Is there interferences between 2 electron wavefunctions?

Question

The 2 slits experiment done with 1 electron shows interference from the "splitted" wavefunctions. EDIT: as precised in an answer, it is after many electrons goes in the experiment that the interferences appear from the spots of all electron

My question is, if we sent 2 electrons simultaneously in the 2 slits and each one can go in only 1 slit because we put a separation between the slits so we ensure that electron e1 goes in slit s1 and electron e2 goes in slit s2 like this :

Does the 2 wavefunction of the 2 electrons would interfere each other ?

EDIT: My question received many answers and I thank you all. The answers are considering the experiment in different ways but I see some key points to take in consideration :

• My question disregards which system is studied. I saw my 2 electrons as 2 systems interfering into each other whereas the correct way to see it is to consider 1 system of 2 particles which may have interference between 2 states.
• Some answers suggest that we should use the same source for both electron which would enforce the idea of 1 system from the source (whereas I find more "challenging" to have 2 different sources but it maybe very tricky to do)
• It is also highlighted that one key point is that we should not know "which way" one electron goes to have interference. So it is important to have no way to distinguish one electron from another (and so if this experiment indeed shows interference in a "not distinguishable electron setup" and afterward we find a way to "mark" an electron from the source and find out this mark on the final spot, it would destroy the interference)
• A study is cited that seems to have some common point with my question : https://doi.org/10.1016/S0006-3495(01)76179-6

(sorry I don't pick an answer as "correct" for the moment as there is a variety of point of view and I don't have the competence to be sure which one are more correct of the other so I did this edit to underline some ideas)

Answer 1

The electrons would interfere with each other in the same way — and under the same circumstances — that photons would interfere with each other in the two-slit experiment. In particular, the electron beams will interfere with each other if the waves entering the slits are coherent, meaning that the phase difference between the waves is constant in time.

The easiest way to do this (for both electrons and photons) is to have the same particle source hitting both of the slits. However, if you could cook up a situation where the electron waves (or light waves) from two different sources were guaranteed to always have the same phase difference when they entered the slits, then the results would be indistinguishable from the case where the slits are illuminated by the same source. Depending on the phase difference between the slits, the interference pattern would shift back & forth. For example, if the waves entering the slits were always in phase with each other, the point equidistant from the slits would be a "bright spot"; if they were 180° out of phase, it would be a "dark spot."

If you were inject a bunch of electrons into the two slits, two at a time, and you didn't make sure that their phase differences were always the same, what would happen? Well, remember that at the screen, you don't see the full interference pattern immediately — you see a couple "flashes" on the screen where individual electrons are detected. The probability distribution of the flashes is higher where the interference pattern is "brighter", and dimmer where it's lower.

If the phase difference between pairs of electrons were always the same, then these individual particle detections would eventually fill out the interference pattern we know & love. But if the phase difference between the slits shifted randomly, then the locations where pair #1 is detected would be governed by a different (shifted) probability distribution than where pair #2 would be detected, which would be governed by a different probability distribution than where pair #3 would be detected, and so on. The net effect is that the "bright" and "dark" spots would wash out, and the net effect would be a uniform distribution on the screen — no interference observed.

Answer 2

In my opinion there will be no interference pattern. I would be very happy to hear valid objections to any of my statements, so I can reconsider.

Edit: I am expanding on my points.

1. You only get an interference pattern if you have 2 slits an electron can even go through. This is not possible in your setup. To be more precise, an electron behaves according to a probability wave, which in this case determines where on the detection screen it lands with which probability. So each of these electrons has its own probability wave and therefore the pattern is the same as if you have 2 single slit experiments using the same detection screen. So you will have a maximum opposed to each slit and the tapering off to each side. They overlap, but not in such a regular way as in the regular double slit experiment.
2. If you only leave one possible slit for an electron to go through, it is like a measurement and the wave function collapses destroying any possible interference pattern. It doesn't matter if we don't know where it landed. We could have a setup where we observe each slit and consider only cases where we can see one electron pass in each of them at the same time, but we will be unable to detect where each of the lands. That's because once we know the location we don't know its momentum (uncertainty). As far as I can see this approach would be identical to yours.
3. It is not the same as a pebble thrown into the water. One single electron can cause an interference pattern with itself. If you have an ordinary double slit 2 electrons will not necessarily interfere with each other, but each single one with itself. This self-interference is not possible if the electron has only one possible slit. I have looked for evidence that electrons would interfere with each other at all, when passing through a double slit, but I haven't found any evidence. With photons it appears to be the rule that they don't interfere with each other at all, so I propose it is the same with electron probability waves. The can interfere with each other as with Coulomb forces, but this is more random and wouldn't cause regular patterns.
4. A direct problem with the experiment is as far as I can tell the Heisenberg uncertainty principle. If 2 electrons pass the 2 slits at the same time you know momentum and location, which doesn't work. To have them possibly interfere with each other they need to have the same momentum, which means wave length.
5. For 2 electrons to create an interference pattern, they would have to be entangled, so they have a common probability wave. This on the other hand would prevent us from claiming that one electron went through either slit. Both electrons would be in a superposition where you are unable to tell where they went.

