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This is something I have read many times that the double slit experiment done with electrons produce the same pattern that we get with light i.e. the electrons undergo superposition similar to that of light waves and thus can interfere constructively or destructively.

But wait! Electrons carry negative charges and so to have two electrons "superpose" shouldn't they be overlapping each other as if being at the same place? And if yes why won't they repel each other?

In general why doesn't the electric charge of an electron matter when we consider the double slit experiment with electrons?

This answer of Michael Seifert starts by saying that two electrons can actually interfere.

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    $\begingroup$ You misunderstood the phenomena, not the multiple electrons at the single place, but single electron at multiple places at once (very cruel analogy, because you have to analyze wave-function instead as @Leo have said). $\endgroup$ Feb 2 at 11:30
  • $\begingroup$ It is a question of scale. If they are close to each other in a dense electron beam confined by an external magnetic field, say, as in a traveling wave tube, then the interference pattern, if any, will change by their interaction but asymptotically at low density without external confinement they are so far apart from each other that their mutual interaction is negligible and the electrons only "interact" individually with the slits. $\endgroup$
    – hyportnex
    Feb 2 at 12:17
  • $\begingroup$ The virtual forces of the EM field "interfere", an excited electron in the electrode is already interacting with the EM field even before it is emitted (also true for photons). All electrons in the apparatus are affected by each other, virtual forces operating at c. The virtual EM field is wavelike ... that's why Huygen's math works. $\endgroup$ Feb 2 at 14:15
  • $\begingroup$ The electron/photon/buckyball molecule is/are dumb ... the EM field knows all. Each photon gets its own "path" .... electrons must be fired one at a time for the electron interference because they do have a strong effect on each other ... but they also get the "path' from the EM field. Neutral particles (buckyballs) have internal E fields (dipoles) and have a strong EM field interaction as they impact the screen. $\endgroup$ Feb 2 at 14:22
  • $\begingroup$ Superposition of two particle states do not depend on any charges. Too visualise what is happening just calculate the squared amplitude of two superimposed states. The product terms of two states would tell you about the interference. Just remember the basics of quantum mechanics that the states are just mathematical tools which exist in Hilbert spaces, while it's squared of the states which have phisical meaning of existential of a particle at infinitely many positions at the same time, just like a wave. These waves interfere with each other just like waves of light hence similar pattern. $\endgroup$
    – Aman pawar
    Feb 2 at 16:11

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In the double slit experiments, you never have two different electrons interacting. It is always the wavefunction of a single electron interfering with itself. You should not confuse the electron (point charged particle) and it's wavefunction (continuous probability amplitude), which behaves as a wave.

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    $\begingroup$ A wave function is a description of an ensemble. When we are talking "single electron wavefunction" what we mean is that the individual systems in the ensemble only have one quantum each. We don't mean an individual quantum measurement. Wave functions don't interfere. They are not even physical. Think of a wave function the way you would about a probability distribution, which is an abstract to describe observed frequencies in repeated experiments. That wave functions live in a linear function space is a consequence of the construction of quantum mechanics as an ensemble theory. $\endgroup$ Feb 2 at 10:00
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    $\begingroup$ @FlatterMann It is up for debate whether the wavefunction is physical, but I would argue that most popular interpretations of quantum mechanics view the wavefunction as real. Viewed that way, it is more than just an ensemble. It is reality. $\endgroup$ Feb 2 at 10:27
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    $\begingroup$ The wave function was designed to be a description of an ensemble by definition. No version of quantum mechanics makes any statement about an individual outcome. We can't say "the third photon from now will land on detector element 37". There is no theory that can predict that. We can only ever predict ensemble averages. This is not an interpretation issue. It's a hard fact that can be observed in any lab that can do quantum measurements. Pilot waves can't tell you individual outcomes, either. $\endgroup$ Feb 2 at 10:38
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    $\begingroup$ @AccidentalTaylorExpansion I don't know how to measure a wave function and I don't mean just up to phase. I mean I just can't perform an infinite number of experiments. I can do a frequentist approximation of a wave function in the same way I can do an approximation of a real number with the fraction of two integers. That rational approximation is not a real number as the mathematicians can very well tell you. The same logic applies to wave functions. Like real numbers they are mathematical abstracts, forever separated from reality by infinity. $\endgroup$ Feb 2 at 10:41
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    $\begingroup$ These considerations are a bit a far stretch from the original question no? To avoid being ambiguous, I would say that my point was simply that there is a single electron in Young's double slit experiment, not two. $\endgroup$ Feb 2 at 11:29
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Electrons are just quanta of the electron field. To say there is one electron in an experiment means more or less the same as to say the number operator of that field has negligible probability to have an electron number greater than one. If there is an electron number greater than one then the different quanta of the field will repel one another. Otherwise there is no repulsion:

https://arxiv.org/abs/2206.09472

In a single particle interference experiment with an electron the field will be a single electron field in the sense explained above and so there will be no repulsion, only quantum interference effects.

If you actually have two electrons they can interfere because of the Pauli exclusion principle, which states that no two electrons can occupy the same quantum state. This leads to a destructive interference effect called anti-bunching:

https://www.nature.com/articles/34611

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  • $\begingroup$ Pauli correlation has nothing to do with interference. $\endgroup$
    – my2cts
    Feb 2 at 16:53
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Electrons would repel each other. If they overlap, they might not for very long.

