## Monday, October 14, 2019

### Can an electron travel through two slits at the same time?

Can an electron travel through two slits at the same time?

Let’s go straight to the source!

In his lectures Richard Feynman described a thought experiment with electrons traveling through two holes

This experiment has been realized later by different teams, e.g. “Demonstration of single-electron buildup of an interference pattern” / A. Tonomura, J. Endo, T. Matsuda, T. Kawasaki, and H. Ezawa / American Journal of Physics 57, 117 (1989); doi: 10.1119/1.16104 (view online: https://doi.org/10.1119/1.16104)

All those experiments were used to support a “classical” interpretation of its results, i.e. “a single electron can pass through both of the slits” (A. Tonomura et al., 1989).

Let us analyze what the author of the experiment thought on this matter.

Feynman stated that electrons are registered in “lumps”.

Then he stated: “Proposition A: Each electron either goes through hole 1 or it goes through hole 2.”

Then he arrived at: “For electrons: P12 ≠ P1+P2.”

And then he finishes: “… since the number that arrives at a particular point is not equal to the number that arrives through 1 plus the number that arrives through 2, as we would have concluded from Proposition A, undoubtedly we should conclude that Proposition A is false. It is not true that the electrons go either through hole 1 or hole 2.”

And yet, in the next chapter

he writes (all bold fonts are mine, not Feynman’s):

1. “when there are two ways for the particle to reach the detector, the resulting probability is not the sum of the two probabilities”

2. “When a particle can reach a given state by two possible routes, the total amplitude for the process is the sum of the amplitudes for the two routes considered separately.

3. “we are going to suppose that the holes 1 and 2 are small enough that when we say an electron goes through the hole, we don’t have to discuss which part of the hole.”

4. “the amplitude for the process in which the electron reaches the detector at x by way of hole 1

5. “the amplitude to go from s to x by way of hole 1 is equal to”

6. “The electron goes from s to 1 and then from 1 to x.”

7. “The electron can go through hole 1, then through hole a, and then to x; or it could go through hole 1, then through hole b, and then to x; and so on.”

8. “amplitude that an electron going through slit 2 will scatter a photon”

9. “the amplitude that an electron goes via slit 2 and scatters a photon”

10. “two factors: first, that the electron went through a hole, and second”

11. “when an electron passes through hole 2

12. “when the electron passes through hole 1

Theses twelve quotes (the lectures have more similar statements) clearly show that Feynman believed that an electron could travel through one hole/slit, or through another one, but he never considered an electron traveling through both holes at the same time; he never made that statement.

He wrote, for instance: “the probability of arrival through both holes”. But “arrival through” is not the same as “traveling through both at the same time”; it means rather “arrival through a screen with two holes”.

The whole idea of a path integral is based on the assumption is that an electron is always located somewhere, i.e. it is always localized, because it is always traveling through this point and then this, and then this, etc. A path does not split, there are no forks (even when a particle circles back making a loop the time keeps running ahead and on each path a particle is always located at one place at a time), hence, there are no instances when an electron is located at to places at the same time.

A path integral was a brilliant idea of a genius: just assign an amplitude to each possible path and add them up! So obvious! After you learn it. That is what many physicists feel - it's natural, and do not think about implications to the fundamentals of quantum mechanics, including the interpretation of the wave-particle duality. And the genius of Feynman was not inventing paths, but assigning an amplitude to each one.

A simple toy with small balls running down a set of pins represents a good model for paths and a path integral.
When a ball drops through a spout, its trajectory through the board is unpredictable. For each trajectory that begins at point A and ends at point B, there is a probability that a ball will travel exactly along that trajectory.

The key words is “probability”. Feynman realized that in the quantum world we can use the same picture, but instead of a probability we have to use a probability amplitude. The one who will explain – why? – deserves the Nobel Prize.

Let us return to our main topic. As we see, the idea of a path integral is based on the assumption is that an electron is always located somewhere, i.e. it is always localized, and, hence, there are no instances when an electron is located at to places at the same time.

This seems contradicts Feynman’s own conclusion about Proposition A.

He wrote: “is not true that the lumps go either through hole 1 or hole 2, because if they did, the probabilities should add”.

But later, as I proved using his own words, in his further analysis he was fine with an electron traveling through one whole or another.

So, what did he really mean?

I believe, when Feynman stated his Proposition A, he simply did not do it as accurate as he should have done.

He should have said: “Proposition A: Each electron either goes through hole 1 or it goes through hole 2 – in a classical sense”.

And this statement is false.

Based on the next chapter (experiments with light), we understand that when he said: “It is not true that the electrons go either through hole 1 or hole 2 ”, he meant “It is not true that we are always able to know if the electrons go either through hole 1 or hole 2 - unless the interference between the two paths is destroyed”.

