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?
Best Answer
If you can tell the particles apart (i.e. they have different mass, charge, etc.) then the state of the two-particle system is just a product of the individual states of the two particles a and b:
$$\psi(\mathbf{r}_1,\mathbf{r}_2)=\psi_a (\mathbf{r}_1) \psi_b (\mathbf{r}_2)$$
If the two particles are utterly identical, then we don't (and can't even in principle) know which one is in which state. So our two-particle wavefunction has to be non-committal as to which particle is in which state. There turns out to be two equally valid ways to do this:
$$\psi_{\pm }(\mathbf{r}_1,\mathbf{r}_2)=A[\psi_a (\mathbf{r}_1) \psi_b (\mathbf{r}_2)\pm \psi_b (\mathbf{r}_1) \psi_a (\mathbf{r}_2)]$$
where $A$ is a normalization factor. Particles that correspond to $\psi_{+}$ are called "bosons" while particles that correspond to $\psi_{-}$ are called "fermions." Two fermions cannot possibly occupy the same state because if $\psi_a=\psi_b$, then $\psi_{-}=0$ and this isn't a square-integrable function (i.e. not a valid wavefunction). This property is dubbed the "Pauli Exclusion Principle."