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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?

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3 Answers 3

up vote 4 down vote accepted

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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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."

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4  
I'm not sure why I was down-voted. This is pretty much straight out of Griffiths'. –  jld Feb 21 '13 at 23:09
    
Not sure either :( But there are three of us who upvoted. There's also a neat explanation along these lines but in the context of interference early in the third volume of the Feynman lectures. –  WetSavannaAnimal aka Rod Vance Oct 6 '13 at 0:03

Suppose that we found two electrons at point A and point B which is located at a significant distance from A. So there would be two spikes in the wave function at point A and B. If we quickly take another measurement not long after we took the first one, we will find two electrons at point A and point B again. And if we keep doing it over and over again quickly, we will get the same result. Now perhaps we can distinguish the electron found at point A as electron A and the one found at B as electron B. Then if we wait long enough, the wavefunctions tend to spread out and eventually overlap. Now we take another measurement and suppose we found two electrons in the region of overlap, once this happens it becomes impossible to determine which one is electron A or which one is electron B. Then the electrons are said to be indistinguishable.

So it is indistinguishable in the sense that we can't tell which one is which, but we can tell that there are two different particles.

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