# What prevents bosons from occupying the same location?

The Pauli exclusion principle states that no two fermions can share identical quantum states. Bosons, one the other hand, face no such prohibition. This allows multiple bosons to essentially occupy the same space, a phenomenon that has been theorized responsible for superconductivity. Bosons do not, however, occupy exactly the same space as can be readily observed by the fact that a Bose-Einstein condensate does not collapse into a singularity.

Both of the rather unusual examples cited above are inherent to low-energy systems. A large collection of $^{12}$C (e.g. in a diamond) does not exhibit particularly unusual behavior. This leads me to hypothesize that the energy distribution of the system is largely responsible for keeping bosons apart. Given the rather basic nature of the question, however, I figured someone here would likely know the "correct" answer. So,

what keeps bosons from occupying the same location?

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Er ... nothing prevents this. That's what a Bose-Einstein condensate is: lots of bosons in the same place and quantum state.

You are observing that the sate is not perfectly localized, but that is a consequence of the state not being exactly zero momentum. Ultimately the Heisenberg principle puts a lower limit on how localized they could be.

If the bosons are composite objects (like Helium atoms, say) then you can write the state in terms of their constituent parts and the Fermionic bits have to obey the Pauli principle.

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So I was partly on the right track with the energy hypothesis. I thought about the constituent nature of the systems in question as well, but couldn't come up with sufficiently succinct wording to include it in the question. Thanks. – AdamRedwine Apr 3 '13 at 17:43

This is really just a comment to dmckee's answer, but it got a bit long for a comment.

The problem with your question:

what keeps bosons from occupying the same location?

is that no particle has a precisely defined position. Remember that when we get down to the sizes of atoms etc particles don't have a position. They are described by a wavefunction that may be localised in space to some extent but never localised down to a single point. As dmckee says, the Heisenberg Uncertainty Principle prevents a particle from being localised down to a point, unless that is you're prepared to allow the momentum to become infinitely uncertain in which case the whole thing turns into a black hole!

In a BEC all the atoms are in the same quantum state, but that state is only localised down to the size of the experimental apparatus. In principle you could make the size of the condensate smaller, but I suspect the increased uncertainty in the momentum would make it hard to keep the condensate coherent and it would break up into individual atoms of different energies.

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I think I follow you, but I don't see how it makes sense to say that reducing the size of the condensate would increase the uncertainty to the point of dissociating the condensate. It seems to me that the position of any individual boson in the condensate would be essentially bound by the condensate as a whole rather than some particular region of the condensate. If my reasoning were correct, that would keep BECs very small because the HUP limits are puny on a macroscopic scale. And yet, clearly, BECs can be contained in relatively sizable macroscopic setups. – AdamRedwine Apr 3 '13 at 17:48
I think in practice the size limits are set by getting the atoms cool enough to form a condensate. You're quite right that it would have to get very small before the HUP became an issue. – John Rennie Apr 3 '13 at 17:58