. 12 we see a similar case at the moment where wave function’s two peaks overlap. How can we observe this interference effect?

An obvious first guess is to measure the position of one of the particles. The result of doing so for particle 1, and repeating the whole experiment many times (just as we always do for the double-slit experiment) is shown in Fig. 13.

Figure 13: If we measure the position of particle 1 at the moment of maximum interference in Fig. 12, and repeat the experiment many times, we will see random dots centered near x=+1, with no interference pattern. (Each new measurement is an orange dot; previous measurements are blue dots.)

There are no interference peaks and valleys at all, in contrast to the case of Fig. 1, which we examined here (in that post’s Fig. 8). Particle 1 always shows up near x1=+1, which is its location where the two peaks intersect (see Figs. 10-12). No interesting structure within or around that peak is observed.

Not surprisingly, if we do the same thing for particle 2, we find the same sort of thing. No interference features appear; there’s just a blob near its pre-quantum location in Fig. 11, x2=-1.

And yet, the quantum interference is plain to see in Fig. 12. If we can’t observe it by measuring either particle’s position, what other options do we have? Where — if anywhere — will we find it? Is it actually observable, or is it just an abstraction?

If you are already tired of hearing about RFK Jr. being the Secretary of Health and Human Services (HHS), buckle in. I get it – it can be exhausting. Don’t we already know everything we need to about what a medical crank he is? While SBM will not neglect the many topics that we cover, we certainly consider it a high priority […]

The post David Geier Hired to Study Vaccines and Autism first appeared on Science-Based Medicine.

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Representative Anna Paulina Luna, the chairwoman of the newly created Task Force on the Declassification of Federal Secrets, has said that the JFK assassination is only the first of several Oversight Committee investigations. Others include the murders of Senator Robert F. Kennedy, the Reverend Dr. Martin Luther King, Jr; the origins of COVID-19; unidentified anomalous phenomena (UAP) and unidentified submerged objects (USOs); the 9/11 terror attack; and Jeffrey Epstein’s client list.

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Photo by History in HD / Unsplash

I have always advocated for a full release of all JFK files. The American people have the right to know what its government knows about the case. Why, however, spend the task force’s time and taxpayer money investigating the assassination? Representative Luna is not leading an honest probe into what happened. Her public statements show she has already made up her mind that the government is hiding a massive conspiracy from the public and she believes she will expose it.

Representative Luna is not leading an honest probe into what happened.

“I believe there are two shooters,” she told reporters last month at a press conference.

She also said she wanted to interview “attending physicians at the initial assassination and then also people who have been on the various commissions looking into—like the Warren Commission—looking into the initial assassination.”

Rep. Anna Paulina Luna, the chairwoman of the newly created Task Force on the Declassification of Federal Secrets, told reporters: “I believe there are two shooters.”

When it was later pointed out to her that all the members of the Warren Commission are dead, as are the doctors who performed the autopsy on JFK, she backpedaled on X to say she was interested in some Warren Commission staff members who were still alive, as well as several physicians who were at Dallas’s Parkland hospital, where Kennedy was taken after he was shot. Rep. Luna told Glenn Beck on his podcast last month, that she thought the single bullet (Oswald’s second shot that struck both Kennedy and Texas Governor John Connally) was “faulty” and the account of the Dallas Parkland doctors who tried saving JFK “reported an entry wound in the neck….we are talking about multiple shots here.”

The Parkland doctors were trying to save Kennedy’s life. They never turned JFK over on the stretcher on which he had been wheeled into the emergency room. And they did a tracheotomy over a small wound in the front of his throat. Some doctors thought that was an entrance wound. Only much later, when they saw autopsy photographs, did they see an even smaller wound in JFK’s high shoulder/neck that was the bullet’s entrance wound. The hole they had seen before obliterating it with the tracheotomy was the exit of the shot fired by Oswald.

Rep. Luna is determined to interview some octogenarian survivors to get their 62-year-old recollections.

That does not appear to be enough to slow Rep. Luna, who is determined to interview some octogenarian survivors to get their 62-year-old recollections. She is planning even for the subcommittee to make a cold case visit to the crime scene at Dealey Plaza. As I wrote recently on X, “The JFK assassination is filled with researchers who think Oliver Stone is a historian. She will find fertile ground in relying on the hazy and ever-changing accounts of ‘original witnesses.’”

The release of 80,000 pages of JFK files in the past week, and the lack of any smoking gun document, has not dissuaded her investigation.

The release of 80,000 pages of JFK files in the past week, and the lack of any smoking gun document, has not dissuaded her investigation. She has reached out to JFK researchers who are attempting furiously to build a circumstantial and X-Files worthy “fact pattern” that the CIA somehow manipulated Oswald before the assassination. All that is missing in the recent document dump is credible evidence for that theory. It has not stopped those peddling it, however. Nor has it slowed Rep. Luna. 

Last week, she showed the extent to which she had fallen into the JFK rabbit hole. She posted on X, “This document confirms the CIA rejected the lone gun theory in the weeks after the JFK assassination. It's called the Donald Heath memo.” Not quite. The memo she cited is not about the Agency deciding who killed or did not kill Kennedy in the weeks after the assassination. It instead is a memo directing Heath, a Miami-based CIA operative, to investigate whether there were assassination links to Cuban exiles in the U.S. Those exiles, who thought that Kennedy was a traitor for the Bay of Pigs fiasco, were on a short list along with Castro’s Cuba and the KGB on the intelligence agency’s early suspect list. 

Government agencies have undoubtedly failed to be fully transparent about the JFK assassination.

Government agencies have undoubtedly failed to be fully transparent about the JFK assassination. The CIA hid from the original Warren Commission its partnership with the Mafia to kill Fidel Castro. And the Agency slow walked for decades information of what it learned about Oswald’s unhinged behavior at the Cuban and Soviet embassies in Mexico City, only six weeks before JFK visited Dallas. I have written that JFK’s murder might have been preventable if the CIA and FBI had shared pre-assassination information about Oswald. However, political theater that is disguised as a fresh investigation, serves no interest other than feeding the en vogue MAGA conspiracy theories that blame everything on the deep state.

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