I know I'm wrong but this is my line of thought: If electrons are indistinguishable, then why do we have an exclusion principle? If we have two electrons in an s orbital, the Pauli exclusion principle says that they can't have the same set of quantum numbers, but then what does that say about electrons being indistinguishable?
So we have these two electrons that are supposed to be indistinguishable, but then we say, no they can't have the same set of quantum numbers, isn't this making them distinguishable then?
Answer
The indistinguishability of particles is expressed by imposing certain symmetry constrains on the state functions and on the observables. As you may know, there can be symmetric and antisymmetric state functions as you interchange two particle coordinates, and all the observables must be invariant under such operations. And this postulate agrees with the experimental data. The particles being identical can be explained as saying that the physical system is unchanged if the particles are interchanged. This formulation can be expressed mathematically in the following way
$\lvert\psi(p(x_1,\dots,x_n))\rvert^2 = \lvert\psi(x_1,\dots,x_n)\rvert^2$
where $p$ is the permutation of the N particle coordinate. However, from the above you can see that the word interchange, here has no physical meaning. The l.h.s and the r.h.s have no separate meaning and it shows the redundancy in the notation. Putting it in other words, the same particle configuration can be expressed in different ways.
As for Pauli exclusion principle, you know that it says that no two identical fermions may occupy the same quantum state. This is because the total wave function for two identical fermions is anti symmetric with respect to the interchange of the particles.
Hope it helps a bit, but for a far better understanding you should look up the Occupation Number Representation. The starting point of this formalism is the notion if indistinguishability.
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