Interference between electrons in double slit experiment

Question

Here I use interfere and interference to refer to an interference pattern of probability waves. With interact I mean all kinds of interactions.

It is experimentally proven that an interference pattern can be observed in the double slit experiment even if only a single electron passes at any given time. The same seems to be true for atoms and even some larger molecules.

What I haven't found clear information for is the question if multiple particles, especially electrons, interfere with each other when passing the double slit simultaniously. Obviously this is not necessary as described above.

It seems clear that in the case of photons they don't interact with each other under most circumstances, except as described at https://en.wikipedia.org/wiki/Two-photon_physics. So in the double slit experiment the wave function describing the probability for the different paths only interferes with itself.

In the case of the electron I would now assume the same thing, even if multiple electrons pass the double slit at the same time. I would even assume that multiple electrons passing simultaniously would cause a less clean interference pattern than when each electron passes the double slit alone. The reason for this is that electrons can interact with each other for example with Coulomb interaction which could mess with the probabilities. But that on the other hand might be rather rare.

Are there any papers or experiments that cleary show that probabities of different electrons interfere with each other? Are there even reasons one would have to assume they would normally interfere with each other? Or is there rather evidence or calculations that show that interference with their own probability waves is the only significant reason for the observation of interference patterns in electrons?

Answer 1

Consider the interference with two electrons and let that these electrons are in different states say one of them $n=1$ and the other $n=2$ then the probability distribution in $x$-space would be $$P(x_1,x_2)=2|\psi_a(x_1,x_2)|^2$$

$$P(x_1,x_2)=2|2^{-1/2}[\psi_1(x_1)\psi_2(x_2)-\psi_2(x_1)\psi_1(x_2)]|^2$$ $$=|\psi_1(x_1)|^2|\psi_2(x_2)|^2+|\psi_2(x_1)|^2|\psi_1(x_2)|^2-[\psi^*_1(x_1)\psi_2(x_1)\psi^*_2(x_2)\psi_1(x_2)+\psi^*_2(x_1)\psi_1(x_1)\psi^*_1(x_2)\psi_2(x_2)]$$ Look at the second term! There is parallel between this situation and the one with the double slit experiment one electron, where the probabilities for finding a particle at a given point $x$ on the screen with both slits open was not the sum of the probabilities with either slit open.

In both cases, the interference terms arise, because in quantum theory, when an event can take place can take place in two or more indistinguishable ways, we add the corresponding amplitudes and not the corresponding probabilities.

Note As $x_1\rightarrow x_2\rightarrow x$ $$P(x_1\rightarrow x,x_2\rightarrow x)\rightarrow 0 \ \ (\text{Pauli principle})$$

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Relevant article

The Double Slit Experiment for Electrons