# Quark model wavefunction separability

When looking at the quark model we say we can write the wavefunction in 4 components, ie $$|\Psi \rangle = |\Psi_{spatial} \rangle|\Psi_{spin} \rangle|\Psi_{flavour} \rangle|\Psi_{colour} \rangle \space \space$$

It isn't clear to me why we can assume that the quantum numbers of the state are completely separable like this. The most general state will be a linear combination of separable states, ie $$|\Psi \rangle = C_{i j kl}|\Psi^{(i)}_{spatial} \rangle|\Psi^{(j)}_{spin} \rangle|\Psi^{(k)}_{flavour} \rangle|\Psi^{(l)}_{colour} \rangle \in \mathcal{H}= \mathcal{H_{spatial}} \otimes \mathcal{H_{spin}} \otimes \mathcal{H_{flavour}}\otimes\mathcal{H_{colour}}$$

Why is the assumption of separability valid for the quark model?

• Since quark model can mean a few different things (according to the Particle Data Group)... Is the assumption of separability a tenet of the quark model that you're referring to? I mean, are you asking for what purpose(s) this separability assumption is adequate, even though the exact QCD single-hadron states probably aren't separable in this way? Nov 14, 2021 at 2:58
• Have you parsed out the complete set of commuting constants of the motion involved? Nov 14, 2021 at 3:08
• @ChiralAnomaly So most notes that I have seen state this separability and then proceed to diagonalize each of the respective individual Hilbert spaces and combine the eigenstates after. I am asking if this is just an approximation or is there a fundamental mathematical or physical reason to expect the quantum numbers of the quark model to separate in this way. Nov 14, 2021 at 9:58
• @CosmasZachos I am not sure what you mean by this Nov 14, 2021 at 9:59

$$|p_\uparrow\rangle= \frac{1}{\sqrt {18}} [ 2| u_\uparrow d_\downarrow u_\uparrow \rangle + 2| u_\uparrow u_\uparrow d_\downarrow \rangle +2| d_\downarrow u_\uparrow u_\uparrow \rangle - | u_\uparrow u_\downarrow d_\uparrow\rangle -| u_\uparrow d_\uparrow u_\downarrow\rangle -| u_\downarrow d_\uparrow u_\uparrow\rangle -| d_\uparrow u_\downarrow u_\uparrow\rangle -| d_\uparrow u_\uparrow u_\downarrow\rangle -| u_\downarrow u_\uparrow d_\uparrow\rangle ].$$ This is an eigenstate of the strong hamiltonian (which, however, has small isospin/flavor breaking terms in it). It is fully color and J invariant. It is not a string of separable states. It is a linear combination of such.