So here’s what the exclusion principle states:
“No 2 fermions( let’s say electrons) can have the same quantum states”
Consider the following hypothetical situation:
Let’s have 2 free electrons in a vacuum. They are also free from gravity. Is it possible to modify the exclusion principle in to the following statement:
“ No 2 fermions (electrons) can occupy the the same position in space and time simultaneously”
I state the above because the electrons are not bound to any atoms. The 2 electrons also have different spins.
Best Answer
You need also to account for spin.
One way to answer your question is rephrase it as the statement that the total wavefunction of a 2-fermion system must be fully antisymmetric. If the spin part of this system is antisymmetric, i.e. $$ \vert\chi\rangle = \frac{1}{\sqrt{2}}\left( \vert +\rangle \vert -\rangle - \vert -\rangle \vert +\rangle\right) \tag{1} $$ then the spatial part can be symmetric and then two fermions could occupy the same position (if they didn't interact) in the sense that the probability density for $\vec r_1=\vec r_2$ would not be $0$. The probability of finding them both at exactly the same point in space is $0$ since the integral of this distribution over an area of size $0$ (i.e. the line $\vec r_1=\vec r_2$) will be $0$.
Of course the electrons would interact through electromagnetic repulsion, which would push them one away from the other.