How to work in the |F,m_F> states in hyperfine structure

[lelouch_v1]

TL;DR Summary

A little confused about the notation in the coupled and uncoupled basis

Suppose that we have two atoms with one proton one electron each, and these electrons interact with each other. The states for the electrons are the singlet(S=0) and the triplet states(S=1). My question is if i have to keep the nuclear spin of the protons parallel when i write the states, for example $$\frac{1}{\sqrt{2}}(|+-\rangle-|-+\rangle)_e\otimes|\Uparrow\rangle_p$$
for the singlet state, and if so, can i use the fact that F=S+I, which in this case whould yield the state |1,1?> ?(S=0,I=1/2+1/2=1,m_s=0,m_I=1)

 

[DrClaude]

If you have two protons, then you also have to write the spin state of the protons in term of singlet and triplet states.

 

[lelouch_v1]

If you have two protons, then you also have to write the spin state of the protons in term of singlet and triplet states.

Thank you for your answer. I thought of that as well, but that would lead to having 16 states in the uncoupled basis, and in the coupled basis we would have $$|2,M_F\rangle \ \ \ , \ \ \ |1,M_F\rangle \ \ \ , \ \ \ |0,0\rangle$$ which yields a 9X9 matrix. The problem with that is that the author of the paper that i am reading (it's on the radical pair mechanism) states that there are 8 eigenvalues of the hyperfine hamiltonian, meaning an 8X8 matrix, so i suppose that he always takes the proton spins to be parallel (either both spins up or both down).

 

[DrClaude]

Can you give the reference?