Wavefunction of a Baryon

Question

How to write the total wavefunction of a Baryon including space part, spin part, isospin part and color part such that the net wavefunction is antisymmetric? What is the difference in wavefunctions of two different baryons but of same quark content say proton $p$ and $\Delta^+$ baryon?

Answer 1

To write the wavefunction of a baryon, you write it as a direct product of the different parts of the wavefunction (just as you would for any other particle):

\begin{equation} \left| \psi \right\rangle = \left| \mbox{spatial} \right\rangle \otimes \left| \mbox{spin} \right\rangle \otimes \left| \mbox{Isospin} \right\rangle \otimes \left| \mbox{color} \right\rangle \end{equation}

Furthermore, the difference between a proton and $ \Delta ^+ $ is that they have different spins and total isospin. The proton is a spin $ 1/2 $ and total isospin $ 1/2 $ object while the $ \Delta ^+ $ is a spin $ 3/2 $ and total isospin $ 3/2 $ object.

Answer 2

You can think of $\Delta^+$ as just the energized state of $p$, due to the spin configuration of the three quarks, and therefore different total spin ($3/2$ vs. $1/2$). This is much the same as the hydrogen atom, where different angular momentum states lead to different energy states.