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Comment by basicplus2 | Hacker News Reader

The Pauli exclusion principle is just a fancy way of saying two electrons (later extended to fermions) can't be in the same place at the same time

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They can, if their spins are opposite. (Speaking of 'place' specifically, different orbitals of the atom overlap in space, so in fact nothing prevents two electrons from occupying the same point in space. A striking example would be an observation that the maximum probability of finding an electron sitting on an s-orbital is inside the atom's nucleus.)

Are you forgetting a Jacobian? The maximum probability isn't inside the nucleus for an s-orbital. E.g. for hydrogen, the maximum probability for the electron is one bohr radius away from the center of the nucleus (and the expectation value of the radius is 1.5 bohr radii away).

Well let me just quote Wikipedia here: ...in three-dimensional space, the maximum probability density occurs at the location of the nucleus and not at the Bohr radius, whereas the radial probability density peaks at the Bohr radius, i.e. when plotting the probability distribution in its radial dependency.

That it is maximum at the nucleus is not of much significance. To quote a professor:[1]

> In some ways it does not provide the best description of the electron distribution, since the region around r= 0, where the wavefunction has its largest values, is a relatively small fraction of the volume accessible to the electron. Larger radii represent larger physical regions since.

If you were to do any kinds of measurements, you are most likely to find it at the Bohr radius, not closer to the nucleus.

[1] http://www.umich.edu/~chem461/QMChap7.pdf

Ah yup you're right, I read your sentence too fast, whoops. The most likely location is at the nucleus, but the most likely radius (integrating over all angles) is at 1 bohr radius.

Yes, the point that is often missed is that Coulomb interactions aren't enough on their own.

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