How are the middle two baryons in the baryon octet obtained?

Question

I understand that it has to be a combination of u,d and s such that it is antisymmetric in two indices, but how is this specific combination obtained?

Answer 1

, so there is hardly any complication.

The Λ is an isosinglet uds, so disconnected from the $\Sigma^\pm $, uus, dds, in terms of isospin raising/lowering operators $I_\pm$, unlike the $\Sigma^0$ (also uds) which is gotten from these via these operators.

These two states are also orthogonal, as all isosinglets are orthogonal to isotriplets.

$\Lambda \sim (uds - dus) \times ... \qquad \hbox{versus} \qquad \Sigma^0\sim (uds+ dus) \times ... .$

For a formal appreciation read up on SU(3) generators.

The over-all baryon wavefunction has to be fully antisymmetric in its three constituents; and color completely antisymmetrizes the 3 quarks. So the combination flavor & spin &space pieces have to be symmetric in total.

Most good texts have the mixed-symmetry wavefunctions of these: they are thus long, messy expressions. Look at this answer.