What will happen if two fermions, like electrons, come together

[touqra]

What will happen if two fermions, like electrons, having the same quantum numbers are brought very close to each other?

 

[Berislav]

If you mean by 'same quantum numbers' what I think you mean, then the propability of finding them close will be small.

P.S.
Quantum objects aren't very easy to manipulate, I think. Especially if you want to keep them in their original state.

 

[Igor_S]

What will happen if two fermions, like electrons, having the same quantum numbers are brought very close to each other?

It depends. If they are tied to some nucleus, each of their energy levels splits up and one of them (irrelevant which one, they're both the same) will go to higher energy state.

If they are, let's say, free and you try to push them closer (I think something like that happens in white dwarfs or neutron stars) you will reach certain limit above which you cannot bring them any closer due to the pauli exclusion principle. You can imagine (at least that's how I do it) that each fermion establishes a "zone" around himself in which other fermions are not allowed to enter which is not effect of any force, but only consequence of their indistinguishability (hope I spelled this correctly .

 

[Berislav]

You can imagine (at least that's how I do it) that each fermion establishes a "zone" around himself in which other fermions are not allowed to enter which is not effect of any force,

If I may elaborate on this for the OP...
Take a simple two fermion state:
Using bra-ket formalism,

$$\mid\psi_1, \psi_2>= \frac{1}{\sqrt2} (\mid\psi_1> \mid\psi_2> - \mid\psi_2> \mid\psi_1> )$$
we clearly see that we can't have two fermions in the same state ( $\psi_1=\psi_2$ ).

But, since you (touqra) are interested in finding fermions in space we should use take a product of a coordinate x (1D, for simplicity) with the state (aka wavefunction):

$$\Psi(x_1, x_2)=\psi(x_1) \psi(x_2) - \psi(x_2) \psi(x_1)$$

Now, the probability of finding a system in some place is the square of the wavefunction. The probability of finding the two fermions will be decrease as the distance between them decreases, but will only be zero when $x_1=x_2$

My point is that there doesn't exist a minimum distance between two fermions in the same state below which we will never find them, that is the "zone" that Igor_S mentions isn't clearly defined.

 

[dextercioby]

It's essential whether the fermions are identical or not. "Bringing very close to each other" usually implies the ultraviolet behavior of the (hopefully) renormalizable field theory which describes their interaction. QED is UV-renormalizable (just like QCD).

Daniel.