Because later he told us that an electron does go through hole 1 or it goes through hole 2 – however, in a different, non-classical sense, with the use amplitudes instead of probabilities.

Feynman wrote a lot. For example, this is what he wrote in "QED: the strange theory of light and matter".
He states that we - everyone in physics - know that light is made of particles, and proves it. And his reasoning is identical to the one about electrons or any other quantum particle. So, there is NO question about the structure of matter - it is made of particles. And a particle can travel only through one hole at a time. And he also called "wave-particle duality" as a "state of confusion".
An article after an article, a book after a book demonstrate that a lot of people writing about quantum mechanics still remain in that state.

If we accept that an electron can travel through a hole – through only one hole, it is not clear yet from Feynman’s discussion what is really happening in a two-hole experiment when no one is watching where exactly an electron gets through the screen?

Naturally, many other physicists jumped on this thought experiment and discussed it in great details in their books.

For example, J. D. Cresser writes (2009; http://physics.mq.edu.au/~jcresser/Phys301/Chapters/Chapter4.pdf; in the following quotes, all bold fonts are mine):
“If electrons are particles, like bullets, then it seems clear that the electrons go either through slit 1orthrough slit 2, because that is what particles would do. The behavior of the electrons going through slit 1 should then not be affected by whether slit 2 is opened or closed as those electrons would go nowhere near slit 2. In other words, we have to expect that P12(x)=P1(x)+P2(x), but this not what is observed. It appears that we must abandon the idea that the particles go through one slit or the other. But if we want to retain the mental picture of electrons as particles, we must conclude that the electrons pass through both slits in some way because it is only by ‘going through both slits’ that there is any chance of an interference pattern forming. After all, the interference term depends on d, the separation between the slits, so we must expect that the particles must ‘know’ how far apart the slits are in order for the positions that they strike the screen to depend on d, and they cannot ‘know’ this if each electron goes through only one slit. We could imagine that the electrons determine the separation between slits by supposing that they split up in some way, but then they will have to subsequently recombine before striking the screen since all that is observed is single flashes of light. So, what comes to mind is the idea of the electrons executing complicated paths that, perhaps, involve them looping back through each slit, which is scarcely believable. The question would have to be asked as to why the electrons execute such strange behavior when there are a pair of slits present, but do not seem to when they are moving in free space. There is no way of understanding the double slit behavior in terms of a particle picture only.”

In the excerpt, the author repeats arguments as old as fifty or even sixty years old – “no way to understand quantum mechanics if particles are only particles”.

And then the author goes on to building an elaborated picture of a wave packet that is a particle and a wave at the same time, etc., etc..

And then, following Feynman, he discusses another mystery, that is - when we know through each hole an electron traveled (e.g. using flashes of light) we destroy the interference. Only when we do not know how exactly electrons travel through the holes, interference exist. Why? No one knows.

The answer, however, lies in the very statement used to prove that electrons cannot travel through one hole or another one.
Let’s read it one more time.

“If electrons are particles, like bullets, then it seems clear that the electrons go either through slit 1orthrough slit 2, because that is what particles would do. The behavior of the electrons going through slit 1 should then not be affected by whether slit 2 is opened or closed as those electrons would go nowhere near slit 2. In other words, we have to expect that P12(x)=P1(x)+P2(x), but this not what is observed. It appears that we must abandon the idea that the particles go through one slit or the other.”

But abandoning “the idea that the particles go through one slit or the other” is not only one logical solution!

Another one is to abandon a previous statement, that said: “The behavior of the electrons going through slit 1 should then not be affected by whether slit 2 is opened or closed as those electrons would go nowhere near slit 2.”

Why should that behavior be not affected? Because this is what we would expect in the classical mechanics from classical particles! But our experiment involves quantum particles! So, why should we impose on them our classical expectations? There is simply no logical reason to do that. So, let’s not do that and see where it will lead us.

If (a) particles do travel through one hole or another (only one hole at a time), and if (b) the interference pattern exists, it means that the statement is wrong.

The statement: “The behavior of the electrons going through slit 1 should then not be affected by whether slit 2 is opened or closed as those electrons would go nowhere near slit 2.” Is wrong.

And that means that the behavior of the electrons going through slit 1 is affected by whether slit 2 is opened or closed even though those electrons would go nowhere near slit 2.

We can make even a more general statement:

Proposition V: when two slits are open, an electron (and a photon, and any quantum particle!) behaves differently than it does when one slit is open.

Proposition V means that when a quantum particle travels to the screen with holes/slits, it already "knows" how many holes are opened there. And under certain circumstances, some aspects of the behavior of those particles exhibit features similar to features of classical waves.

