Distance and the HUP

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I've been thinking about what Bose-Einstein Condensates means in terms of the Heisenberg Uncertainty Principle.

When a cluster of atoms is made very cold the particles lose momentum, relative to the other particles in the cluster. This causes the position of the particles to become uncertain, relative to each other, and when the uncertainty in position becomes greater than the distance to the other particles, the atoms lose their individual identities and become part of the condensate. Is that more or less correct?

In that case, would it be correct to say that the distance function between particles is a multi-valued function? Such a distance function would be based on the positions of the two particles and their momentums. The smaller the momentums,the wider the variation in the values the distance function returns.

I don't think this is the usual way of describing the Uncertainty Principle, but is it wrong?
 

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