Electrons can overlap. They do so in an He atom. Both electrons occupy the lowest energy orbital. Being Fermions, they cannot overlap if they have the same state. So they must differ in at least one quantum number. In the s orbital, that number is spin.

The electrons are attracted to the nucleus, which is why they are so close together. But you should not think of them as staying on opposite sides of the nucleus as they orbit. They don't have precise positions and momenta. They don't orbit.

The wave function does not describe where they are. It describes where their positions are likely to be if you measured it precisely. Each electron has a "presence" everywhere. It is difficult to get away from the idea that the electron is a point particle somewhere inside the wave function, and the wave function just tells you where it is likely to be.

The wave function is concentrated near the nucleus, so the presences is concentrated near the nucleus. That is all the position the electrons have.

You could calculate the expectation value of the potential energy of their mutual repulsion. You would have to do a double integral where each electron could be anywhere in all space. There is a contribution from where they are both very close together, but it is small because most of the integral has them farther apart.

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  • $\begingroup$ electrons overlap in any atom $\endgroup$
    – fraxinus
    Feb 2 at 19:54
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Electrons do not interfere with themselves nor with each other. It is the wave function that exhibits interference effects. For a successful demonstration of electron interference a low enough bundle intensity must be used so that repulsion effects can be neglected and a one electron wave function can be used.

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Interference or diffraction are effects of the interaction of the particle with the field they pass through (see Casimer effect) and which is found in the holes, what we see on the screen: diffraction or interference is that the image of the distribution of particles in this quantified field, we can say that the evolution of physical ideas: particle, wave, field, is a snake that has bitten its tail.

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The reason why the electron double slit works the same way as the photon double slit and the neutron double slit and the helium atom double slit is because all of these are measurements on distributions of quanta of energy. Energy behaves the same, no matter what kind of system it describes.

What undergoes "superposition" are not the individual systems. Superposition is a property of our theory that describes our uncertainty about their "state".

Think of quantum experiments the way you would think about dice. How does the physics of dice work? Dice can be either resting on the table or they can be rolling. Resting dice show one of six faces. They have zero kinetic energy in the rest system of the table. When we pick them up and we roll them they acquire kinetic energy. Rolling dice do not have a value assigned to them. They have to come to rest, first, which they do by shedding energy through friction on the table surface. Only after all of their kinetic energy is shed do they have a defined value, again. We can't talk about the physical state of rolling dice in the language that we use for their outcomes. We can assign a probability distribution to the ensemble (infinite repetition of the same dice roll) of the dice, though. For fair dice that distribution predicts that we will see a roughly equal number of outcomes for every value on average (law of large numbers). We don't have information about the individual outcome of a throw, but we have a good idea that "3" will appear with a similar frequency as "1".

Quantum systems work in a similar way (but as a warning: the math is different and so are the physical consequences, so the analogy only takes you so far): a quantum system is either in a well defined state or it is evolving. While it is evolving we can't assign a final state to the individual system. It is undetermined, just like the rolling dice. When we make a quantum measurement we take a certain amount of energy out of the quantum system. This is analogous to letting the dice lose their kinetic energy on the table. It's an irreversible energy transfer from the quantum system to the measurement system. The total amount of energy that gets transferred is called "the quantum" (of energy). The numerical amount of that energy determines what we call the new state of the system. If we repeat this process over and over again we can build up knowledge about the ensemble of the quantum system, just like we did with the probability distribution of the dice.

It turns out that the math of the function that describes the ensemble, called the wave function, is similar to that of vectors in a vector space. We can perform linear addition, multiplication with constants and operations like decomposition in base vectors with this quantity. These operations on this mathematical abstract are what gives rise to the term "superposition". This is all well defined, but we have to be careful not to assign this function and its mathematical properties to the individual quantum system. It was never intended to be used that way, just like the probability function of individual dice was never intended to be a statement about a single dice throw. All of these qauntities are ensemble averages.

One of the main differences between the dice example and quantum systems is that dice have their "measurement device" built in by having six faces. We are always performing the same "measurement" on them. That is not so with quantum systems. Here "the measurement" is defined by the external system that absorbs the energy of the quantum systems during the irreversible energy transfer process of the measurement. From that it follows that we can perform different measurements on the same ensemble of quantum systems by changing to a different measurement system. What we can never do is to perform two different measurements on an individual copy of the quantum system. It has only one amount of energy and when that energy is shed in one kind of measurement, then it's irreversibly gone and we can't use it again for a second or third measurement. That's no different from dice throws, again. One individual dice throw can be carried out only once. The next time we throw the dice it's a different physical process, just using the same dynamics.

So what does that mean for your electrons? It means that one electron gives you one amount of energy somewhere in your detectors. No interference. Just one measurement result, similar to rolling a "4" in an individual dice roll. Repeat and a different electron will land somewhere else, just like the dice might now show "2". Rinse, repeat many times and a pattern builds up. With dice it's a pattern of equal outcomes. With electrons, photons, neutrons, helium atoms, cows, asteroids and Porsche 911s it's the same diffraction pattern. OK, don't try it with cows, asteroids and cars. It won't work with macroscopic objects, but not because they don't have quantum behavior. We just can't measure it at room temperature and with the instruments that we can build with our current technology. Point is... the system doesn't matter, as long as we are performing the same kind of measurement on its energy.

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