Particles are not waves. But their behavior may be wave-like.

Let us step for a moment away from the main matter and make this note on the nature of waves.

All classical waves are NOT specific individual physical objects. A wave is a specific form/state of a substance described by a mathematical object called “a field” (more on definitions in "On a Definition Of Science").  A field is a mathematical description of a state of a substance distributed over a large region of space. A substance has structure and composed of a vast number of small and usually identical "blocks" (atoms, molecules, balls and springs). Thinking about a classical wave as of one undivided large object is simply wrong. But even an electromagnetic field has quantum structure – photons. So, when one says this word “a wave” – what does one actually mean?

Let us assume that a wave-function is an actual physical wave. A particle is a wave-pocket traveling in space. Fine. Does it have a definitive size; a boundary between the region filled with matter and energy and the rest of the universe? If it does - so, it is just a large particle? If not, if all the mass and energy asymptotically "smeared" over the whole universe (a mathematical cut-off exists, like "effective radius", but it is mathematical - like a half-life for a radioactive element), how does all that mass and energy get smeared over the whole universe the moment a particle leaves an atom and then "collapses" back when it hits a screen? And if light is a composition of photons, and the double slit interference experiment for light should be explained in the same way it is explained for electrons - how to make a physical wave-pocket for it - it needs to travel at the speed of light in a non-relativistic theory.

These and other questions make this picture too complicated - it does not worth to be fought for.

But in that case one needs a different, simpler model. And that model exists - a particle is always a particle, it just is not classical, hence behaves in a non-classical way described by SchrÃ¶dinger's equation. And that behavior - statistically - resembles some elements of the behavior of classical waves.

But quantum particles are NOT waves.

And the two-slit experiment does not give us any proof to the statement that particles are also waves. The wave-particle duality is NOT about this.

What the two-slit experiment shows us is that the configuration of the screen (one hole, two holes, three holes, etc.) affects the motion of the electrons, photons, all particles traveling toward that screen.

In the classical world, a particle does not know anything about the screen it travels to until it hits it.

But an electron “knows”/“feels” if the hole 2 is open or closed. If we shine a light on an electron, it actually “forgets” about the existence of another hole and travels like the only one hole exists – hence, the destruction of interference.

The real question now is: how do quantum particles “know” how a screen is built and react to its structure?

That is the true mystery of quantum mechanics.

This question requires a new discussion.

In general, the answer is – quantum particles “know” about the features of a screen in the same way they “know” about states of each other when they have been prepared in an entangled way.

The double-slit experiment and quantum entanglement are two very close phenomena.

Let’s go straight to the source – the famous EPR paper.
It has many layers, more than just the thought experiment they use to claim that quantum mechanics is not a complete theory (e.g. click on this link and scroll down to Appendix III). If someone talks to you about entanglement, ask if he/she rad this paper. If not - does not worth your time (more an entanglement in "On the Entanglement Between SuprFluidity, SuperConductivity and Entanglement").

The fact of the matter is that this experiment does show that quantum mechanics is different from classical mechanics (as EPR put it – “incomplete”).

When this matter is accepted, one has a choice: (a) follow the strategy "shut up and calculate" and do not spend any time on trying to make the theory "complete", or (b) spend some time on trying to make the theory "complete".
In the latter case, one can be inventing different approaches - some are mentioned in the four pieces about a cat:

But the simplest (thank you - Occame!) way to resolve all the mysteries of quantum mechanics would be to assume that - yes, "spooky action at a distance" exists, and it exists due to facter than light interactions!

Naturally, Einstein would never accepted this solution, but no one is infallible.

Particles that travel faster than light have been proposed, and named tachyons.

Tachyons are responsible for that "spooky action at distance".

There is a whole world of particles that cannot travel slower than the speed of light! And that world interacts with our world, where particles cannot travel faster than the speed of light.

Simple!

Imagine a sea of tachyons. Every known particle can have its counterpart in that sea: tachyo-electron, tachyo-proton, etc. Due to fluctuations, for a teeny-tiny instant of time, those tachyons may enter our world, become a so-called virtual particle, and interact with our-world particles. But even more interesting process happens when our-world particles can disappear from our world and enter the world of tachyons, spend there a teeny-tiny instant of time and come back again - but at a different location, or with a different speed, or both, or in general in a different state.

When two particles are entangled, they keep interacting via tachyons. And that is why making one particle to accept a certain state (e.g. by imposing a magnetic field) it makes another particle – that one that was entangle with the first one – to immediate accept a corresponding state
(more on entanglement in

Some of the entanglement experiments (thought or real) could have been explained even without the use of tachyons. The distances between the particles would allow photons to make the particles “feel” each other. But tachyons are just so much cooler!

