Understanding Spin in Composite Systems

[aaaa202]

Suppose you have an electron and a proton, which both can have either spin up or down. The possible combinations are:

$\uparrow\downarrow, \downarrow\uparrow, \uparrow\uparrow, \downarrow\downarrow$

But for some reason my book only allows a linear combination of the two first since they both have m=0 with the argument that m should advance in integer steps. Now this is a bit weird for me, wasn't that a rule applying for each electrons spin that we are now using on the total spin of the system - how can we just do that? It seems for me that we should either choose that the rule that the spin number advances in integer steps for each electrons spin or for the total spin - because doesn't it create problems for either if we choose both?

 

[haael]

The latter two cases are an excited hydrogen atom. Maybe your book considered only non-excited atoms, that's why it was restricted to the former two cases.