However I found this article: https://doi.org/10.1016/S0006-3495(01)76179-6 . I am not sure if this work is validated. There it appears that particles can interfere even if the path is known as long as many particles have "phase-correlated wave distributions".

Answer 3

I'm thinking the two wave packets could interfere if they were identical and overlapped n the region beyond the slits.

Answer 4

The 2 slits experiment done with 1 electron shows interference from the "splitted" wavefunctions.

You can't observe or detect this interference by using just 1 electron. There would be one dot on the screen and no interference can be seen from that. Only if many electrons have been sent sequentially an interference pattern can be shown.

if we sent 2 electrons simultaneously in the 2 slits and each one can go in only 1 slit because we put a separation between the slits so we ensure that electron e1 goes in slit s1 and electron e2 goes in slit s2... Does the 2 wavefunction of the 2 electrons would interfere each other ?

You assume there are two wavefunctions, for each electron one. That is not how quantum theory works - if the system consists of $N$ particles, we should describe it with single wave function of $N$ variables (in non-relativistic theory), or there is single field whose state corresponds to "$N$ particles". So whichever theory we use, there is single concept that describes all the "interesting" electrons in the system.

Sending just two electrons at a time into the two slits at the same time (with error much smaller than $\frac{L}{\sqrt{\frac{2E}{m}}}$ where $L$ is distance separating the slits and $E$ is energy of the electron) is difficult but seems possible.

I did not try such experiment nor did I analyze this in detail, but intuitively I would say that if we managed to put many pairs of electrons through the slits in the above manner, we could observe interference similar to the standard two-slit experiment.

Why? For two reasons.

1. All electrons in the experiment should be described by single concept, either wave function $\psi$ or operator field. Boundary conditions for this field are such that the field can be "strong" in both slits at the same time. Single field that has such phase relation between two positions will interfere. So mathematically things look very similar to the standard explanation of two-slit experiment with single electron beam, which predicts intereference. Same math, same prediction.

2. By making sure the electrons are in the slits at the same time, we lose the capability to find out which screen mark was due to which slit electron. The electron that falls on the screen at position $X$ could have come from any slit of the two, like in the standard two-slit experiment with one electron beam. If we believe the argument that impossibility to say which way the electron went means "the electron wave/field went both ways", then these two possibilities should interfere in the calculation and the measured pattern should manifest this interference.

In addition to interference, since we have two electrons in the system at the same time, there should be some repulsion during their flight, so on the symmetry axis there should also be a decrease of the pattern intensity.

This is a very interesting idea to investigate experimentally. We are quite sure that interference of separate sources would be present for EM field, because we can observe it for low frequency sources (radio, microwaves) and by extrapolation we believe it would be present even for visible or UV radiation. But for electrons I think this is much less investigated. It would be great if somebody tried to do this experiment.

Even from theoretical viewpoint, this is interesting because EM interference, even from two different sources, is always described as "photon interferes just with itself, not with other photons". If there are two sources, still just one photon interferes with itself. But here with two electrons, the interference would be due to two electrons present at the same time, so it is quite a different situation.

Answer 5

There will be little two slit interference. The wave function is an expansion of Slater determinants dominated by determinants where each electron goes through a different slit, or rather, where each orbital travels through a different slit. Two slit interference results when orbitals pass through both slits. Because of Coulomb repulsion the contribution of such orbitals is suppressed.