Wave function when there is coupling between spin and position

[Happiness]

Why can't the general state, in the presence of coupling, take the form $$\psi_-(r)\chi_++\psi_+(r)\chi_-$$ where $\psi_+(r)$ and $\psi_-(r)$ are respectively the symmetric and anti-symmetric part of the wave function, and $\chi_+$ and $\chi_-$ are respectively the spinors representing spin up and spin down? In other words, why must the "coefficient" of the spin-up spinor be symmetric, in the presence of coupling?

Reference: Intro to QM, David J Griffiths, p210

 

[mfb]

You just changed the orientation of the z axis (by swapping what up and down is). Why would it be symmetric and antisymmetric? I don't know, needs more context, I don't have Griffiths around.

 

[Happiness]

The part of the spatial wave function associated/coupled with $\chi_+$ does not need to be symmetric, right?

On second thought, I think $\psi_+$ does not represent a symmetric wave function, but it’s just to denote the part of the spatial wave function associated with $\chi_+$. Throughout the book, Griffiths has been using $\psi_+$ to mean a symmetric wave function, but I think it does not mean that in this particular case, hence the confusion.

 

[mfb]

That makes more sense.