Of course, until tachyons are found, they are just a theory, a mathematical abstract. But so was the Higgs boson.

By employing tachyons, we replace several difficult problems with one difficult problem – finding tachyons.

Tachyons, or in general the world of faster than light particles, can also explain such intriguing quantum phenomenon as tunneling.

A classical particle cannot escape a potential well - when it has not enough energy. But a quantum particle can "tunnel" through. Why? Because due to interactions with tachyons it may "accidentally" (a scientific name – via fluctuations) gain energy enough to get "over the well".
And, finally, back to the double-slit electron diffraction experiment.

A screen is also made of particles. A sea of tachyons between a flying electron and a screen makes those two objects interact and their evolution correlate. Of course, the evolution of a screen is simple – being there. But the evolution of a traveling electron is affected by the structure of the screen. In a way, this picture is similar to the “pilot-wave”theory.

There is a mechanical model that may help to visualize the phenomenon.

Imagine a small ball floating in water. It has a little motor that spins a fan and makes it move. But it also has inside a small of-center spinner, that makes the ball vertically oscillate in water. Those oscillations travel away and when they reach an obstacle, for example a screen, they get reflected and act back on the ball. Of course, the reflected waves will depend on the structure of a screen (one hole or two). And that may affect the motion of the ball.

The sea of tachyons should bring back some version of a “hidden-variables” theory, because the particle-tachyon interaction does not obey the limits imposed by the Von Neumann’s theorem (although, some physicists claim that the theorem has flaws anyway).

The next step is the development of an appropriate mathematical model – and the Nobel!

You’re welcome!

Naturally, I am not the first one who talks about superlight particles. But those particles have been originally treated as a "bad thing" (a sign that quantum mechanics is incomplete), and later as a technical element - one of the options for existing particles. I believe those particles should be considered as the means for explaining the fundamental properties of quantum mechanics. The easiest of the possible means.

And finally, the answer to the question in the title - no, an electron cannot travel through two slits at the same time.

But it does NOT have to!

Disclaimer: the bulk of this post originally were published as appendixes to other posts on this page: Fundamentals of Quantum Physics

Appendix I

All existing analyses of the double-slit electron experiment are based on two statements:
1. When only one slit (hole) is open electrons reach the screen  and form a Bell-shaped pattern.
2. When two slits are open, electrons form an interference pattern that does not represent a simple composition of two Bell-shaped patterns.
But all those existing analyses make the same logical mistake.
The assume that when electrons travel through a single slit, they behave like classical particles.
However, there is absolutely no reason for that assumption.
An analogy with the light traveling through one or two slits (holes) shows that the pattern formed by electrons depends on the size of the opening.
When there is only one opening, but it is large (large enough, in a certain sense), then electrons will be forming a classical-like Bell-shaped pattern. But in this case, even with two openings, we should expect a classical-like pattern, and no interference.
But when the opening is small, electrons should form a single-slit interference pattern. When another opening becomes available, electrons form an interference pattern as well, but this should not be a surprise anymore, because electrons have already formed an interference pattern with only one opening.
And in that case the real question is why do electrons form an interference pattern when they travel through a single small opening?
All existing analyses of the double-slit electron experiments simply combine two incompatible pictures, the classical picture of particles traveling through one hole, and a quantum picture of particles traveling through two holes.
Of course, when you use inconsistent logic, you get confusing results.

Appendix II

When I sent a copy of this article to arXiv, I knew it would be rejected.

My blog, as it’s said in a title, is an experiment. This time I was wishing for reasons my article would be rejected.

I expected to see something like “wrong format”, “loose language”, “absence of citations/references”, “a wrong arXiv section”.

But the result of my fishing exceeded my expectations.

I was told, quote: “article does not contain sufficient original or substantive scholarly research” (the full letter is at the end of this appendix)

It makes one put things in perspective.

When authors write an article where they apply a theory beyond the area of its applicability, trying to use a standard quantum mechanical formalism to a classical system, which is like applying Newton's 2nd law to relativistic particles, but including references, scientifically sound terminology, and cool mathematical symbols - it is considered a research worth to be published (“Quantum theory cannot consistently describe the use of itself”; Appendix II of this piece provides deeper analysis).

When an author offers a critical analysis of a logical structure of statements made about fundamental quantum mechanical phenomenon offering an alternative interpretation – that’s not a “substantive scholarly research”.

This is a typical example of a narrow-minded formatted thinking in science.

Originally, a scientific magazine was an instrument for (1) exchange of scientific ideas, and (2) reporting the results of a scientific research.

Of course, the format matters! But that's what editors are for, or moderators.

I'm not sure how many people on average read every arXiv paper, but in four days since its publication, my article was read by more than 30 people (that doesn't count reads on Academia and Researchgate).
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