It's simple to write down a Bose-Einstein Condensate wavefunction---in the position basis,

\(\Psi(r_1,\ldots,r_N) = \psi(r_1)\cdots\psi(r_N)\) (1)

But in experiments the boson is never an elementary particle, it's something like Rubidium-87 which is only a boson if you zoom out beyond the substructure of nucleus + electrons. For example, if \(|r_1 - r_2|=10\text{pm}\), so that the electron shells of two atoms are overlapping, then I expect that equation (1) above cannot be right.

So, I expect there should be a more precise equation that reduces to equation (1) in a certain limit ... or perhaps equation (1) is a the first term in a series, and there are correction terms. How does this work?