Let’s go straight to the source!
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 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, following Feynman, he discusses another mystery, that is - when we know through each hole an electron travelled (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 ravel 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.
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 Schrödinger's equation. And that behavior - statistically - resembles some elements of the behavior of classical waves.
But quantum particles are NOT waves.
Naturally, Einstein would never accepted this solution, but no one is infallible.
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!
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